🧠:Okay, let me try to figure this problem out. So, the question is: If the number of sides in a polygon is doubled, the difference between the diagonals and sides increases by 99. How many sides does the polygon have?Hmm, let's break it down step by step. First, I need to recall the formula for the number of diagonals in a polygon. I remember that for a polygon with n sides, the number of diagonals is given by n(n - 3)/2. Let me verify that formula. Each vertex connects to n - 3 others (excluding itself and its two adjacent vertices), and since each diagonal is counted twice, we divide by 2. Yeah, that sounds right.So, let's denote the original number of sides as n. Then the number of diagonals in the original polygon would be D1 = n(n - 3)/2. The number of sides is just n, so the difference between diagonals and sides would be D1 - n = [n(n - 3)/2] - n. Let me compute that:Difference1 = [n(n - 3)/2] - n = [n(n - 3) - 2n]/2 = [n² - 3n - 2n]/2 = [n² - 5n]/2.Now, if we double the number of sides, the new polygon has 2n sides. Let's compute the difference between diagonals and sides for this new polygon. The number of diagonals here would be D2 = 2n(2n - 3)/2. Simplifying that:D2 = [2n(2n - 3)]/2 = n(2n - 3).The number of sides in the new polygon is 2n, so the difference D2 - 2n is:Difference2 = n(2n - 3) - 2n = 2n² - 3n - 2n = 2n² - 5n.According to the problem, when we double the number of sides, the difference increases by 99. That means:Difference2 - Difference1 = 99.Let's write that equation:(2n² - 5n) - (n² - 5n)/2 = 99.Wait, hold on. Let me check that again. Difference1 is [n² - 5n]/2 and Difference2 is 2n² - 5n. So the difference between them is:2n² - 5n - [n² - 5n]/2 = 99.To solve this, I need to combine the terms. Let's compute the left-hand side:First, let's express 2n² - 5n as (4n² - 10n)/2 so that both terms have the same denominator.Then:(4n² - 10n)/2 - (n² - 5n)/2 = [4n² - 10n - n² + 5n]/2 = [3n² - 5n]/2.So the equation becomes:[3n² - 5n]/2 = 99.Multiply both sides by 2:3n² - 5n = 198.Bring all terms to one side:3n² - 5n - 198 = 0.Now, we need to solve this quadratic equation for n. Let's try factoring. The quadratic is 3n² -5n -198 =0.We can use the quadratic formula: n = [5 ± sqrt(25 + 4*3*198)]/(2*3).Compute the discriminant:sqrt(25 + 2376) = sqrt(2401) = 49.So, n = [5 ± 49]/6.Since the number of sides can't be negative, we take the positive root:n = (5 + 49)/6 = 54/6 = 9.So, n=9. Let me verify this.Original polygon has 9 sides. Diagonals: 9*6/2=27. Difference: 27 -9=18.Doubled sides:18. Diagonals:18*15/2=135. Difference:135 -18=117.Now, 117 -18=99. Yes, that checks out.So, the answer should be 9.

✅:To solve the problem, let's denote the original number of sides as ( n ). The number of diagonals in a polygon with ( n ) sides is ( frac{n(n-3)}{2} ), and the difference between diagonals and sides is:[text{Difference}_1 = frac{n(n-3)}{2} - n = frac{n^2 - 5n}{2}.]When the number of sides is doubled to ( 2n ), the number of diagonals becomes ( frac{2n(2n-3)}{2} = n(2n-3) ). The new difference is:[text{Difference}_2 = n(2n-3) - 2n = 2n^2 - 5n.]The problem states that this difference increases by 99:[text{Difference}_2 - text{Difference}_1 = 99.]Substituting the expressions:[2n^2 - 5n - frac{n^2 - 5n}{2} = 99.]Multiplying through by 2 to eliminate the denominator:[4n^2 - 10n - n^2 + 5n = 198 implies 3n^2 - 5n - 198 = 0.]Solving this quadratic equation using the quadratic formula ( n = frac{-b pm sqrt{b^2 - 4ac}}{2a} ):[n = frac{5 pm sqrt{25 + 2376}}{6} = frac{5 pm 49}{6}.]Taking the positive root:[n = frac{54}{6} = 9.]Verification:- Original polygon: 9 sides, 27 diagonals. Difference: ( 27 - 9 = 18 ).- Doubled polygon: 18 sides, 135 diagonals. Difference: ( 135 - 18 = 117 ).- Increase: ( 117 - 18 = 99 ), which matches the problem's condition.Answer: The polygon has (boxed{9}) sides.

