🧠:Okay, let's try to figure out how to prove that when nine polygons each with area 1 are placed inside a square of area 5, at least two of them must intersect by at least 1/9. Hmm, this seems like a problem that might involve the pigeonhole principle. Let me recall that the pigeonhole principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. So maybe if I divide the square into regions and apply this principle?First, the square has an area of 5, so each side must be of length sqrt(5), right? Because area of a square is side squared, so side length is the square root of the area. So the square is sqrt(5) by sqrt(5). Now, nine polygons each of area 1 are placed inside this square. The total area of the polygons is 9*1=9, which is more than the area of the square (which is 5). So there's definitely overlap somewhere, but we need to show that at least two of them overlap by at least 1/9.Wait, the total area of all polygons is 9, and the square can only hold 5, so the excess is 9 - 5 = 4. This excess has to be distributed among the overlaps. But how does this relate to the minimum overlap between two polygons?Maybe using the principle that if the total excess area is distributed over the possible overlaps, then there must be at least one overlap that's large enough. But how do we formalize this?Let me think. Suppose that each pair of polygons overlaps by less than 1/9. Then, the total overlapping area would be less than the number of pairs multiplied by 1/9. How many pairs are there? With nine polygons, the number of pairs is C(9,2) = 36. So if each pair overlaps by less than 1/9, the total overlapping area would be less than 36*(1/9) = 4. But wait, the total overlapping area is exactly the excess area, which is 4. So if each overlap is less than 1/9, then the total overlap would be less than 4. But since the total overlap must be exactly 4, this is a contradiction. Therefore, there must be at least one pair that overlaps by at least 1/9.Wait, is that correct? Let me check again. The total area covered without any overlaps would be 9, but the square can only hold 5. So the overlapping area is 9 - 5 = 4. Now, if each pair overlaps by less than 1/9, then summing over all pairs, the total overlap would be less than 36*(1/9) = 4. But overlapping areas can't just be summed directly because when three or more polygons overlap the same region, that area is counted multiple times in the total. For example, if three polygons overlap in a region, that region is counted three times (once for each pair). Therefore, the actual total overlapping area (counted with multiplicity) is more than or equal to 4. But the sum over all pairs of their overlaps is an upper bound on the total overlapping multiplicity?Wait, this is getting a bit confusing. Let me recall that in inclusion-exclusion principle, the total area covered by the polygons is the sum of the areas of each polygon minus the sum of the areas of their pairwise intersections plus the sum of the areas where three overlap, and so on. So the total area covered is:Total = Sum(areas) - Sum(pair overlaps) + Sum(triple overlaps) - ... But in our case, the total area covered is 5, and the sum of the areas is 9, so:5 = 9 - Sum(pair overlaps) + Sum(triple overlaps) - ... Therefore, Sum(pair overlaps) - Sum(triple overlaps) + ... = 9 - 5 = 4.But if we ignore the higher-order overlaps (triple, etc.), which can be positive or negative, the Sum(pair overlaps) is at least 4. Because if we have Sum(pair overlaps) - (other terms) = 4, then Sum(pair overlaps) = 4 + (other terms). But the other terms could be positive or negative. For example, if all overlaps are pairwise, then Sum(pair overlaps) = 4. If there are triple overlaps, then Sum(pair overlaps) would be more than 4 because we subtract those triple overlaps. Wait, no. The inclusion-exclusion formula alternates signs. So Sum(pair overlaps) is subtracted first. Let me write the formula properly:Total area = Sum(individual areas) - Sum(pair overlaps) + Sum(triple overlaps) - Sum(quadruple overlaps) + ... Therefore, rearranged:Sum(pair overlaps) = Sum(individual areas) - Total area + Sum(triple overlaps) - Sum(quadruple overlaps) + ... So Sum(pair overlaps) = 9 - 5 + Sum(triple overlaps) - Sum(quadruple overlaps) + ... Therefore, Sum(pair overlaps) = 4 + [Sum(triple overlaps) - Sum(quadruple overlaps) + ...]But since the terms in the brackets can be positive or negative, we can't be sure. However, if all overlaps are only pairwise, then Sum(pair overlaps) = 4. If there are higher-order overlaps, the Sum(pair overlaps) must be greater than 4, because for example, a triple overlap would contribute three times to the pair overlaps but is subtracted once in the inclusion-exclusion. Wait, no. Let's take an example. If three polygons overlap in some area A, then in the inclusion-exclusion formula, that area is counted three times in the individual areas, subtracted three times in the pair overlaps, and added once in the triple overlaps. Therefore, the total contribution is 3A - 3A + A = A. So the triple overlap area is added back in. Therefore, if you have triple overlaps, the Sum(pair overlaps) is 4 + Sum(triple overlaps) - ... So the Sum(pair overlaps) could be more or less than 4 depending on the higher-order terms. But since we don't have information about higher-order overlaps, maybe we can't directly use this.Alternatively, maybe we can use an inequality. The principle that the total pairwise overlap is at least the excess area. Wait, but the problem is that overlapping regions can be counted multiple times. So if two polygons overlap, that's counted once. If three overlap, that overlapping region is counted three times (once for each pair). So the total pairwise overlaps can't be less than the total excess area. Hmm, but the total excess area is 4. So is the total pairwise overlap at least 4? Because each unit of overlapping area is counted once for each pair that overlaps there. So if you have a region where k polygons overlap, that contributes C(k,2) to the total pairwise overlaps. Therefore, the total pairwise overlaps would be the sum over all regions of C(k,2) * area(region). But the total excess area is Sum_{k>=2} (k-1) * area(region). Because for each region covered by k polygons, the excess is (k-1)*area(region). Therefore, total excess area E = Sum_{k>=2} (k-1)*area(region) = 4.But the total pairwise overlaps is Sum_{k>=2} C(k,2)*area(region). Since C(k,2) = k(k-1)/2, which is greater than or equal to (k-1) when k >= 2. Therefore, the total pairwise overlaps is >= 2 * E = 8. Wait, let's check:For each region with k overlaps, C(k,2) = k(k-1)/2, and (k-1) is the contribution to E. So C(k,2)/ (k-1) ) = k/2. So if k >=2, then C(k,2) >= (k-1) because k/2 >=1 when k >=2? Wait, no. For k=2, C(2,2)=1, which is equal to (2-1)=1. For k=3, C(3,2)=3, which is 3 compared to (3-1)=2. So C(k,2) = (k(k-1))/2. So compared to (k-1), which is the term in E, we have C(k,2) = (k/2)*(k-1). So the total pairwise overlaps is (k/2)*(k-1)*area(region) summed over all regions, and E is (k-1)*area(region) summed over all regions. Therefore, total pairwise overlaps = (k/2) * E_terms. Therefore, since k >=2 in each term, total pairwise overlaps >= (2/2)*E_terms = E_terms = E =4. So total pairwise overlaps is at least 4.Therefore, the sum of all pairwise intersection areas is at least 4. Since there are C(9,2)=36 pairs, if each pair intersects in less than 1/9, then the total pairwise intersections would be less than 36*(1/9)=4. But we just concluded that the total pairwise intersections must be at least 4. Therefore, it's impossible for all pairs to intersect in less than 1/9. Hence, there must be at least one pair that intersects in at least 1/9. That's the contradiction. Therefore, proved.Wait, let me verify once again. If each pairwise intersection is less than 1/9, total intersections < 36*(1/9) =4. But the total intersections must be >=4, hence contradiction. Therefore, at least one pair must intersect in at least 1/9. That seems correct. So the key idea is using the pigeonhole principle on the total overlapping area required versus the maximum possible if all overlaps are small.Alternatively, another approach: divide the square into smaller regions and use pigeonhole. But since the square has area 5, and each polygon has area 1, maybe dividing into 5 regions? But there are 9 polygons. Wait, if we divide the square into 5 regions, each of area 1, then by pigeonhole, at least two polygons must be in the same region, hence overlapping. But the problem is that the square can't be divided into 5 regions each of area 1, since the total area is 5. But how would you do that? Maybe not straightforward.Alternatively, divide the square into 9 smaller squares, each of area 5/9. Then, by pigeonhole principle, each polygon of area 1 must intersect at least two of these smaller squares, because the area of each small square is 5/9 <1. Wait, but I'm not sure. If a polygon is placed inside the big square, how much can it be contained within a small square? If a polygon has area 1, and the small squares have area 5/9, then the polygon cannot be entirely contained within a single small square, since 1 >5/9. Therefore, each polygon must intersect at least two small squares. Then, by some argument, maybe two polygons must intersect the same pair of small squares, leading to an overlap. Hmm, not sure.Alternatively, use the idea of the union of the polygons. The total area they cover is at most 5, but their total area is 9, so the average number of overlaps per point is 9/5 = 1.8. But how does that help?Wait, maybe using the probabilistic method? If we consider the expected overlap between two random polygons. But not sure.Alternatively, think about it as an integral over the square of the number of polygons covering each point. The integral over the square of the number of covers is 9. But the square has area 5, so the average coverage is 9/5. Then, by some averaging argument, there must be a point covered by at least 9/5 polygons. But 9/5 is 1.8, so there must be a point covered by at least two polygons. But that just tells us that overlaps exist, not the area.Alternatively, using measure theory. The measure (area) of the union is 5. The sum of the measures is 9. Therefore, the measure of the intersections is at least 9 -5 =4. But this is the total overlapping area counting overlaps of two or more polygons. However, we need the overlap between two specific polygons. So if the total overlapping area is 4, spread across C(9,2)=36 possible pairs, then by averaging, the average pairwise overlap is 4/36=1/9. Therefore, there must be at least one pair with overlap at least the average, i.e., 1/9. That's the key!Yes, that seems right. If the total overlapping area is 4, and there are 36 pairs, then the average overlap per pair is 4/36=1/9. Therefore, by the averaging principle (or the pigeonhole principle), at least one pair must have overlap at least the average, which is 1/9. Hence, proved.But wait, does this total overlapping area of 4 correspond exactly to the sum over all pairwise overlaps? Because if there are regions where three or more polygons overlap, those regions are counted multiple times in the total pairwise overlaps. For example, a region covered by three polygons contributes three times to the pairwise overlaps. Therefore, the total pairwise overlaps is actually more than 4. So if the total overlapping area (counted with multiplicity) is more than 4, then the average pairwise overlap is more than 4/36=1/9, hence there must exist a pair with overlap at least 1/9. Therefore, the previous argument still holds.Alternatively, even if the total overlapping area with multiplicity is more than 4, when we take the average over 36 pairs, it's still at least 4/36, so the conclusion remains.Therefore, the main idea is:1. Total area of polygons: 9.2. Area of the square:5.3. Therefore, overlapping area (counted with multiplicity) is 9 -5=4.4. The total overlapping area counted across all pairs is at least 4.5. There are C(9,2)=36 pairs.6. Therefore, average overlap per pair is at least 4/36=1/9.7. Hence, some pair must have overlap at least 1/9.Yes, that seems to be a solid argument. So the key was realizing that the total overlapping area, even when considering higher-order overlaps, is at least 4 when counted across all pairs, and dividing by the number of pairs gives the required lower bound on at least one pair's overlap.Alternatively, another way to think about it is using the probabilistic method. Suppose that we pick two polygons at random. The expected value of their overlapping area is total overlapping areas divided by number of pairs, which is 4/36=1/9. Therefore, there must exist at least one pair where the overlapping area is at least the expected value, i.e., 1/9.Therefore, regardless of the approach, the conclusion is that there must be at least two polygons overlapping by at least 1/9. Therefore, the proof is complete.Final Answerboxed{dfrac{1}{9}}

✅:To prove that some two of the nine polygons, each with an area of 1, must intersect with an area of at least ( frac{1}{9} ) when placed in a square of area 5, we can use the following reasoning:1. Total Area of Polygons: The total area of the nine polygons is ( 9 times 1 = 9 ).2. Area of the Square: The area of the square is 5.3. Excess Area: The excess area due to overlapping is ( 9 - 5 = 4 ).4. Total Pairwise Overlaps: The total overlapping area (counted with multiplicity) must be at least 4. This is because the overlapping areas contribute to the excess area when considering the inclusion-exclusion principle.5. Number of Pairs: There are ( binom{9}{2} = 36 ) pairs of polygons.6. Average Overlap per Pair: The average overlap per pair is ( frac{4}{36} = frac{1}{9} ).7. Conclusion: By the pigeonhole principle, at least one pair of polygons must have an overlap area of at least the average value, which is ( frac{1}{9} ).Thus, we conclude that some two of the polygons must intersect with an area of at least ( frac{1}{9} ).[boxed{dfrac{1}{9}}]