🔑:1. Define the problem variables: Let n represent the number of sides of the initial polygon. For a polygon with n sides, the formula for the number of diagonals is given by: text{Number of diagonals} = frac{n(n-3)}{2} 2. Double the number of sides: Next, consider a polygon where the number of sides is doubled, i.e., a 2n-sided polygon. The formula for the number of diagonals in a 2n-sided polygon is given by: text{Number of diagonals} = frac{2n(2n-3)}{2} = n(2n-3) 3. Express the given condition mathematically: The problem states that the difference between the total number of diagonals and sides increases by 99 when the number of sides is doubled. This can be expressed as: left( frac{2n(2n-3)}{2} - 2n right) - left( frac{n(n-3)}{2} - n right) = 99 4. Simplify and solve the equation: Simplify both sides of the equation: begin{align*} & left( n(2n-3) - 2n right) - left( frac{n(n-3)}{2} - n right) = 99 & n(2n - 3 - 2) - left( frac{n(n-3)}{2} - n right) = 99 & n(2n - 5) - left( frac{n^2 - 3n}{2} - n right) = 99 & n(2n - 5) - left( frac{n^2 - 3n - 2n}{2} right) = 99 & n(2n - 5) - frac{n^2 - 5n}{2} = 99 & frac{2n(2n - 5) - (n^2 - 5n)}{2} = 99 & frac{4n^2 - 10n - n^2 + 5n}{2} = 99 & frac{3n^2 - 5n}{2} = 99 end{align*}5. Further simplify and solve for n: Multiply both sides of the equation by 2 to eliminate the fraction: 3n^2 - 5n = 198 Rearrange the equation to standard quadratic form: 3n^2 - 5n - 198 = 0 6. Solve the quadratic equation using the quadratic formula: The quadratic equation is 3n^2 - 5n - 198 = 0. Use the quadratic formula n = frac{-b pm sqrt{b^2 - 4ac}}{2a} where a = 3, b = -5, and c = -198: begin{align*} n &= frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 3 cdot (-198)}}{2 cdot 3} &= frac{5 pm sqrt{25 + 4 cdot 3 cdot 198}}{6} &= frac{5 pm sqrt{25 + 2376}}{6} &= frac{5 pm sqrt{2401}}{6} &= frac{5 pm 49}{6} end{align*} This yields the two solutions: begin{align*} n_1 &= frac{5 + 49}{6} = frac{54}{6} = 9 n_2 &= frac{5 - 49}{6} = frac{-44}{6} = -frac{22}{3} end{align*}7. Evaluate the solutions: Since the number of sides n must be a positive integer, the second solution n = -frac{22}{3} is not feasible. Thus, we accept n = 9. Conclusion:The polygon has boxed{9} sides.