🔑:1. Let's denote the square with an area of ( S = 5 ).2. There are nine polygons, each with an area of ( S_i = 1 ) for ( i = 1, 2, ldots, 9 ).3. Let ( S_{ij} ) denote the area of the intersection between the ( i )-th and ( j )-th polygons, where ( i neq j ). Our goal is to show that there exist at least two polygons whose intersection area is at least ( frac{1}{9} ).4. Start with the inequality concerning the total area covered by all polygons and their intersections: [ S geq sum_{i=1}^{9} S_i - sum_{1 leq i < j leq 9} S_{ij} ]5. Given that ( S = 5 ) and ( S_i = 1 ) for all ( i ), we can substitute these values into the inequality: [ 5 geq 1 + 1 + ldots + 1 - sum_{1 leq i < j leq 9} S_{ij} ] Simplifying, the sum of areas of nine polygons is ( 9 ): [ 5 geq 9 - sum_{1 leq i < j leq 9} S_{ij} ]6. Rearrange the inequality to solve for the sum of the intersections: [ sum_{1 leq i < j leq 9} S_{ij} geq 9 - 5 ] This simplifies to: [ sum_{1 leq i < j leq 9} S_{ij} geq 4 ]7. The number of distinct pairs ((i, j)) from 9 polygons is given by ( binom{9}{2} ): [ binom{9}{2} = frac{9 times 8}{2} = 36 ]8. If we assume that no two polygons intersect with an area greater than or equal to ( frac{1}{9} ), then each pair's intersection area ( S_{ij} ) would be less than ( frac{1}{9} ). Therefore, the total intersection area sum would be: [ sum_{1 leq i < j leq 9} S_{ij} < 36 times frac{1}{9} = 4 ]9. However, we previously established that: [ sum_{1 leq i < j leq 9} S_{ij} geq 4 ]10. The inequality ( sum_{1 leq i < j leq 9} S_{ij} < 4 ) contradicts ( sum_{1 leq i < j leq 9} S_{ij} geq 4 ).11. Therefore, there must be at least one pair ((i, j)) for which the intersection area ( S_{ij} ) is greater than or equal to ( frac{1}{9} ). Conclusion:[boxed{text{There exist at least two polygons with an intersection area of at least } frac{1}{9}.}]

🧠:Okay, let's try to figure out how many push-ups Sophia did on the last day. So, the problem says she did push-ups every day for 7 days, and each day after the first, she did 5 more than the day before. The total push-ups over the seven days are 175. We need to find out how many she did on the last day, and the options are given from 35 up to 55.First, I need to recall how to handle sequences where each term increases by a constant number. That's an arithmetic sequence, right? So, the number of push-ups each day forms an arithmetic progression where the difference between consecutive days is 5 push-ups.Let me denote the number of push-ups she did on the first day as, let's say, a₁. Then, each subsequent day would be a₁ + 5, a₁ + 10, and so on, up to the seventh day. The seventh day would be a₁ + 5*(7-1) = a₁ + 30. Because the difference is 5 each day, so for the nth day, it's a₁ + 5(n-1).Now, the total number of push-ups over 7 days is the sum of this arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is Sₙ = n/2 * (2a₁ + (n-1)d), where d is the common difference. Alternatively, it's also written as Sₙ = n*(a₁ + aₙ)/2, since the sum can be calculated as the average of the first and last term multiplied by the number of terms.In this case, n is 7 days, d is 5 push-ups per day, and S₇ is 175. Let me write that out:Sum = 7/2 * [2a₁ + (7-1)*5] = 175Simplifying the equation step by step:First, compute (7-1)*5: that's 6*5 = 30.So, the equation becomes:7/2 * [2a₁ + 30] = 175Multiply both sides by 2 to eliminate the denominator:7*(2a₁ + 30) = 350Divide both sides by 7:2a₁ + 30 = 50Subtract 30 from both sides:2a₁ = 20Divide by 2:a₁ = 10So, the first day she did 10 push-ups. Then each subsequent day increases by 5. Let's list out the days:Day 1: 10Day 2: 15Day 3: 20Day 4: 25Day 5: 30Day 6: 35Day 7: 40Wait, but the options given include 40 as option E, and 45 as D, 50 as C. Hmm. Wait, according to this calculation, on the seventh day she did 40 push-ups, which is option E. But let me check the total to make sure.Let's add them up: 10 + 15 + 20 + 25 + 30 + 35 + 40. Let's compute step by step:10 + 15 = 2525 + 20 = 4545 + 25 = 7070 + 30 = 100100 + 35 = 135135 + 40 = 175. Yes, that's correct. So the total is 175, which matches the problem statement. Therefore, the seventh day is 40 push-ups, which is option E.But wait, hold on. Let me cross-verify using another approach to be sure. Maybe using the formula Sₙ = n/2*(a₁ + aₙ). Here, n=7, S₇=175, so:175 = 7/2*(a₁ + a₇)Multiply both sides by 2: 350 = 7*(a₁ + a₇)Divide by 7: 50 = a₁ + a₇So, a₁ + a₇ = 50. But we also know that a₇ = a₁ + 6d = a₁ + 30 (since d=5). Therefore:a₁ + (a₁ + 30) = 502a₁ + 30 = 502a₁ = 20 => a₁=10, which matches the previous result. Then a₇ = 10 + 30 = 40. So, that's the same answer. So option E, 40.But wait, the options given are (A)55, (B)35, (C)50, (D)45, (E)40. So 40 is option E. But let me check again because sometimes there might be a miscalculation. Let me try another way.Alternatively, maybe set up the equations again. Let a₁ be the first day's push-ups. Then the number of push-ups each day is:a₁, a₁+5, a₁+10, a₁+15, a₁+20, a₁+25, a₁+30.Adding these together:a₁ + (a₁ +5) + (a₁ +10) + (a₁ +15) + (a₁ +20) + (a₁ +25) + (a₁ +30) =7a₁ + (5 +10 +15 +20 +25 +30)Compute the sum of the increments: 5+10=15, 15+15=30, 30+20=50, 50+25=75, 75+30=105. So total sum is 7a₁ +105 = 175.So 7a₁ = 175 -105 =70 => a₁=10. Then the last day is a₁ +30=40. So again, 40, option E.But wait, the problem is presented as Sophia doing push-ups for 7 days, increasing by 5 each day after the first, total 175. So answer should be 40. But let me check the options again. Yes, E is 40. So that's correct.But why is there an option C)50? Maybe a common mistake is to miscalculate the number of terms. Let me see: if someone thought that on day 7, the increase is 5*7=35, so a₁ +35, but that's incorrect because the first day is a₁, then each subsequent day adds 5. So day 2 is a₁ +5, day 3 a₁+10, ..., day 7 is a₁ +6*5= a₁+30. So if someone added 5*7=35, they might get a₁+35=10+35=45, which is option D. But that's a mistake because the nth term is a₁ + (n-1)d. So day 7 is a₁ +6*5.Alternatively, maybe another approach where someone uses the average. The total is 175 over 7 days, so average per day is 25. Since the sequence is evenly spaced, the average is also the average of the first and last term. So (a₁ + a₇)/2 =25, so a₁ +a₇=50, which we had earlier. Then since a₇ =a₁ +30, solving gives a₁=10, a₇=40. So that's correct.But perhaps someone might think that the fourth day is the middle day, which would be the average. So day 4 would be 25, then day 5:30, day6:35, day7:40. So yes, that's correct.Alternatively, maybe someone miscalculates the total increments. Let's say if someone thought the total increment is 5 per day for 7 days, so 5*7=35, and then set up the equation 7a₁ +35=175, leading to a₁=20. Then the last day would be 20 +30=50, which is option C. But that's wrong because the increments are not 5 each day added to the first day, but each day after the first. The total increments are 0 for day1, 5 for day2, 10 for day3, ...,30 for day7. So total increments are 5*(0+1+2+3+4+5+6)=5*(21)=105, which is why the equation is 7a₁ +105=175. So if someone forgets that day1 has 0 increments, and instead adds 5 each day starting from day1, they would get 5*7=35, leading to the wrong answer. So that's a possible mistake leading to option C.But in our case, the correct answer is E)40. Therefore, the answer is E.

✅:Sophia did push-ups for 7 days, increasing her count by 5 each day after the first. The total push-ups were 175. To find the number on the last day:1. Arithmetic Sequence Setup: Let ( a_1 ) be the push-ups on day 1. The sequence is: ( a_1, , a_1 + 5, , a_1 + 10, , dots, , a_1 + 30 ) (for day 7).2. Sum Formula: The sum of the sequence is: [ S_7 = frac{7}{2} times [2a_1 + 6 times 5] = 175 ] Simplifying: [ 7(2a_1 + 30) = 350 implies 2a_1 + 30 = 50 implies a_1 = 10 ]3. Last Day Calculation: On day 7, push-ups ( = a_1 + 30 = 10 + 30 = 40 ).4. Verification: Daily counts: 10, 15, 20, 25, 30, 35, 40. Summing these confirms ( 10 + 15 + dots + 40 = 175 ).Final Answer: (boxed{E}) (40)

🔑:1. Identify the sequence and the conditions: Sophia did push-ups every day for 7 days, with each subsequent day involving 5 more push-ups than the previous day. The total number of push-ups over these 7 days is 175.2. Define the first term and common difference: Let (a) be the number of push-ups Sophia did on the first day. Each subsequent day, she did 5 more push-ups than the previous day, so the common difference (d = 5).3. Establish the mathematical representation for each day: Representing the push-ups done on each of the 7 days in terms of (a) (the first day's push-ups), we have: [ a, quad a + 5, quad a + 10, quad a + 15, quad a + 20, quad a + 25, quad a + 30 ]4. Sum the arithmetic series: The sum of push-ups over the 7 days should add up to 175. Thus, [ a + (a + 5) + (a + 10) + (a + 15) + (a + 20) + (a + 25) + (a + 30) = 175 ]5. Combine and simplify the sum: Combine like terms: [ a + a + a + a + a + a + a + (5 + 10 + 15 + 20 + 25 + 30) = 175 ] This simplifies to: [ 7a + (5 + 10 + 15 + 20 + 25 + 30) = 175 ] Calculating the constant sum gives: [ 5 + 10 + 15 + 20 + 25 + 30 = 105 ] Therefore, [ 7a + 105 = 175 ]6. Solve for (a): Isolate (a): [ 7a = 175 - 105 ] [ 7a = 70 ] [ a = 10 ]7. Determine the push-ups on the last day: To find the number of push-ups on the last day (the 7th day), use the formula: [ a_7 = a + 6d ] Substituting in the values (a = 10) and (d = 5): [ a_7 = 10 + 6 times 5 ] [ a_7 = 10 + 30 ] [ a_7 = 40 ]Conclusion: The number of push-ups Sophia did on the last day is (boxed{40}).

🧠:Okay, so I need to find the radius of a circle that's tangent to the given circle (radius 10) and the two perpendicular chords AB and AC drawn through point A. The chords are mutually perpendicular, and AB is 16 units long. Hmm, let me start by visualizing this problem.First, there's a circle with radius 10. Point A is on this circle. From point A, two chords AB and AC are drawn such that they are perpendicular to each other. Since AB is 16, I should probably find the length of AC as well, but maybe that's not necessary right away.The circle we need to find is tangent to the original circle and the two chords. So this circle must be inside the original circle, touching it from the inside, and also touching both chords AB and AC. Because the chords are perpendicular, the tangent circle is likely in the corner formed by AB and AC near point A. Wait, but point A is on the original circle, so the tangent circle can't be at point A itself because that's already on the circumference. Instead, it must be somewhere near the corner formed by the two chords but inside the original circle.Let me sketch this mentally. The original circle has center O, radius 10. Point A is on the circumference. Chords AB and AC are perpendicular. Let me recall that the length of a chord is related to the distance from the center. For chord AB of length 16, the distance from the center O to AB can be calculated using the formula: distance = √(r² - (chord/2)²). Wait, but chord length is 2√(r² - d²), where d is the distance from the center to the chord. So rearranging, d = √(r² - (chord/2)²). Let's compute that.For chord AB: length 16, so half-length is 8. Then, the distance from O to AB is √(10² - 8²) = √(100 - 64) = √36 = 6. So the center O is 6 units away from chord AB.Similarly, chord AC is perpendicular to AB. Since AB and AC are perpendicular and both pass through A, which is on the original circle, we need to find the length of AC. Wait, but maybe the problem doesn't give AC's length? The problem states that AB and AC are mutually perpendicular chords through A, but only AB is given as 16. So perhaps we need to find AC's length first.Wait, maybe we can find AC's length using the fact that OA is the radius, which is 10. Since AB and AC are chords through A, and they are perpendicular, triangle BAC is a right-angled triangle at A. So BA = 16, AC = let's say x, then BC would be the hypotenuse. But BC is another chord. However, maybe there's another way to relate OA to the distances.Alternatively, since AB and AC are perpendicular chords intersecting at A, which is on the circle, maybe we can use coordinate geometry. Let me try setting up coordinates.Let me place point A at (10, 0) on the coordinate system, so the original circle has center at (0,0) with radius 10. Then, chord AB is 16 units long. Let me find the coordinates of point B.Since AB is a chord of the circle starting at A (10,0) and length 16. The coordinates of B can be found by moving 16 units from A along the chord AB. But since AB is a chord, the direction of AB is important. Wait, but maybe this is complicated. Alternatively, since we know the distance from the center to chord AB is 6, as calculated earlier.Wait, if the distance from the center (0,0) to chord AB is 6, then chord AB is a line that is 6 units away from the center. Since point A is on both the chord AB and the original circle, we can find the equation of chord AB.Let me recall that the distance from the center (0,0) to the line AB is 6. The line AB passes through point A (10,0). Let me find the equation of line AB.Let’s denote the slope of AB as m. The equation of AB is then y - 0 = m(x - 10), so y = m(x - 10). The distance from (0,0) to this line is |0 - m*0 + (-10m)| / sqrt(m² + 1) ) = | -10m | / sqrt(m² + 1) ) = 10|m| / sqrt(m² + 1). This distance is given as 6. So:10|m| / sqrt(m² + 1) = 6Square both sides:100m² / (m² + 1) = 36Multiply both sides by (m² + 1):100m² = 36(m² + 1)100m² = 36m² + 3664m² = 36m² = 36/64 = 9/16So m = ±3/4Therefore, the slope of AB is ±3/4. Since AB is a chord of the circle passing through (10,0), with slope 3/4 or -3/4. Let's check both possibilities.First, if m = 3/4, then the equation of AB is y = (3/4)(x - 10). Let's find the other endpoint B.We know that AB has length 16. The distance from A (10,0) to B along the line y = (3/4)(x - 10). Let’s parametrize this line. Let’s set t as a parameter. Starting at A (10,0), moving along the line with direction vector (4,3) since the slope is 3/4. Wait, the direction vector can be (4,3) because for a slope of 3/4, over 4 units in x, up 3 in y.But the length from A to B is 16. The length of the direction vector (4,3) is 5. So moving 16 units from A along this direction would be (10,0) + (4,3)*(16/5). Wait, the unit vector in the direction (4,3) is (4/5, 3/5), so moving 16 units along that direction would be (10,0) + (4/5 *16, 3/5 *16) = (10 + 64/5, 0 + 48/5) = (10 + 12.8, 9.6) = (22.8, 9.6). But this point must lie on the circle of radius 10. Wait, but (22.8)^2 + (9.6)^2 = (22.8)^2 is way more than 100. That can't be. So something's wrong here.Wait a minute, perhaps my approach is incorrect. Because chord AB is of length 16, so the distance from the center is 6, as computed earlier. But if we parametrize the chord AB, which is 16 units long, then the other endpoint B must lie on the circle. Wait, but point A is already on the circle. So chord AB is a chord starting at A (10,0) and ending at another point B on the circle. The length of AB is 16, so we need to find coordinates of B such that distance from A (10,0) to B is 16 and B lies on the circle x² + y² = 100.So let's set up equations. Let B = (x,y). Then (x - 10)^2 + (y - 0)^2 = 16^2 = 256, and x² + y² = 100.Subtracting the second equation from the first:(x - 10)^2 + y² - (x² + y²) = 256 - 100 => x² - 20x + 100 + y² - x² - y² = 156 => -20x + 100 = 156 => -20x = 56 => x = -56/20 = -14/5 = -2.8.So x-coordinate of B is -2.8. Then y² = 100 - x² = 100 - (14/5)^2 = 100 - 196/25 = (2500 - 196)/25 = 2304/25. So y = ±2304/25^(1/2) = ±48/5 = ±9.6.Therefore, the coordinates of B are (-14/5, 48/5) and (-14/5, -48/5). Since the chord AB is 16 units, and given the original circle, these are the two possible positions for B.But since AB and AC are mutually perpendicular chords at A, and AC is also a chord through A, we need to find the coordinates of C such that AC is perpendicular to AB.Assuming AB has slope m, then AC has slope -1/m. Since AB has slope (from A (10,0) to B (-14/5, 48/5)), let's compute the slope of AB.Slope of AB is (48/5 - 0)/(-14/5 - 10) = (48/5) / (-14/5 - 50/5) = (48/5) / (-64/5) = 48/-64 = -3/4. Similarly, if B is (-14/5, -48/5), the slope would be (-48/5 - 0)/(-14/5 - 10) = (-48/5)/(-64/5) = 48/64 = 3/4. Wait, so depending on which B we take, the slope is either -3/4 or 3/4. Let's pick one to work with. Let's take B as (-14/5, 48/5), so slope is -3/4. Then, AC, being perpendicular, would have slope 4/3.Therefore, chord AC has slope 4/3 and passes through A (10,0). Let's find the coordinates of point C. Similarly, chord AC must also be a chord of the original circle, so point C is another intersection of the line AC with the circle.The equation of AC is y - 0 = (4/3)(x - 10), so y = (4/3)x - 40/3.Find intersection points of this line with the circle x² + y² = 100.Substitute y = (4/3)x - 40/3 into the circle equation:x² + [(4/3)x - 40/3]^2 = 100Compute [(4/3)x - 40/3]^2:= (16/9)x² - 2*(4/3)*(40/3)x + (1600/9)= (16/9)x² - (320/9)x + 1600/9So total equation:x² + (16/9)x² - (320/9)x + 1600/9 = 100Multiply all terms by 9 to eliminate denominators:9x² + 16x² - 320x + 1600 = 900Combine like terms:25x² - 320x + 1600 = 90025x² - 320x + 700 = 0Divide by 5:5x² - 64x + 140 = 0Use quadratic formula:x = [64 ± √(64² - 4*5*140)] / (2*5)Compute discriminant:64² = 40964*5*140 = 2800So discriminant = 4096 - 2800 = 1296√1296 = 36Thus, x = [64 ± 36]/10So x = (64 + 36)/10 = 100/10 = 10, which is point A, and x = (64 - 36)/10 = 28/10 = 14/5 = 2.8Therefore, the coordinates of point C are (14/5, y). Compute y:y = (4/3)(14/5) - 40/3 = (56/15) - (200/15) = (-144/15) = -48/5 = -9.6Thus, point C is (14/5, -48/5). So AC goes from A (10,0) to C (14/5, -48/5). Let me check the length of AC.Distance between A (10,0) and C (14/5, -48/5):Δx = 14/5 - 10 = 14/5 - 50/5 = -36/5Δy = -48/5 - 0 = -48/5Length AC = √[(-36/5)^2 + (-48/5)^2] = √[(1296 + 2304)/25] = √[3600/25] = √144 = 12So AC is 12 units long. Okay, so AB is 16, AC is 12, and they are perpendicular at A. Now, we need to find a circle tangent to the original circle (radius 10) and tangent to both chords AB and AC.This circle will be located in the "corner" formed by chords AB and AC, near point A, but inside the original circle. The center of this tangent circle must be equidistant from both chords AB and AC, and the distance from the center to each chord will be equal to its radius, say r. Additionally, the distance from the center of this tangent circle to the original circle's center (O, at (0,0)) must be 10 - r, since they are tangent internally.So let's denote the center of the desired circle as (h,k). Then:1. The distance from (h,k) to chord AB is r.2. The distance from (h,k) to chord AC is r.3. The distance from (h,k) to (0,0) is 10 - r.So we need to write equations based on these three conditions.First, let's find the equations of chords AB and AC.We already have AB's equation. From earlier, AB is from A (10,0) to B (-14/5, 48/5), which we found had slope -3/4. The equation of AB is y = -3/4 x + c. Let's verify. At point A (10,0):0 = -3/4*10 + c => c = 30/4 = 15/2. Wait, but earlier when we calculated using distance, we found the equation of AB as y = m(x - 10) with m = -3/4. So that would be y = -3/4 x + 30/4, which simplifies to y = -3/4 x + 15/2. Let me confirm with point B (-14/5, 48/5):y = -3/4*(-14/5) + 15/2 = (42/20) + 15/2 = (21/10) + 75/10 = 96/10 = 48/5. Correct.Similarly, chord AC has equation y = 4/3 x - 40/3, as we found earlier. Let me verify with point C (14/5, -48/5):y = 4/3*(14/5) - 40/3 = 56/15 - 200/15 = -144/15 = -48/5. Correct.So equations of AB and AC are:AB: y = -3/4 x + 15/2AC: y = 4/3 x - 40/3The distance from a point (h,k) to AB is |(-3/4)h - k + 15/2| / sqrt( ( (-3/4)^2 + (-1)^2 )) = | -3h/4 - k + 15/2 | / sqrt(9/16 + 1) = | -3h/4 - k + 15/2 | / sqrt(25/16) = | -3h/4 - k + 15/2 | / (5/4) ) = (4/5)| -3h/4 - k + 15/2 |.Similarly, distance from (h,k) to AC is |4/3 h - k - 40/3| / sqrt( (4/3)^2 + (-1)^2 ) = |4h/3 - k - 40/3| / sqrt(16/9 + 1) = |4h/3 - k - 40/3| / sqrt(25/9) = |4h/3 - k - 40/3| / (5/3) ) = (3/5)|4h/3 - k - 40/3|.Since both distances must equal the radius r:(4/5)| -3h/4 - k + 15/2 | = r(3/5)|4h/3 - k - 40/3 | = rAdditionally, the distance from (h,k) to (0,0) is sqrt(h² + k²) = 10 - r.This gives us three equations. Since the circle is tangent to both chords and inside the original circle, the center (h,k) should lie in the region where both distances to the chords are positive, and the center is closer to A than the original center. Given that the chords are near A (10,0), the tangent circle is likely in the first quadrant near A. Wait, but A is at (10,0), which is on the x-axis. The chords AB and AC are going upwards and downwards? Wait, point B was at (-14/5, 48/5) which is (-2.8, 9.6), so AB goes from (10,0) to (-2.8,9.6), which is upwards and to the left. AC goes from (10,0) to (2.8, -9.6), so downwards and to the left. Therefore, the corner formed by AB and AC at A is like a "right angle" pointing towards the origin. So the tangent circle should be in that corner, between the two chords, closer to A.Therefore, the center (h,k) is in the region near A, but offset from A towards the interior. So the signs in the absolute value equations might be determined by the position. Let's assume that the expressions inside the absolute value are negative for the distance to AB and positive for the distance to AC? Wait, not sure. Let's think.For the distance to AB: The line AB is above the center (h,k) if the center is below AB. Since AB is going from (10,0) to (-2.8,9.6), the line AB is rising to the left. The region where the center (h,k) would be located is below the line AB and above the line AC (since AC is going down to the right). Therefore, the distance formulas would have certain signs.Alternatively, since the center is in the region between the two chords near A, let's consider the direction of the normals.But maybe instead of dealing with absolute values, we can consider the orientation. Let's suppose that the center (h,k) is such that:For line AB: the normal vector pointing towards the interior (where the tangent circle is) would be in the direction of the normal vector. The equation of AB is y = -3/4 x + 15/2. The standard form is 3x + 4y - 30 = 0. The normal vector is (3,4), pointing upwards to the right. Since the center is inside the circle, maybe we need to take the negative of the normal vector direction. Wait, the distance formula's sign depends on which side of the line the point is.Alternatively, let's plug in a point that's inside the circle near A. Let's take a point close to A, say (10 - ε, 0), which is just to the left of A along the x-axis. Plugging into AB's equation: 3x + 4y - 30. At (10 - ε,0): 3*(10 - ε) + 0 - 30 = 30 - 3ε - 30 = -3ε < 0. So points just left of A are on the negative side of AB's equation. Similarly, for line AC, which is y = 4/3 x - 40/3. Standard form: 4x - 3y - 40 = 0. At point (10 - ε,0): 4*(10 - ε) - 0 - 40 = 40 - 4ε - 40 = -4ε < 0. So points just left of A are on the negative side of AC's equation as well.But the center of the tangent circle needs to be in the region where both distances to AB and AC are positive (i.e., the same side as the normals). Wait, no. The distance formula gives absolute value, so regardless of the side, it's positive. But the actual value inside the absolute value could be positive or negative. So to remove the absolute value, we need to determine the sign.Given that the center (h,k) is near A, in the region where x < 10 (since it's inside the circle), and near (10,0). Let's suppose that for line AB: 3x + 4y - 30. At the center (h,k), since it's inside the circle and near A, plugging into 3h + 4k - 30 would likely be negative, as for points just left of A, as we saw. Similarly, for line AC: 4h - 3k - 40. At points near A (10,0), plugging h slightly less than 10, k near 0: 4*(10 - ε) - 0 - 40 = -4ε < 0. Therefore, both expressions inside the absolute values are negative. Therefore, |expression| = -expression. Therefore:For AB: (4/5)( - ( -3h/4 - k + 15/2 )) = rWait, let's clarify. The standard form of AB is 3x + 4y - 30 = 0, so the distance from (h,k) is |3h + 4k - 30| / 5. Earlier, we derived it as (4/5)| -3h/4 - k + 15/2 |. Let me verify:Original equation of AB: y = -3/4 x + 15/2 => rearranged: 3x + 4y - 30 = 0. Therefore, distance is |3h + 4k - 30| / 5. So that's the correct expression. Similarly, the equation of AC is 4x - 3y - 40 = 0 (since y = 4/3 x - 40/3 => 4x - 3y - 40 = 0). So distance from (h,k) to AC is |4h - 3k - 40| / 5.Therefore, our two distance equations are:|3h + 4k - 30| / 5 = r|4h - 3k - 40| / 5 = rGiven that (h,k) is near A (10,0) and inside the circle, let's evaluate the expressions inside the absolute values.For AB: 3h + 4k - 30. At (10,0): 3*10 + 4*0 -30 = 0. But the center is near A but slightly inside. Let's take h = 10 - t, k = 0 + s, where t and s are small positive numbers. Then 3(10 - t) + 4s - 30 = -3t + 4s. Depending on t and s, this could be negative or positive. But given that the center is in the region formed by the two chords, which are going left-up and left-down from A, the center is likely to the left of AB and below AC. Wait, but AB is going up-left, AC is going down-left. The region near A between AB and AC would be a sort of angle between the two chords. If we are inside that angle, then the sign of the distance expressions might be negative.Alternatively, since we found earlier that just left of A, both expressions are negative, and the center is in that area, then both |3h + 4k -30| and |4h -3k -40| would be equal to -(3h + 4k -30) and -(4h -3k -40). Therefore:( -(3h + 4k -30) ) /5 = r( -(4h -3k -40) ) /5 = rTherefore:-3h -4k +30 = 5r-4h +3k +40 = 5rSo now we have two equations:1) -3h -4k +30 = 5r2) -4h +3k +40 = 5rSet them equal:-3h -4k +30 = -4h +3k +40Simplify:-3h -4k +30 = -4h +3k +40Bring all terms to left side:-3h -4k +30 +4h -3k -40 = 0( -3h +4h ) + ( -4k -3k ) + (30 -40) = 0h -7k -10 = 0Thus, h =7k +10So we have a relationship between h and k: h =7k +10.Now, substitute h =7k +10 into one of the equations. Let's take the first one:-3h -4k +30 =5rPlug h =7k +10:-3*(7k +10) -4k +30 =5r-21k -30 -4k +30 =5r-25k =5rThus, r = -5kSince radius r is positive, this implies that k is negative. So k = -r/5. Wait, but r is positive, so k is negative. That makes sense because if the center is near A (10,0), but inside the circle, and given the direction of the chords, the center would be slightly below the x-axis. Wait, but AC is going downwards, so being below the x-axis might be in the direction of AC. But let's see.Wait, if k is negative, then the center is below the x-axis, which is towards where chord AC is going. However, given that both chords are left of A, maybe the center is to the left and slightly below A.Now, we also have the third equation: the distance from (h,k) to (0,0) is 10 - r. So:sqrt(h² +k²) =10 -rBut we have h =7k +10, and r= -5k. Let's substitute h and r in terms of k.So h =7k +10, r= -5kThus, sqrt( (7k +10)^2 +k² ) =10 - (-5k) =10 +5kSquare both sides:(7k +10)^2 +k² = (10 +5k)^2Expand left side:49k² +140k +100 +k² =50k² +140k +100Right side:100 +100k +25k²Wait, wait, let's compute both sides correctly.Left side:(7k +10)^2 +k² = 49k² +140k +100 +k² =50k² +140k +100Right side:(10 +5k)^2 =100 +100k +25k²Set equal:50k² +140k +100 =25k² +100k +100Subtract right side from left side:25k² +40k =0Factor:k*(25k +40)=0Solutions: k=0 or 25k +40=0 =>k= -40/25= -8/5= -1.6But k=0 would imply r=0, which is invalid. So k= -8/5= -1.6Thus, k= -1.6, which is -8/5.Then, h=7k +10=7*(-8/5)+10= -56/5 +50/5= (-56+50)/5= -6/5= -1.2Wait, h= -6/5= -1.2? But the center is supposed to be near A (10,0). Wait, this can't be. If h= -1.2, that's way to the left of the center (0,0). But the original circle has radius 10, so the center at (-1.2, -1.6) would be inside the original circle, but is it near A (10,0)? No, it's on the opposite side. This must mean there's an error in my assumptions.Wait, this is a problem. If we follow the equations, we get the center at (-6/5, -8/5), which is (-1.2, -1.6). But this is not near point A (10,0) at all. It's in the third quadrant if the original circle is centered at (0,0). That doesn't make sense because the circle tangent to both chords near A should be near A, not on the opposite side.Therefore, my earlier assumption about the signs must be wrong. Let me re-examine the distance equations.When I set up the equations, I assumed that both expressions inside the absolute values were negative, leading to:-3h -4k +30 =5r-4h +3k +40 =5rBut if the center is near A (10,0), which is on the circle, but the tangent circle is supposed to be near A. However, when we solved the equations, we got a center at (-1.2, -1.6), which is inside the original circle but far from A. That suggests that there's another solution where the expressions inside the absolute values are positive.Let me redo the equations without assuming the signs. The distance formulas are:|3h +4k -30| /5 =r|4h -3k -40| /5 =rThese absolute values can have two cases each. Let's consider possible combinations.Case 1: Both expressions are positive.Thus:3h +4k -30 =5r4h -3k -40 =5rThen, subtract the two equations:(3h +4k -30) - (4h -3k -40)=0- h +7k +10=0 => h=7k +10Then, substitute into first equation:3*(7k +10) +4k -30 =5r21k +30 +4k -30 =5r =>25k=5r =>r=5kSince r>0, k>0.Then, the distance from center (h,k) to origin is sqrt(h² +k²)=10 -r=10 -5kThus,sqrt( (7k +10)^2 +k² )=10 -5kSquare both sides:49k² +140k +100 +k²=100 -100k +25k²50k² +140k +100=25k² -100k +10025k² +240k=0k*(25k +240)=0Solutions k=0 or k= -240/25= -9.6But k>0 from Case 1, so no solution here. Hence, Case 1 gives no valid solution.Case 2: First expression positive, second negative.Thus:3h +4k -30=5r- (4h -3k -40)=5r => -4h +3k +40=5rSo, equations:3h +4k -30=5r-4h +3k +40=5rSet equal:3h +4k -30= -4h +3k +407h +k -70=0 =>7h +k=70Express h in terms of k:h=(70 -k)/7=10 -k/7Substitute into first equation:3*(10 -k/7) +4k -30=5r30 -3k/7 +4k -30=5r(25k)/7=5r =>r=5k/7Then, distance from center to origin:sqrt( h² +k² )=10 -r=10 -5k/7Compute h=10 -k/7Thus,sqrt( (10 -k/7)^2 +k² )=10 -5k/7Square both sides:(100 - (20k)/7 +k²/49) +k²=100 - (100k)/7 +25k²/49Multiply all terms by 49 to eliminate denominators:49*100 - 20k*7 +k² +49k²=49*100 -100k*7 +25k²Compute each term:Left side:4900 -140k +k² +49k²=4900 -140k +50k²Right side:4900 -700k +25k²Set equal:4900 -140k +50k²=4900 -700k +25k²Subtract right side:0 +560k +25k²=0Factor:25k² +560k=0k*(25k +560)=0Solutions: k=0 or k= -560/25= -22.4Again, k=0 invalid (r=0), and k=-22.4 which is way outside the original circle (radius 10). So no solution here.Case3: First expression negative, second positive.Thus:- (3h +4k -30)=5r => -3h -4k +30=5r4h -3k -40=5rSet equal:-3h -4k +30=4h -3k -40-7h -k +70=0 =>7h +k=70Same as Case2. Wait, no:Wait, equate the two expressions:-3h -4k +30=4h -3k -40Bring all terms to left:-3h -4k +30 -4h +3k +40=0-7h -k +70=0 =>7h +k=70Same as Case2. So substituting h=10 -k/7 into the first equation:-3*(10 -k/7) -4k +30=5r-30 +3k/7 -4k +30=5r(-25k)/7=5r =>r= -5k/7But r>0 implies k<0. Then, distance from center (h,k) to origin:sqrt(h² +k²)=10 -r=10 -(-5k/7)=10 +5k/7But since k is negative, let me set k= -m, m>0. Then h=10 -(-m)/7=10 +m/7.Thus,sqrt( (10 +m/7)^2 +m² )=10 +5*(-m)/7=10 -5m/7But sqrt(...) must be positive, but RHS is 10 -5m/7. For RHS to be positive:10 -5m/7 >0 =>5m/7 <10 =>m <14But m>0. So m ∈ (0,14)Compute sqrt( (10 +m/7)^2 +m² )=10 -5m/7Square both sides:(10 +m/7)^2 +m²=(10 -5m/7)^2Expand left side:100 + (20m)/7 +m²/49 +m²=100 + (20m)/7 + (50m²)/49Right side:100 - (100m)/7 +25m²/49Set equal:100 +20m/7 +50m²/49=100 -100m/7 +25m²/49Subtract right side:(20m/7 +100m/7) + (50m²/49 -25m²/49)=0120m/7 +25m²/49=0Multiply both sides by 49:120m*7 +25m²=0840m +25m²=0m*(840 +25m)=0Solutions: m=0 or m= -840/25= -33.6But m>0, so no solution. Thus, no valid solution in Case3.Case4: Both expressions negative.Thus:-3h -4k +30=5r-4h +3k +40=5rWhich was our initial assumption, leading to h=7k +10, r= -5k. Then we found k= -8/5, h= -6/5, r=8. But this center is at (-6/5, -8/5), which is (-1.2, -1.6), inside the original circle but on the opposite side from A. This circle would be tangent to AB and AC, but not near A. Wait, but maybe it's tangent on the other side. Wait, the original circle is radius 10, centered at (0,0). The circle at (-1.2, -1.6) with radius 8 is quite large. Wait, 8 is close to 10, but distance from (-1.2, -1.6) to (0,0) is sqrt(1.44 +2.56)=sqrt(4)=2. So the distance between centers is 2, and since the original circle has radius 10 and the tangent circle has radius 8, 10 -8=2, which matches. So this circle is internally tangent to the original circle and has radius 8, located at (-1.2, -1.6). But does this circle also tangent to both chords AB and AC?Let me check. The distance from (-1.2, -1.6) to AB should be equal to r=8.Compute distance from (-6/5, -8/5) to AB: |3*(-6/5) +4*(-8/5) -30| /5= |(-18/5 -32/5 -30)| /5= |(-50/5 -30)| /5= | -10 -30| /5=40/5=8. Correct. Similarly, distance to AC: |4*(-6/5) -3*(-8/5) -40| /5= |(-24/5 +24/5 -40)| /5= |0 -40| /5=40/5=8. Correct. So this circle is indeed tangent to both chords and the original circle, but it's located diametrically opposite to point A. However, the problem states that the circle is tangent to the given circle and the constructed chords. There might be two such circles: one near A and one opposite. But in the problem statement, it's possible that only the smaller circle near A is considered. However, according to our calculations, the only solution is radius 8. But 8 is quite large. Wait, radius 8 circle inside a radius 10 circle, separated by 2 units between centers. That seems possible. But the problem might be expecting a different answer. Did I make a mistake in the coordinate setup?Wait, let me re-examine. The chords AB and AC are both drawn through point A, which is on the original circle. The circle we're looking for is tangent to the original circle and to both chords. The problem doesn't specify on which side of the chords the circle lies, so both solutions might be valid. However, the circle with radius 8 is quite large and located opposite to A, but the problem might be referring to the smaller circle near A. However, in our equations, we only found one solution. Therefore, perhaps my coordinate choice affected the result.Wait, perhaps I made a mistake in the coordinate system setup. I placed point A at (10,0), but maybe a different coordinate system would make this clearer. Alternatively, maybe there's another circle near A with a smaller radius.Alternatively, maybe I missed something in the problem statement. Let me re-read it."Through point A of a circle with a radius of 10, two mutually perpendicular chords AB and AC are drawn. Calculate the radius of a circle that is tangent to the given circle and the constructed chords, given AB = 16."So the circle is tangent to the original circle (radius 10) and to the two chords AB and AC. The chords are through point A, which is on the original circle. The problem doesn't specify the position of the tangent circle, so there might be two solutions. However, in our calculation, we only found one solution with radius 8. But 8 is larger than half of 10, which might seem counterintuitive for a circle tangent to two chords near the edge. But maybe that's correct.Alternatively, perhaps there's an error in the sign when we derived h =7k +10. Let me re-examine the equations.We had:From the two distance equations when both expressions are negative:-3h -4k +30 =5r ...(1)-4h +3k +40 =5r ...(2)Subtract (1) - (2):-3h -4k +30 +4h -3k -40 =0h -7k -10 =0 =>h=7k +10Then, substituting into (1):-3*(7k +10) -4k +30=5r-21k -30 -4k +30=5r-25k=5r =>r= -5kSince radius r is positive, k must be negative. Then, the distance from center to origin is sqrt(h² +k²)=10 -r=10 -(-5k)=10 +5kBut k is negative, so 10 +5k must be positive. Therefore:10 +5k >0 =>k >-2Given that k= -8/5= -1.6, which is greater than -2, so this is valid.So center at (h,k)= (-6/5, -8/5), radius=8. This circle is located at (-1.2, -1.6), radius 8, which is tangent to the original circle (distance between centers=2=10-8), and tangent to both chords AB and AC at distance 8. This seems correct.But why is this circle not near point A? Because when you have two perpendicular chords intersecting at a point on the circumference, the circle tangent to both chords and the original circle can be on either side of the chords. However, near point A, the distance from A to the center of the tangent circle would need to be such that the circle also fits inside the original circle. Given that the chords are 16 and 12, maybe the circle near A would have a smaller radius.Wait, but according to the equations, there's only one solution. This suggests that perhaps there's only one circle that satisfies the conditions, which is the one we found with radius 8. But that seems counterintuitive. Let me think.Alternatively, perhaps the problem requires the circle to be tangent to the chords and the original circle externally. But the original circle is given, and the constructed chords are inside it. The tangent circle must be inside the original circle, tangent to it internally. Our solution is internally tangent, which is correct.But if I consider the two chords AB and AC, and the circle tangent to them near A, the radius should be smaller. But according to the equations, there's no such solution. Why?Wait, perhaps my coordinate system choice is leading to this. Let me try a different approach without coordinates.Let’s consider that the two perpendicular chords AB and AC through point A on the circle with radius 10. Let’s denote the center of the original circle as O. The circle we need is tangent to the original circle and to both chords. Let’s denote the center of the desired circle as P, radius r.Since P is equidistant from both chords AB and AC, and those chords are perpendicular, the center P must lie along the angle bisector of the angle formed by AB and AC at point A. However, since AB and AC are chords intersecting at A, which is on the original circle, the angle bisector would be the line that is equidistant to both chords. Given that the chords are perpendicular, the angle bisector would be the line that makes 45 degrees with both chords.However, since the center P must also be at a distance of 10 - r from O, perhaps we can use this geometric relationship.Alternatively, consider the distance from P to AB and AC is r, and the distance from P to O is 10 - r.Let’s model this with coordinate geometry again, but perhaps choose a different coordinate system.Let me place point A at (0,10), so the original circle is centered at (0,0) with radius 10. Then chords AB and AC are perpendicular, passing through A (0,10). Let’s assume AB is horizontal for simplicity, so AC would be vertical. But given that AB and AC are chords of the circle, let's find their lengths.Wait, if AB is horizontal through A (0,10), then the equation of AB is y=10. But a horizontal line at y=10 would touch the circle x² + y²=100 only at (0,10), so that can't be a chord. Therefore, AB cannot be horizontal. This suggests that placing A at (0,10) might not be helpful. Maybe better to place A at (10,0) as before.Wait, in my initial coordinate system, placing A at (10,0), but solving gave a center at (-1.2, -1.6). If instead, I had placed A at (0,10), maybe the solution would appear in a different quadrant. However, regardless of the coordinate system, the problem is symmetric, so the radius should be the same.Alternatively, perhaps there's a formula for the radius of such a tangent circle. If we have two perpendicular lines (chords) intersecting at a point on a circle of radius R, then the radius r of a circle tangent to both lines and the original circle is given by... Hmm, I don't recall the exact formula, but maybe we can derive it.Let’s consider the general case. Let’s have a circle of radius R, and two perpendicular chords intersecting at a point A on the circle. The circle tangent to both chords and the original circle will have its center along the bisector of the right angle formed by the chords, at a distance r from each chord, and its center will be at a distance of R - r from the original center.Let’s model this in coordinates. Let’s place point A at (R,0). The two chords are perpendicular, so one can be taken as making an angle θ with the x-axis, and the other θ + 90 degrees. However, without loss of generality, we can consider one chord along a certain line and the other perpendicular.However, the distance from the center of the original circle (O at (0,0)) to each chord can be computed based on the chord lengths.Given chord AB has length 16 in our problem, so the distance from O to AB is d1 = sqrt(R² - (AB/2)^2) = sqrt(10² -8²)=6, as before. Similarly, chord AC has length 12 (calculated earlier), so distance from O to AC is d2= sqrt(10² -6²)=8.Wait, this is interesting. The distances from the center O to the two perpendicular chords AB and AC are 6 and 8, which are the legs of a right triangle, and the distance from O to point A is the hypotenuse, which is the radius 10. Indeed, 6² +8²=36+64=100=10². This makes sense because the point A is the intersection of the two chords, and the distances from O to each chord are d1 and d2, which are legs of a right triangle with hypotenuse OA=10.Therefore, this configuration forms a right triangle with legs 6 and 8, hypotenuse 10. This is key.Now, the center P of the tangent circle must be located at a distance r from each chord AB and AC, and at a distance R - r =10 -r from O.Let’s consider the coordinates of P. Since the chords are perpendicular and the distances from O to the chords are 6 and 8, we can model this in a coordinate system where chord AB is located at a distance 6 from O, and chord AC at distance 8 from O.But perhaps it's easier to use coordinate axes aligned with the chords. Let’s set up a coordinate system such that chord AB is along the line d1=6, and chord AC along d2=8. Wait, this might not be straightforward. Alternatively, let's use the fact that the distances from O to the chords are 6 and 8, and the angle between the chords is 90 degrees.The center P of the tangent circle lies at distances r from both chords, so it lies at the intersection of the lines parallel to AB and AC, each at a distance r from them. Since AB and AC are perpendicular, the locus of points at distance r from both is a square corner, and the center P lies along the angle bisector, which is the line at 45 degrees to the chords.But given that the original circle's center O is at distances 6 and 8 from the chords, we can model this using coordinate geometry where O is at (6,8), wait, no. Wait, maybe another approach.Let’s consider the problem in terms of coordinate geometry with O at (0,0), and the two chords AB and AC such that the distance from O to AB is 6 and to AC is 8, and AB and AC are perpendicular. Then, the equations of AB and AC can be written as:AB: (x/6) + (y/8) =1 (just a guess, but actually, the distance from O to AB is 6, and to AC is8). Wait, no. The equation of a line at distance d from O is x cosθ + y sinθ =d. Since AB and AC are perpendicular, their angles θ differ by 90 degrees.Alternatively, since the distances are 6 and 8, and OA=10, which is the hypotenuse of the right triangle formed by the distances to the chords.Assume that AB is the line with distance 6 from O, and AC the line with distance8 from O, and they are perpendicular. Let’s assign coordinates such that AB is horizontal and AC is vertical, but this might not be possible since their distances are 6 and8. Alternatively, use the following setup.Let’s let AB have the equation ax + by +c=0 with distance 6 from O, and AC have equation -bx + ay +d=0 to be perpendicular, with distance8 from O. But this might complicate.Alternatively, use vector approach. The two chords are perpendicular, and their distances from O are6 and8. The position of point A is the intersection of the two chords, and lies on the circle of radius10.From the distances, we can use the formula that if two lines are at distances p and q from the origin and are perpendicular, then the distance from the origin to their intersection point is sqrt(p² +q²). Wait, in our case, point A is the intersection of AB and AC, which are at distances6 and8 from O, and are perpendicular. Then, the distance from O to A should be sqrt(6² +8²)=10, which matches since A is on the circle of radius10. This is consistent.Therefore, this forms a right triangle with legs6 and8, hypotenuse10, where A is the vertex opposite the hypotenuse. Therefore, the coordinates of A can be considered as (6,8) scaled to the hypotenuse10. Wait, but how?Actually, if the distances from O to AB and AC are6 and8, and OA=10, then the coordinates of A can be found using the direction ratios. The coordinates of A are (d1*cosθ +d2*sinθ, d1*sinθ -d2*cosθ) or something like that, but this might be more complex.Alternatively, realize that the coordinates of A can be represented as (d1, d2) rotated by some angle. But perhaps this is not necessary.Given the complexity, perhaps returning to the initial coordinate system where we found the solution r=8 is correct. Even though it's counterintuitive that the circle is on the opposite side from A, mathematically it satisfies all conditions: tangent to both chords and the original circle. However, the problem statement says "a circle that is tangent to the given circle and the constructed chords", without specifying the location, so this is a valid solution.But in the problem, it's specified that the chords are drawn through point A, which is on the original circle. The tangent circle we found is not near A, but on the opposite side. Perhaps the problem expects this answer, radius8, which is one of the options. However, given the original chords are length16 and12, which are quite long, the circle of radius8 might be plausible.Alternatively, maybe I made a miscalculation earlier. Let me verify the center (-6/5, -8/5) which is (-1.2, -1.6). The distance from this center to AB is |3*(-1.2) +4*(-1.6) -30| /5= | -3.6 -6.4 -30| /5= | -40| /5=8. Correct. Distance to AC is |4*(-1.2) -3*(-1.6) -40| /5= | -4.8 +4.8 -40| /5= | -40| /5=8. Correct. Distance to O is sqrt( (-1.2)^2 + (-1.6)^2 )=sqrt(1.44+2.56)=sqrt(4)=2, which is10 -8=2. Correct. So this circle is valid.Therefore, despite being counterintuitive, the correct radius is8. However, this feels large, but given the configuration, it's correct. Perhaps the problem expects this answer.Final AnswerThe radius of the tangent circle is boxed{8}.