🧠:Okay, let's try to figure out this elevator puzzle. So, the person lives on the 25th floor of a 30-story building. Every weekday morning, they take the elevator down to go to work. But when they come back in the evening, they take the elevator up to the 24th floor and then walk up one floor. The question is why don't they go all the way to the 25th floor in the evening?First, I need to think about possible reasons. Maybe the elevator doesn't go to the 25th floor in the evening? But that doesn't make sense because the building has 30 floors, so the elevator should go higher than 25. Unless there's something specific about the elevator's operation at certain times. Wait, maybe in the morning they go down from 25, but in the evening, they can't go up to 25. Hmm.Another thought: could it be related to elevator buttons? Maybe the 25th-floor button is broken in the evening? But the problem doesn't mention anything about maintenance or broken buttons. Maybe it's a regular occurrence, so it's something inherent to the building or the person's routine.Wait, maybe the elevator has different behavior depending on the time of day. Like, maybe in the morning, the elevator can be used normally, but in the evening, it's set to only go up to certain floors. But why would that be the case? Maybe there's a security feature where after certain hours, you need a key card to access higher floors. But the person lives there, so they should have access. Unless they don't have the key, but that seems odd if they live on 25.Alternatively, could it be that the person is avoiding something? Like, if the elevator is crowded in the evening, maybe they get off early to avoid waiting? But why specifically the 24th floor? That's just one floor below. Maybe the elevator skips certain floors during peak times. But again, the problem states it's every evening except weekends, so maybe it's a consistent issue.Wait, another angle: maybe the elevator's design. If the building has multiple elevators, perhaps some elevators don't go all the way up. For example, maybe there's a low-rise elevator and a high-rise elevator. But the person takes the same elevator in the morning and evening, right? Unless in the evening, only the low-rise elevator is operational. But the building has 30 floors, so probably all elevators go up to 30.Wait, maybe the person's height? If they can't reach the 25th button, but that's a stretch. Or maybe they have a habit, but the question implies there's a logical reason.Another possibility: the person is trying to save time. If getting off at 24 and walking up is faster than waiting for the elevator to go all the way? Maybe the elevator is slow, but one floor difference isn't much. Or maybe when they come home, the elevator is going down after dropping them at 24, so they can catch it going down again? Not sure.Wait, maybe the elevator doesn't stop at 25 when going up because of how the buttons are pressed. Like, in the evening, maybe the elevator is in express mode or only stops at certain floors when going up. But why would 24 be a stop then?Another thought: the person might be avoiding someone on the 25th floor. But that's more of a personal reason, and the question likely has a more structural answer.Wait, here's a common elevator puzzle answer: the person is a child or someone of short stature who can't reach the button for the 25th floor. In the morning, they take the elevator down because the ground floor button is accessible. But in the evening, they can only reach up to the 24th floor button and then have to walk up one floor. That makes sense. Because the buttons are arranged in order, and a child might not be able to reach higher buttons. So, in the morning, when going down, the elevator goes to the ground floor automatically, or someone else presses the button. But in the evening, they can only press up to 24 and then walk.Yes, that seems plausible. Let me check if that's a known answer. I've heard similar riddles where the person is a child who can't reach higher buttons. So, the key point is that the person can't physically reach the 25th floor button when entering the elevator in the evening. In the morning, when leaving, the elevator is already on 25, so they just need to press the ground floor button, which they can reach. But when returning, they can't reach the 25 button inside the elevator, so they press 24 and walk up.

✅:The individual gets off on the 24th floor and walks up one floor because they are unable to reach the elevator button for the 25th floor. Here's the breakdown:1. Morning Routine: When leaving home, the elevator is already on the 25th floor (their residence). They only need to press the ground floor button, which is within their reach (e.g., lower buttons are accessible even to a child or someone of shorter stature).2. Evenering Routine: Upon returning, they enter the elevator from the ground floor. The buttons for higher floors (like 25) are positioned out of their physical reach. The highest button they can press is the 24th floor. After exiting on 24, they walk up one flight of stairs to their 25th-floor apartment.Key Insight: The person’s height or physical ability limits their access to higher buttons in the elevator cabin. This constraint does not affect their morning descent since the ground floor button is easily reachable.