✅:Given a circle with radius 10 and point ( A ) on the circle, two mutually perpendicular chords ( AB ) and ( AC ) are drawn. We need to find the radius of a circle tangent to the given circle and the chords ( AB ) and ( AC ), given ( AB = 16 ).1. Distance from the center to the chords: - The distance from the center ( O ) to chord ( AB ) is calculated using the formula for the distance from the center to a chord: [ text{Distance} = sqrt{r^2 - left(frac{text{chord length}}{2}right)^2} = sqrt{10^2 - 8^2} = 6 ] - Similarly, the length of chord ( AC ) is found to be 12, and the distance from ( O ) to ( AC ) is 8.2. Equations of the chords: - The equation of chord ( AB ) is derived as ( 3x + 4y - 30 = 0 ). - The equation of chord ( AC ) is derived as ( 4x - 3y - 40 = 0 ).3. Setting up the equations for the tangent circle: - The center of the tangent circle ((h, k)) must be equidistant from both chords ( AB ) and ( AC ), and the distance from ((h, k)) to the center ( O ) must be ( 10 - r ). - The distances from ((h, k)) to the chords ( AB ) and ( AC ) are given by: [ frac{|3h + 4k - 30|}{5} = r quad text{and} quad frac{|4h - 3k - 40|}{5} = r ] - The distance from ((h, k)) to ( O ) is: [ sqrt{h^2 + k^2} = 10 - r ]4. Solving the system of equations: - By considering the absolute values and solving the equations, we find that the center of the tangent circle is ((-1.2, -1.6)) and the radius ( r ) is 8.5. Verification: - The distance from the center ((-1.2, -1.6)) to the chords ( AB ) and ( AC ) is 8, and the distance to the center ( O ) is 2, which confirms the radius is ( 10 - 8 = 2 ).Thus, the radius of the circle tangent to the given circle and the chords ( AB ) and ( AC ) is (boxed{8}).

🔑:1. Let ( O ) be the center of the given circle with radius ( 10 ).2. Let ( A ) be a point on the circle, and through ( A ), two mutually perpendicular chords ( AB ) and ( AC ) are drawn. Given ( AB = 16 ).3. Since ( AB ) is a chord passing through ( A ) with midpoint ( K ), using the properties of chords in circles, the perpendicular from the center ( O ) to the chord ( AB ) will bisect it. Therefore, the length of ( AK ) or ( KB ) is: [ AK = KB = frac{AB}{2} = frac{16}{2} = 8 ]4. In the right triangle ( OAK ) formed by the radius ( OA ) and the perpendicular ( OK ) from ( O ) to ( AB ), using the Pythagorean theorem: [ OA^2 = OK^2 + AK^2 ] Here, ( OA = 10 ) and ( AK = 8 ), so: [ 10^2 = OK^2 + 8^2 ] Simplify and solve for ( OK ): [ 100 = OK^2 + 64 implies OK^2 = 100 - 64 implies OK^2 = 36 implies OK = 6 ]5. Now, considering the point ( C ) where ( AC ) is perpendicular to ( AB ) and also a chord passing through ( A ). Let ( AC ) have length ( AC ).6. As both chords are perpendicular to each other and their midpoints share the symmetry about the center ( O ): [ AC^2 = OA^2 - OK^2 ] Plugging in the known distances: [ AC^2 = 10^2 - 6^2 implies AC^2 = 100 - 36 implies AC^2 = 64 implies AC = sqrt{64} = 8 ]7. Therefore, the distance from the center of the original circle ( O ) to the new circle ( O_1 ) (the circle that is tangent to both chords and the original circle): [ O_1O = sqrt{(OK + KA)^2 - AC^2} ] Given that ( K ) is the midpoint of ( AB ) and considering right angles: [ O_1O = sqrt{left(8-6right)^2} = 2 ]8. Finally, using the geometric touch properties given by the problem and calculated radius estimates: [ O_1 = 16 - 8 ] Conclusion:The radius of the second circle that is tangent to the first circle and constructed chords is:[ boxed{8} ]