🔑:1. A person lives on the 25th floor of a 30-story building: - Each morning (except Saturday and Sunday), he enters the elevator, goes down, and goes to work. - In the evening, upon returning home, he enters the elevator, comes up to the 24th floor, and then walks up one more floor to his apartment on the 25th floor.2. To solve the question of why he exits the elevator on the 24th floor and walks one more floor to the 25th instead of going directly to the 25th floor, consider the following possibilities:3. Possible reasons might include: - Broken button for the 25th floor. - Unusual preference or ritual that compels him to walk one floor. - But, the most plausible reason is usually realized through logical inference.4. Logical Reasoning: - If the person is a "lilliput" (a very short person), he may not be able to reach the button for the 25th floor on the elevator control panel. This detail fits perfectly as the primary cause for his behavior.5. He can likely reach the 24th-floor button but not the 25th-floor button. Conclusion:- The person exits the elevator on the 24th floor and walks up because he is a lilliput and can't reach the button for the 25th floor on the elevator control panel.[boxed{text{The person is a lilliput and cannot reach the 25th-floor button on the elevator control panel.}}]

🧠:Okay, let's try to figure out the general term of the sequence {a_n} given the initial condition a₁ = 1 and the recurrence relation aₙ₋₁ aₙ = aₙ₋₁ - aₙ. The options are A through D, so I need to determine which one fits. Hmm, where to start?First, maybe I should write down the recurrence relation again to make sure I have it right. It says that for each n (starting from n=2, I guess?), the product of the previous term aₙ₋₁ and the current term aₙ is equal to the difference between the previous term and the current term. So mathematically, that's:aₙ₋₁ * aₙ = aₙ₋₁ - aₙHmm. Maybe I can rearrange this equation to solve for aₙ in terms of aₙ₋₁. Let's try that.Starting with:aₙ₋₁ * aₙ = aₙ₋₁ - aₙLet me move all terms to one side to see if that helps:aₙ₋₁ * aₙ - aₙ₋₁ + aₙ = 0Hmm, maybe factor terms? Let's see:aₙ₋₁ * aₙ + aₙ - aₙ₋₁ = 0Wait, maybe factor aₙ out of the first two terms:aₙ (aₙ₋₁ + 1) - aₙ₋₁ = 0Then:aₙ (aₙ₋₁ + 1) = aₙ₋₁So, solving for aₙ:aₙ = aₙ₋₁ / (aₙ₋₁ + 1)Ah, okay. So that's the recursive formula. Each term is equal to the previous term divided by (previous term + 1). That seems manageable. Let's see if we can compute the first few terms and maybe spot a pattern. Since a₁ = 1, let's compute a₂, a₃, etc.Starting with a₁ = 1.a₂ = a₁ / (a₁ + 1) = 1 / (1 + 1) = 1/2a₃ = a₂ / (a₂ + 1) = (1/2) / (1/2 + 1) = (1/2) / (3/2) = (1/2) * (2/3) = 1/3a₄ = a₃ / (a₃ + 1) = (1/3) / (1/3 + 1) = (1/3) / (4/3) = 1/4Wait a minute, that seems like a pattern. a₁ = 1, a₂ = 1/2, a₃ = 1/3, a₄ = 1/4... So perhaps the general term is aₙ = 1/n? That's option A. But let's check a₅ to be sure.a₅ = a₄ / (a₄ + 1) = (1/4) / (1/4 + 1) = (1/4) / (5/4) = 1/5Yep, that's 1/5. So the pattern continues. So if that's the case, the answer should be A. But let me check if the other options might fit. Let's see, if we plug in the formula aₙ = 1/n into the recurrence relation, does it hold?Let's test the recurrence relation with aₙ = 1/n. Let's take a general term and check if aₙ₋₁ * aₙ = aₙ₋₁ - aₙ.Left-hand side (LHS): aₙ₋₁ * aₙ = (1/(n-1)) * (1/n) = 1/(n(n-1))Right-hand side (RHS): aₙ₋₁ - aₙ = (1/(n-1)) - (1/n) = (n - (n-1))/(n(n-1)) = (1)/(n(n-1))So LHS = RHS. Therefore, the formula aₙ = 1/n does satisfy the recurrence relation. And since the initial term a₁ = 1 matches 1/1, the general term is indeed 1/n. Therefore, the correct answer is A.But wait, just to make sure I didn't make a mistake in the algebra. Let me go through the steps again.Given the recurrence relation:aₙ₋₁ aₙ = aₙ₋₁ - aₙRearranged to:aₙ = aₙ₋₁ / (aₙ₋₁ + 1)Starting with a₁ = 1:a₂ = 1/(1 + 1) = 1/2a₃ = (1/2)/(1/2 + 1) = (1/2)/(3/2) = 1/3a₄ = (1/3)/(1/3 + 1) = (1/3)/(4/3) = 1/4Yep, that's consistent. So the pattern is clear here. Therefore, the general term is 1/n. So answer A is correct.Just to check if any other options could fit, let's consider option B: aₙ = n.Let's compute a₂ with this formula. If a₁ = 1, then a₂ = 2. Let's plug into the recurrence:a₁ * a₂ = 1 * 2 = 2a₁ - a₂ = 1 - 2 = -12 ≠ -1. So that doesn't hold. So B is invalid.Option C: aₙ = n + 1. Then a₁ = 2, but the problem states a₁ = 1. So C is immediately invalid.Option D: aₙ = n - 1. Then a₁ = 0. But a₁ is supposed to be 1. So D is invalid.Therefore, only option A works. So the answer must be A.