🧠:Okay, let's try to solve this problem. So, the function is f(x) = x² - a x + 2a, and we know that its zeros are integers. We need to find the sum of all possible values of the coefficient a. The answer choices are given, but I need to work through it step by step.First, since the zeros are integers, let's recall that for a quadratic equation x² + bx + c = 0, the sum of the roots is -b and the product is c. Wait, but here the quadratic is written as x² - a x + 2a. So comparing to the standard form x² + bx + c, here b is -a and c is 2a. So, the sum of the roots would be -b = a, and the product of the roots is c = 2a.So, if the roots are integers, let's denote them as p and q. Then we have:p + q = aandp * q = 2aSo, our goal is to find all integer pairs (p, q) such that their sum is a and their product is 2a. Then, we can find all possible a's and sum them up.Hmm, so we have two equations:1. p + q = a2. p * q = 2aWe need to solve these equations for integers p and q, and then find the corresponding a's.Let me try to substitute a from the first equation into the second. Since a = p + q, substitute into the second equation:p * q = 2(p + q)So, p*q - 2p - 2q = 0Hmm, maybe we can factor this. Let's add 4 to both sides:p*q - 2p - 2q + 4 = 4Then, factor:(p - 2)(q - 2) = 4Ah, that seems like a useful step. Let me check:Left side: (p - 2)(q - 2) = p*q - 2p - 2q + 4, which is exactly the left side of the equation above. So, yes, (p - 2)(q - 2) = 4.So now, since p and q are integers, (p - 2) and (q - 2) must be integer divisors of 4. Let's list all pairs of integers (d1, d2) such that d1 * d2 = 4.The integer divisors of 4 are: 1, 2, 4, -1, -2, -4.So possible pairs (d1, d2):(1, 4), (4, 1), (2, 2), (-1, -4), (-4, -1), (-2, -2)Each of these pairs will give us (p, q) by adding 2 back to each component.Let's go through each pair:1. (1, 4): Then p = 1 + 2 = 3, q = 4 + 2 = 6. Then a = p + q = 3 + 6 = 92. (4, 1): p = 4 + 2 = 6, q = 1 + 2 = 3. Similarly, a = 6 + 3 = 93. (2, 2): p = 2 + 2 = 4, q = 2 + 2 = 4. a = 4 + 4 = 84. (-1, -4): p = -1 + 2 = 1, q = -4 + 2 = -2. Then a = 1 + (-2) = -15. (-4, -1): p = -4 + 2 = -2, q = -1 + 2 = 1. a = -2 + 1 = -16. (-2, -2): p = -2 + 2 = 0, q = -2 + 2 = 0. a = 0 + 0 = 0Wait, so these are the possible pairs. Let me verify each a value and check if the roots are integers.First, for a = 9: The quadratic is x² -9x + 18. Let's factor: (x - 3)(x - 6) = 0, which gives roots 3 and 6. Correct, integers.For a = 8: Quadratic is x² -8x + 16. Which factors as (x -4)², so root 4 (double root). Still an integer.For a = -1: Quadratic is x² - (-1)x + 2*(-1) = x² + x -2. Factoring: (x + 2)(x -1), roots -2 and 1. Integers.For a = 0: Quadratic is x² -0x +0 = x². So root 0 (double root). Integer.So, the possible a values are 9, 8, -1, 0. Wait, but in the pairs above, a = 9, 8, -1, 0. Are there duplicates?Looking at the pairs:(1,4) and (4,1) both give a =9.Similarly, (-1, -4) and (-4, -1) both give a = -1.(2,2) gives a=8.(-2, -2) gives a=0.So the distinct a values are 9,8,-1,0.Wait, but the answer options don't include 0. Let me check if a=0 is valid.If a=0, then the quadratic is x² +0x +0 = x², which has a double root at 0. 0 is an integer, so yes, a=0 is valid. But let me check the product and sum again for a=0.Sum of roots is a=0, product is 2a=0. So roots must sum to 0 and multiply to 0, which is satisfied by 0 and 0. Correct.But looking at the answer choices: (A)7, (B)8, (C)16, (D)17, (E)18.Hmm, 9 +8 + (-1) +0 =16. So the sum would be 16. But 16 is option C. However, the answer options include 16, but let me check again.Wait, maybe I missed some possible divisor pairs? Let me check again.The divisors of 4 are 1, 2, 4, -1, -2, -4. So the possible pairs (d1, d2) where d1*d2=4:(1,4), (4,1), (2,2), (-1,-4), (-4,-1), (-2,-2). I think that's all. So 6 pairs, leading to a values 9,9,8,-1,-1,0. So distinct a's are 9,8,-1,0. Sum them: 9 +8 + (-1) +0=16. So 16 is the sum. But let me check the answer options. Option C is 16. So the answer should be C.Wait, but the problem says "the sum of all possible values of the coefficient a". So if possible a's are 9,8,-1,0, their sum is 16. So answer C.But let me check again if I missed any possible pairs. For example, maybe ( -2, -2 ) gives a=0, which we considered. What about ( -1, -4 ) and ( -4, -1 ), which both give a= -1. So that's covered. So the possible a's are 9,8,-1,0. Sum is 16. So answer is C. But let me check again with a different approach to confirm.Alternative approach:Given that the quadratic x² -a x +2a has integer roots. So discriminant must be a perfect square. The discriminant D is a² - 8a. So D = a² -8a. For roots to be integers, discriminant must be a perfect square.So, a² -8a = k², where k is integer. Then, a² -8a -k²=0. Hmm, this might be a different path, but maybe more complicated.Alternatively, since the roots are integers, they must satisfy p + q = a and pq = 2a. So as before, we get pq = 2(p + q). Which leads to (p -2)(q -2)=4. So same as before. So maybe my first approach was correct.Alternatively, perhaps the answer is 16. But let me check the answer options again. The choices are A)7, B)8, C)16, D)17, E)18.So 16 is an option. So if my answer is 16, then option C is correct.Wait, but in my possible a's, 0 is included, but maybe the problem is expecting only non-zero a's? Let me check the problem statement again: "the sum of all possible values of the coefficient a". It says "all possible values", so 0 should be included if it's valid. Since when a=0, the function becomes x², which has a root 0 (with multiplicity 2), which is an integer. So 0 is valid. So sum is 16.But let me check the answer options again. 16 is there. So the answer should be C)16.But wait, the original answer options include (E)18. Maybe I made a mistake in my calculation? Let me double-check the possible a's.From (1,4) and (4,1): a=9From (2,2): a=8From (-1,-4) and (-4,-1): a=-1From (-2,-2): a=0So the possible a's are 9,8,-1,0. So sum is 9+8+(-1)+0=16. Yes. So 16 is correct, so answer C.But wait, let me check if there are other factor pairs. For example, (d1, d2) could also be ( -2, -2 ), which gives a=0. Wait, that's covered. Are there any other divisor pairs? 4 can be written as ( -1, -4 ), ( -4, -1 ), ( -2, -2 ), etc. I think that's all.Alternatively, maybe I miscalculated a for some pair.Wait, when (d1, d2) = (-1, -4):p = d1 +2= -1 +2=1q = d2 +2= -4 +2= -2So a= p + q=1 + (-2)= -1. Correct.Similarly, (d1,d2)=(-4,-1):p= -4+2=-2q= -1+2=1a= -2 +1=-1. Correct.For (d1,d2)=(-2,-2):p=0, q=0, a=0. Correct.So all a's are accounted for. So sum is 16. So answer is C. But I want to confirm once more.Alternatively, maybe I missed a possible divisor pair. Let's list all possible integer pairs (d1, d2) such that d1*d2=4.Positive pairs:(1,4), (2,2), (4,1)Negative pairs:(-1, -4), (-2, -2), (-4, -1)Is that all? Yes. So 6 pairs. So 6 pairs leading to 4 distinct a's. So sum 16. So the answer is C.But why is 0 included? If a=0, the quadratic is x². It has a double root at 0, which is an integer. So it's acceptable.But the answer choices include 16, which is the sum of 9+8+(-1)+0=16. So that's correct.Therefore, the correct answer is C)16.

✅:The zeros of ( f(x) = x^2 - ax + 2a ) are integers. Let the roots be ( p ) and ( q ). Then:[p + q = a quad text{and} quad pq = 2a.]Substituting ( a = p + q ) into ( pq = 2a ):[pq = 2(p + q) implies pq - 2p - 2q = 0 implies (p - 2)(q - 2) = 4.]The integer factor pairs of 4 are ((1, 4)), ((4, 1)), ((2, 2)), ((-1, -4)), ((-4, -1)), ((-2, -2)). Adding 2 to each component to find ( p ) and ( q ):- ((1+2, 4+2) = (3, 6)) ⟹ ( a = 3 + 6 = 9 )- ((4+2, 1+2) = (6, 3)) ⟹ ( a = 9 )- ((2+2, 2+2) = (4, 4)) ⟹ ( a = 8 )- ((-1+2, -4+2) = (1, -2)) ⟹ ( a = 1 + (-2) = -1 )- ((-4+2, -1+2) = (-2, 1)) ⟹ ( a = -1 )- ((-2+2, -2+2) = (0, 0)) ⟹ ( a = 0 )Distinct values of ( a ) are ( 9, 8, -1, 0 ). Summing these:[9 + 8 + (-1) + 0 = 16.]Answer: (boxed{C})

🔑:Given the function ( f(x) = x^{2} - ax + 2a ), we need to find the sum of all possible values for the coefficient ( a ) such that the roots of the quadratic equation are integers.Let the roots of the quadratic equation ( f(x) = 0 ) be two integers. According to Vieta’s formulas, if the roots of the polynomial ( x^2 - ax + b ) are ( p ) and ( q ), then:1. ( p + q = a )2. ( pq = 2a )Let's denote the roots by ( p ) and ( q ). Therefore, we have:[ p + q = a ][ pq = 2a ]By substitution and transforming the system of equations, we get:[ pq = 2(p + q) ]Rearranging this equation, we find:[ pq - 2p - 2q = 0 ]Adding 4 to both sides:[ pq - 2p - 2q + 4 = 4 ]Factoring the left side:[ (p - 2)(q - 2) = 4 ]Next, we solve ((p - 2)(q - 2) = 4) by considering the factor pairs of 4:- ( (p - 2, q - 2) = (4, 1) ) yields ( p - 2 = 4 ) and ( q - 2 = 1 ) [ p = 6, q = 3 ] Thus, ( a = p + q = 6 + 3 = 9 )- ( (p - 2, q - 2) = (2, 2) ) yields ( p - 2 = 2 ) and ( q - 2 = 2 ) [ p = 4, q = 4 ] Thus, ( a = p + q = 4 + 4 = 8 )- ( (p - 2, q - 2) = (-4, -1) ) yields ( p - 2 = -4 ) and ( q - 2 = -1 ) [ p = -2, q = 1 ] Thus, ( a = p + q = -2 + 1 = -1 )- ( (p - 2, q - 2) = (-2, -2) ) yields ( p - 2 = -2 ) and ( q - 2 = -2 ) [ p = 0, q = 0 ] Thus, ( a = p + q = 0 + 0 = 0 )Summarizing the possible integer values for ( a ), we have:[ a = 9, 8, -1, 0 ]Adding these values together gives:[ 9 + 8 + (-1) + 0 = 16 ]Thus, the sum of all possible values of ( a ) is:[ boxed{16} ]

🧠:Okay, let's tackle this problem step by step. We need to figure out which of the four given sets is the empty set. The options are A, B, C, D. Let me go through each one carefully.Starting with option A: {x | x + 3 = 3}. Hmm, this set contains all real numbers x such that when you add 3 to them, you get 3. Let me solve the equation x + 3 = 3. Subtract 3 from both sides, and we get x = 0. So the set A is actually {0}, which has one element. Therefore, set A is not empty. Got that.Moving on to option B: {(x, y) | y² = -x², x, y ∈ ℝ}. Alright, this is a set of ordered pairs (x, y) where the square of y is equal to the negative square of x. Since both x and y are real numbers, their squares are non-negative. So, y² is always ≥ 0, and -x² is always ≤ 0. The only way these two can be equal is if both sides are zero. That means y² = 0 and -x² = 0, which implies both x and y must be 0. So the only solution is (0, 0). Therefore, set B is {(0, 0)}, which is not empty. So B is not the empty set either.Next is option C: {x | x² ≤ 0}. Let's think. The square of any real number x is always non-negative, right? So x² is always ≥ 0. The inequality x² ≤ 0 would only be true when x² is exactly 0. That happens when x = 0. Therefore, set C is {0}, which is again a singleton set, not empty. So C is out.Now, option D: {x | x² - x + 1 = 0, x ∈ ℝ}. This is the set of real numbers x that satisfy the quadratic equation x² - x + 1 = 0. To determine if there are real solutions, I need to check the discriminant of the quadratic. The discriminant D is given by D = b² - 4ac. Here, a = 1, b = -1, c = 1. Plugging in, D = (-1)² - 4*1*1 = 1 - 4 = -3. Since the discriminant is negative, the quadratic equation has no real roots. Therefore, there are no real numbers x that satisfy the equation, which means set D is empty.Wait a second, let me double-check each step to make sure I didn't make a mistake. For set A, solving x + 3 = 3 gives x = 0. That seems straightforward. For set B, the equation y² = -x². Since squares are non-negative, the only solution is x = 0, y = 0. Correct. For set C, x² ≤ 0 only when x = 0. Right. For set D, discriminant is indeed (-1)^2 - 4(1)(1) = 1 - 4 = -3, which is negative. So no real solutions. Therefore, set D is the empty set. Looks like all my steps are correct. I don't think I made any errors here. So the answer should be D.

✅:The empty set is determined by analyzing each option:- Set A: Solving ( x + 3 = 3 ) yields ( x = 0 ). Thus, ( {0} ), not empty. - Set B: ( y^2 = -x^2 ) implies ( x = y = 0 ) (since squares are non-negative). Thus, ( {(0, 0)} ), not empty. - Set C: ( x^2 leq 0 ) requires ( x = 0 ). Thus, ( {0} ), not empty. - Set D: The quadratic ( x^2 - x + 1 = 0 ) has a discriminant ( D = (-1)^2 - 4(1)(1) = -3 ). Since ( D < 0 ), there are no real solutions. Hence, this set is empty. Answer: (boxed{D})

🔑:1. Option (A): Consider the set ({x mid x + 3 = 3}). [ x + 3 = 3 implies x = 3 - 3 implies x = 0 ] Thus, ({x mid x + 3 = 3} = {0}). This set is not empty.2. Option (B): Consider the set (left{(x, y) mid y^2 = -x^2, x, y in mathbf{R}right}). For any real number (x), [ y^2 = -x^2 ] Since (x^2 geq 0), (-x^2 leq 0). The only way for (y^2) to equal (-x^2) is if both sides are zero (since (y^2 geq 0) always). Therefore, [ -x^2 = 0 implies x^2 = 0 implies x = 0 ] And, [ y^2 = 0 implies y = 0 ] Thus, (left{(x, y) mid y^2 = -x^2, x, y in mathbf{R}right} = {(0, 0)}). This set is not empty.3. Option (C): Consider the set (left{x mid x^2 leq 0right}). Since (x^2 geq 0) for all (x in mathbf{R}), the only solution to (x^2 leq 0) is [ x^2 = 0 implies x = 0 ] Thus, (left{x mid x^2 leq 0right} = {0}). This set is not empty.4. Option (D): Consider the set (left{x mid x^2 - x + 1 = 0, x in mathbf{R}right}). The quadratic equation (x^2 - x + 1 = 0) has discriminant [ Delta = b^2 - 4ac = (-1)^2 - 4 cdot 1 cdot 1 = 1 - 4 = -3 ] Since the discriminant is negative, the equation has no real solutions. Consequently, [ left{x mid x^2 - x + 1 = 0, x in mathbf{R}right} = emptyset ] Conclusion:The set that represents the empty set is option (D).[boxed{mathrm{D}}]

🧠:Okay, so I need to solve this system of equations with 1000 variables, right? Each equation is like x_i squared equals a times the next x plus 1, and the last one loops back to x1. Hmm, this seems cyclic. Since all the equations are similar, maybe there's a pattern or a symmetric solution. Let me start by considering if all the variables are equal. That might simplify things.Let's assume that x1 = x2 = ... = x1000 = x. If that's the case, then each equation becomes x² = a x + 1. So, solving for x, we get x² - a x - 1 = 0. The quadratic equation would give solutions x = [a ± sqrt(a² + 4)] / 2. Since |a| > 1, the discriminant sqrt(a² + 4) is real, so these are real solutions. That seems straightforward. But wait, are these the only solutions? Maybe there are other solutions where the variables aren't all equal. But with such a symmetric system, symmetric solutions are often the first to look for.But let me check if the system could have non-symmetric solutions. Suppose the variables are not all equal. Then, each x_i squared depends on the next x. Maybe there's a periodic pattern, like x1, x2, x3, ..., repeating every certain number of steps. But with 1000 variables, which is a large number, it's hard to imagine. Maybe periodicity with a smaller cycle? For example, if it repeats every 2 steps, then x1 = x3 = x5..., and x2 = x4 = x6..., but since 1000 is even, this might be possible. Let me try that.Assume x1 = x3 = x5 = ..., and x2 = x4 = x6 = .... Then, for the first equation: x1² = a x2 + 1. The second equation: x2² = a x3 + 1 = a x1 + 1. Then the third equation: x3² = a x4 + 1 = a x2 + 1, which is the same as the first equation. So, this creates a system where x1 and x2 satisfy:x1² = a x2 + 1 x2² = a x1 + 1Subtracting these two equations: x1² - x2² = a(x2 - x1) Factor left side: (x1 - x2)(x1 + x2) = -a(x1 - x2) If x1 ≠ x2, then we can divide both sides by (x1 - x2), getting x1 + x2 = -a If x1 = x2, then we have the same equation as before: x1² = a x1 + 1, which brings us back to the symmetric solution.So, if x1 ≠ x2, then x1 + x2 = -a. Then, let's use this with the original equations. Let me write the equations again with x1 + x2 = -a.From x1² = a x2 + 1 and x2² = a x1 + 1. Let's substitute x2 = -a - x1 into the first equation.x1² = a(-a - x1) + 1 x1² = -a² - a x1 + 1 x1² + a x1 + (a² - 1) = 0 Quadratic in x1: x1² + a x1 + (a² - 1) = 0 Solutions: x1 = [-a ± sqrt(a² - 4(a² - 1))]/2 Simplify discriminant: sqrt(a² - 4a² + 4) = sqrt(-3a² + 4) Wait, but |a| > 1, so sqrt(-3a² + 4) would be imaginary if 3a² > 4. Since |a| >1, 3a² >4 when |a| > 2/sqrt(3) ≈1.1547. But since |a| >1, if |a| is greater than 2/sqrt(3), the discriminant becomes negative, leading to complex solutions. But we need real solutions. So, this case of x1 ≠ x2 would only give real solutions if -3a² +4 ≥0, i.e., a² ≤4/3. But since |a|>1, this would require 1 < |a| ≤ 2/sqrt(3) ≈1.1547. So, only in that interval would there be real solutions for the non-symmetric case. However, the problem states that |a| >1, not specifying further. Therefore, depending on the value of a, there might be two types of solutions: symmetric ones (always present) and non-symmetric ones (only when |a| ≤2/sqrt(3)). But since the problem just says |a|>1, maybe we need to consider all possible cases? Wait, but the problem says "given number satisfying |a|>1". So, perhaps we need to present solutions that work for any |a|>1. But the non-symmetric solutions would only exist if |a| ≤2/sqrt(3). Hmm. So, maybe the symmetric solutions are the only ones valid for all |a|>1. The non-symmetric solutions exist only in a subset of |a|>1. Therefore, perhaps the system has symmetric solutions for all |a|>1, and additional solutions when |a| is in (1, 2/sqrt(3)].But the problem asks to solve the system given that |a|>1. So, depending on the value of a, the solutions may vary. However, since the problem doesn't specify further constraints on a, maybe the answer is to present all possible solutions. But maybe the only real solutions for all |a|>1 are the symmetric ones. Let me verify.For the symmetric solution, x = [a ± sqrt(a² +4)]/2. Let's check if these are real. Since a² +4 is always positive, sqrt(a² +4) is real, so x is real. Therefore, the symmetric solutions are always real, regardless of |a|>1. For the non-symmetric solutions, when |a|>1 but ≤2/sqrt(3), we can have real solutions. But for |a|>2/sqrt(3), those solutions become complex. Therefore, in the context of the problem, which doesn't specify whether solutions should be real or complex, but usually, systems like this expect real solutions unless stated otherwise. Therefore, if the problem is expecting real solutions, then the non-symmetric solutions only exist when 1 < |a| ≤ 2/sqrt(3). But the problem states |a|>1, so depending on the value of a, there might be 2 symmetric solutions and potentially 2 non-symmetric solutions (but only when |a| is in that interval). However, with 1000 variables, the non-symmetric solution would require a 2-cycle, so variables alternate between x1 and x2. But since there are 1000 variables (even number), this would work. So, in total, possible solutions are:1. All variables equal to x = [a + sqrt(a² +4)]/2.2. All variables equal to x = [a - sqrt(a² +4)]/2.3. For 1 < |a| ≤2/sqrt(3), variables alternate between two values x and y, satisfying x + y = -a, x² = a y +1, y² = a x +1.But since the problem says "solve the system", perhaps we need to present all possible solutions. However, the problem might be expecting the symmetric solutions as the primary ones, given that non-symmetric ones exist only in a subset. But maybe in the problem's context, even if non-symmetric solutions exist for certain a, they should be mentioned.Wait, but the problem is stated as "given number satisfying |a|>1". So, a is fixed, and we need to solve the system for that given a. So, depending on a, we might have different solutions. However, unless told otherwise, the solution would include all real solutions for that a. So, perhaps the answer should include both symmetric solutions and, if applicable, the non-symmetric ones.But since the problem is presented in a general context (without specifying whether a is in that smaller interval), perhaps the main solutions are the symmetric ones, and the non-symmetric ones are additional when possible. However, given that 1000 is even, the 2-cycle would work. But perhaps there are cycles of other lengths as well? For example, maybe cycles of length dividing 1000. But this complicates things. However, given the system's cyclic nature, perhaps the only solutions are the symmetric ones and the 2-cycle ones (when possible). Because higher cycles would lead to more complex systems.Alternatively, maybe all solutions must be symmetric. Let's check with a small n first. Suppose n=2. Then the system is:x1² = a x2 +1 x2² = a x1 +1In this case, we have the same as before. Solutions are either x1=x2 with x² = a x +1, or x1≠x2 with x1 + x2 = -a. So, for n=2, there are both symmetric and non-symmetric solutions (when possible). So, for n=1000, which is even, perhaps the same applies. That is, we can have solutions where variables alternate between x and y, forming a 2-cycle. However, with n=1000, this alternation would cycle through all variables, so x1, x2, x3,... would alternate x, y, x, y,... up to x1000, which is y, and then x1000² = a x1 +1, which would be y² = a x +1. So, same as the n=2 case. Therefore, the 2-cycle solutions are valid for even n. Therefore, in the case of n=1000, which is even, we can have both symmetric solutions and 2-cycle solutions (when |a| ≤2/sqrt(3}).Therefore, the system has the symmetric solutions x_i = [a ± sqrt(a² +4)]/2 for all i, and if |a| ≤2/sqrt(3), also the 2-cycle solutions where x_{2k-1} = x, x_{2k} = y, with x and y satisfying x + y = -a, x² = a y +1, y² = a x +1.But the problem states |a| >1. So, depending on the value of a, these 2-cycle solutions may or may not exist. Therefore, the complete solution set is:- For any |a| >1, the symmetric solutions x_i = [a + sqrt(a² +4)]/2 and x_i = [a - sqrt(a² +4)]/2 for all i.- Additionally, if 1 < |a| ≤ 2/sqrt(3), there are 2-cycle solutions where the variables alternate between two values x and y satisfying the aforementioned equations.But since the problem says "solve" without specifying constraints on a beyond |a|>1, maybe we are to present all possible solutions. However, in typical math problems, unless specified, they might expect the symmetric solutions. But given the system's cyclic nature, it's possible to have other solutions. But maybe given the high number of variables (1000), it's impractical to consider all possible cycles. However, the 2-cycle is a natural extension from the n=2 case.But let's test with a specific a. Let's pick a=1.2, which is greater than 1 and less than 2/sqrt(3)≈1.1547. Wait, 1.2 is actually greater than 1.1547 (since 2/sqrt(3)≈1.1547). Wait, 2/sqrt(3) is approximately 1.1547, so a=1.2 would satisfy |a|>1.1547, so non-symmetric solutions would not exist. Wait, no. The condition for non-symmetric solutions is |a| ≤2/sqrt(3), so approximately 1.1547. So, if |a| is between 1 and 1.1547, then non-symmetric solutions exist. If |a|>1.1547, they don't. Therefore, for |a|>1.1547, only symmetric solutions exist. For 1 < |a| ≤1.1547, both symmetric and non-symmetric solutions exist.Therefore, depending on the given a, the solutions vary. But since the problem states |a|>1 without further restriction, we need to present all solutions valid for any a in that range. So, the answer must include the symmetric solutions, and mention the non-symmetric ones when applicable. However, since the problem doesn't specify whether a is in the range allowing non-symmetric solutions, perhaps we should present all possible solutions with conditions.But the problem says "solve the system of equations, in which a is a given number satisfying |a|>1". So, the value of a is given, fixed, but we don't know its exact value beyond |a|>1. Therefore, the solution must cover all possible cases based on a. Hence, the answer should state:The system has the symmetric solutions x_i = [a ± sqrt(a² +4)]/2 for all i. Additionally, if 1 < |a| ≤ 2/sqrt(3), there exist solutions where the variables alternate between two values x and y satisfying x + y = -a, x² = a y +1, and y² = a x +1.However, since the problem might expect only the symmetric solutions, given the high number of variables and the lack of specification regarding a, but given that it's a cyclic system, it's prudent to mention both possibilities.But maybe there's another approach. Let's try to consider the general case. Suppose we have x_{i}^2 = a x_{i+1} +1 for i=1,...,1000, with x_{1001}=x1. Let's try to express all variables in terms of x1. For example, x2 = (x1² -1)/a. Then x3 = (x2² -1)/a = [( (x1² -1)/a )² -1]/a. This quickly becomes complicated, leading to a high-degree equation. With 1000 variables, this would result in an equation of degree 2^1000, which is intractable. Therefore, such an approach isn't feasible. Hence, symmetric solutions or solutions with periodic patterns are the only feasible ones to find analytically.Given that, the symmetric solutions are straightforward. For the 2-cycle solutions, as previously mentioned, they exist only if the discriminant is non-negative, which happens when |a| ≤2/sqrt(3). However, since |a|>1 is given, the overlapping interval is 1 < |a| ≤2/sqrt(3). Therefore, for a in that interval, there are two types of solutions: symmetric and 2-cycle. For |a|>2/sqrt(3), only symmetric solutions exist.But how does the problem want the answer? Since it's a system with 1000 equations, and given the user's instruction is to provide the solution, likely the symmetric solutions are expected. However, to be thorough, we should mention both cases.Therefore, summarizing:The system has two symmetric solutions where all variables are equal:x_i = [a + sqrt(a² + 4)] / 2 for all i,andx_i = [a - sqrt(a² + 4)] / 2 for all i.Additionally, if 1 < |a| ≤ 2/sqrt{3}, there are two more solutions where the variables alternate between two values x and y, which satisfy:x + y = -a,x² = a y + 1,y² = a x + 1.These alternating solutions can be found by solving the quadratic equations derived from the above system.But since the problem states |a| >1, we need to specify that these alternating solutions exist only when 1 < |a| ≤2/sqrt{3} ≈1.1547.But the problem might be expecting only the symmetric solutions as the answer, especially since it's a high-dimensional system, and non-symmetric solutions may not be required unless specified. However, given the problem's cyclic nature and the possibility of periodicity, it's safer to present both possibilities.Wait, but let me check the 2-cycle solutions. Suppose we have x and y alternating. Then for the last equation, x1000² = a x1 +1. Since 1000 is even, x1000 = y, and x1 = x. Therefore, y² = a x +1, which is consistent with the second equation. So, the system reduces to two equations:x² = a y +1,y² = a x +1,and x + y = -a.So, solving these, we can find x and y. Let's attempt to solve them.From x + y = -a, we have y = -a -x. Substitute into the first equation:x² = a(-a -x) +1 = -a² -a x +1,which rearranged is x² + a x + (a² -1) =0,as before. Similarly, substituting x = -a -y into the second equation gives the same quadratic. The solutions are:x = [ -a ± sqrt( a² -4(a² -1) ) ] / 2= [ -a ± sqrt( -3a² +4 ) ] /2.Therefore, real solutions exist when -3a² +4 ≥0 ⇒ a² ≤4/3 ⇒ |a| ≤2/sqrt(3)≈1.1547.Given that the original condition is |a|>1, the overlap is 1 < |a| ≤2/sqrt(3). Therefore, these solutions exist only in that interval.Therefore, the complete solution set is:1. For any a with |a|>1, the symmetric solutions:x_i = [a + sqrt(a² +4)]/2,andx_i = [a - sqrt(a² +4)]/2,for all i=1,…,1000.2. Additionally, if 1 < |a| ≤2/sqrt(3), there are solutions where the variables alternate between two values x and y given by:x = [ -a + sqrt(-3a² +4) ] /2,y = [ -a - sqrt(-3a² +4) ] /2,and vice versa (x and y swapped).Therefore, each variable with an odd index is x, and even index is y, or the other way around.However, since 1000 is even, both configurations are valid (starting with x or y).Therefore, for such a, there are two more solutions (the two permutations of x and y).But in total, considering the symmetric solutions and the alternating ones when possible, the system has:- Two solutions for |a|>1.- Four solutions for 1 < |a| ≤2/sqrt(3).But since swapping x and y gives different solutions, but in the context of the entire system, since the variables are in a cycle, starting with x or y might lead to distinct solutions. However, in terms of the set of solutions, swapping all x and y would produce the same system, so they are distinct.But in terms of explicit writing, the solutions are:For the symmetric case, all variables equal to [a ± sqrt(a² +4)]/2.For the alternating case, variables alternate between x and y where x and y are [ -a ± sqrt(-3a² +4) ] /2.Therefore, the final answer should mention both possibilities, specifying the conditions on a.However, given the problem's phrasing, it asks to "solve the system of equations" given |a|>1. So, to solve it means to find all real solutions. Therefore, we must present both the symmetric solutions (which always exist) and the alternating ones (when they exist).Hence, the boxed solutions would be the symmetric ones, and mention the alternating ones conditionally. But since the user might expect all solutions, but given the format requires putting the final answer in boxes, perhaps only the symmetric solutions are required. However, to be precise, we should check if the problem expects all solutions.But given the system's symmetry and the standard approach to such cyclic systems, the primary solutions are the symmetric ones. The alternating solutions are additional but only in specific cases. However, since the problem doesn't specify to find all solutions, just to solve the system, which could mean finding all possible solutions. Therefore, the answer should include both.But how to present this in a box? The problem is in Chinese, maybe, but the instructions here are in English. Typically, for such systems, the symmetric solutions are considered the main ones. However, to be thorough, the answer should include both. But given the formatting constraints, perhaps we box the symmetric solutions and mention the others in text. However, the user instruction says to put the final answer within boxed{}.But the system might have multiple solutions, so maybe multiple boxed answers. However, standard practice is to present all solutions. Given that, perhaps the answer is two boxed expressions for the symmetric solutions, and conditional boxed expressions for the alternating ones. But since the conditions on a can't be represented in a box, maybe the answer is expected to be the symmetric solutions.Alternatively, since the problem states |a|>1, and doesn't restrict further, but the non-symmetric solutions exist only for a subset, maybe the answer is just the symmetric solutions. However, the problem might be designed to have only symmetric solutions, and the non-symmetric ones are extraneous. But given the mathematical analysis, both exist when allowed.But perhaps I need to confirm with a test value. Let's take a=1.5, which is greater than 2/sqrt(3)≈1.1547. For a=1.5, the symmetric solutions are:x = [1.5 ± sqrt(1.5² +4)]/2 = [1.5 ± sqrt(2.25 +4)]/2 = [1.5 ± sqrt(6.25)]/2 = [1.5 ±2.5]/2.Thus, x = (1.5 +2.5)/2=4/2=2, and x=(1.5 -2.5)/2= (-1)/2=-0.5.So, solutions are 2 and -0.5. Plugging back, x=2: 2²=4=1.5*2 +1=3+1=4. Correct. x=-0.5: (-0.5)^2=0.25=1.5*(-0.5)+1=-0.75+1=0.25. Correct.For a=1.5, non-symmetric solutions would require solving x + y = -1.5 and x²=1.5 y +1, y²=1.5 x +1. Let's see if real solutions exist. The discriminant is -3*(1.5)^2 +4= -6.75 +4= -2.75 <0. Therefore, no real solutions. Hence, only symmetric solutions exist for a=1.5.For a=1.1, which is within 1 <|a| <2/sqrt(3). Let's compute discriminant: -3*(1.1)^2 +4= -3*(1.21)+4= -3.63 +4=0.37>0. Therefore, real solutions exist. So, non-symmetric solutions exist here.Thus, the answer must include both cases.However, given the problem's requirement to present the final answer boxed, and considering that solutions may depend on a, but the problem states that a is given with |a|>1, the answer is likely expected to present all possible solutions, conditional on a.But in Chinese math problems, especially in olympiad style, cyclic systems often expect symmetric solutions unless non-symmetric are explicitly required. However, given the analysis, both exist when possible. But how to present this?Perhaps the answer is:All variables are equal and satisfy ( x = frac{a pm sqrt{a^2 + 4}}{2} ). If ( 1 < |a| leq frac{2}{sqrt{3}} ), additional solutions exist where the variables alternate between two values.But translating this into boxed form, since there are two symmetric solutions, they can be boxed. For the alternating case, since they depend on solving quadratics, perhaps expressing them in terms of a, but it's more complex.Alternatively, if the problem expects only the symmetric solutions, then the answer is simply the two boxed expressions.Given that the problem is presented without restriction on the solution type and the natural solutions are symmetric, I think the expected answer is the two symmetric solutions. Therefore, the final answer is:( boxed{x_i = dfrac{a + sqrt{a^2 + 4}}{2}} ) and ( boxed{x_i = dfrac{a - sqrt{a^2 + 4}}{2}} ) for all ( i ).But to confirm, let's see. If the variables are all equal, it satisfies each equation, and these solutions are valid for any a (with |a|>1). The other solutions exist only for specific a, but since the problem states "given a" with |a|>1, and doesn't ask for conditions on a, the complete solution includes both symmetric solutions regardless of a, and the alternating ones if applicable. However, since the alternating solutions can't be expressed without piecewise conditions, and the problem might expect the symmetric ones, I think the answer is the two boxed solutions above.

✅:The system of equations is cyclic and symmetric, suggesting that a symmetric solution where all variables are equal may exist. Assuming ( x_1 = x_2 = cdots = x_{1000} = x ), each equation reduces to ( x^2 = ax + 1 ). Solving this quadratic equation gives:[x = frac{a pm sqrt{a^2 + 4}}{2}]These solutions are real for all ( a ) with ( |a| > 1 ). Additionally, non-symmetric solutions where the variables alternate between two values ( x ) and ( y ) exist if ( 1 < |a| leq frac{2}{sqrt{3}} ). However, these solutions require solving a more complex system and are only valid for a specific range of ( a ).Given the problem's requirement to solve the system for any ( a ) with ( |a| > 1 ), the symmetric solutions are always valid and thus the primary solutions. Therefore, the solutions are:[boxed{x_i = dfrac{a + sqrt{a^2 + 4}}{2}} quad text{and} quad boxed{x_i = dfrac{a - sqrt{a^2 + 4}}{2}} quad text{for all } i = 1, 2, ldots, 1000]

🔑:Consider the system of equations given by:[begin{aligned}x_1^2 & = a x_2 + 1, x_2^2 & = a x_3 + 1, & vdots x_{999}^2 & = a x_{1000} + 1, x_{1000}^2 & = a x_1 + 1.end{aligned}]where (a) is a given number satisfying (|a| > 1). We need to find the solutions of this system.We first consider the case (a > 1). The equations involve squares on the left-hand sides, which are nonnegative, so (x_i^2 geq 0). Thus, we have:[ x_i geq -frac{1}{a} quad text{for } i = 1, 2, ldots, 1000.]Suppose, without loss of generality, that (x_1 = max { x_i }). This implies:[ x_1 geq x_2, x_3, ldots, x_{1000}.]# Case 1: (x_1 geq 0)If (x_1 geq 0), we start by examining the first equation:[ x_1^2 = a x_2 + 1.]Since (x_1 geq x_2), squaring both sides, we get:[ x_1^2 geq x_2^2.]Thus:[ a x_2 + 1 geq a x_3 + 1 implies x_2 geq x_3.]Using similar arguments repeatedly, we conclude:[ x_1 geq x_2 geq x_3 geq ldots geq x_{1000} geq x_1.]Thus:[ x_1 = x_2 = cdots = x_{1000}.]Let (x_1 = x_2 = cdots = x_{1000} = x). Then from the first equation:[ x^2 = a x + 1.]Solving this quadratic equation:[ x^2 - a x - 1 = 0,]we get:[ x = frac{a pm sqrt{a^2 + 4}}{2}.]Since (a > 1), the positive root will be:[ x = frac{a + sqrt{a^2 + 4}}{2}.]# Case 2: (x_1 < 0)Now, consider (x_i < 0) for every (i). From:[ x_1^2 = a x_2 + 1,]because (x_1 geq x_2), squaring, we get:[ x_1^2 leq x_2^2,]implying:[ a x_2 + 1 leq a x_4 + 1 implies x_2 leq x_4.]Using similar arguments repeatedly:[ x_1 geq x_3 geq x_5 geq ldots geq x_{999} geq x_1]and[ x_2 leq x_4 leq ldots leq x_{1000} leq x_2.]Thus:[ x_1 = x_3 = x_5 = ldots = x_{999} ]and[ x_2 = x_4 = ldots = x_{1000}.]Let (x_1 = x_3 = cdots = x_{999} = x_A) and (x_2 = x_4 = cdots = x_{1000} = x_B). Then, the system reduces to:[ x_A^2 = a x_B + 1,][ x_B^2 = a x_A + 1.]Subtracting these equations:[ x_A^2 - x_B^2 = a x_B - a x_A implies (x_A - x_B)(x_A + x_B + a) = 0.]This yields:1. (x_A = x_B), then:[ x_A = x_B = frac{a pm sqrt{a^2 + 4}}{2},]where the negative solution is:[ x = frac{a - sqrt{a^2 + 4}}{2}.]2. (x_A + x_B + a = 0), resulting in:[ x_A^2 + a x_A + (a^2 - 1) = 0.]The discriminant of the quadratic equation:[ (a^2 - 4 + 3 a^2),]The discriminant ((a^2 - 4 + 3 a^2) = -3a^2 + 4),Thus, if (a > frac{2}{sqrt{3}}), this equation will have no real solutions.Conclusively, the solutions are:[ boxed{x = frac{a + sqrt{a^2 + 4}}{2} text{or} x = frac{a - sqrt{a^2 + 4}}{2}}]blacksquare