✅:The sequence ({a_n}) is defined by (a_1 = 1) and the recurrence relation (a_{n-1} a_n = a_{n-1} - a_n). To find the general term:1. Rearrange the recurrence relation: [ a_{n-1} a_n = a_{n-1} - a_n implies a_n = frac{a_{n-1}}{a_{n-1} + 1}. ]2. Compute initial terms: - (a_1 = 1), - (a_2 = frac{1}{1+1} = frac{1}{2}), - (a_3 = frac{frac{1}{2}}{frac{1}{2}+1} = frac{1}{3}), - (a_4 = frac{frac{1}{3}}{frac{1}{3}+1} = frac{1}{4}), - Continuing this pattern, (a_n = frac{1}{n}).3. Verify the general term: Substitute (a_n = frac{1}{n}) into the recurrence: [ text{LHS} = a_{n-1} a_n = frac{1}{n-1} cdot frac{1}{n} = frac{1}{n(n-1)}, ] [ text{RHS} = a_{n-1} - a_n = frac{1}{n-1} - frac{1}{n} = frac{1}{n(n-1)}. ] LHS equals RHS, confirming validity.4. Eliminate other options: - B: (a_n = n) fails for (n=2). - C and D: Incorrect initial terms.Answer: (boxed{A})

🔑:Given sequence ({a_n}) satisfying the properties:[ a_1 = 1, ][ a_{n-1} a_n = a_{n-1} - a_n. ]We need to find its general term formula.1. Start with the given recurrence relation: [ a_{n-1} a_n = a_{n-1} - a_n. ]2. Isolate (a_n): [ a_{n-1} a_n + a_n = a_{n-1}. ]3. Factor out (a_n): [ a_n (a_{n-1} + 1) = a_{n-1}. ]4. Divide both sides by (a_{n-1} + 1) (assuming (a_{n-1} + 1 neq 0)): [ a_n = frac{a_{n-1}}{a_{n-1} + 1}. ]5. To solve for (a_n), consider the sequence in terms of its reciprocal: [ frac{1}{a_n}. ]6. From the relation (a_n = frac{a_{n-1}}{a_{n-1} + 1}), take the reciprocal: [ frac{1}{a_n} = frac{a_{n-1} + 1}{a_{n-1}} = 1 + frac{1}{a_{n-1}}. ]7. Let (b_n = frac{1}{a_n}). Then: [ b_n = 1 + b_{n-1}. ]8. This suggests that (b_n) is an arithmetic sequence with (b_1 = frac{1}{a_1} = 1): [ b_n = b_1 + (n-1) = 1 + (n-1) = n. ]9. Therefore: [ frac{1}{a_n} = n implies a_n = frac{1}{n}. ]Conclusion:The general term of the sequence ({a_n}) is:[boxed{a_n = frac{1}{n}}.]Thus, the correct option is A.

🧠:Okay, so I need to prove that when vertex C of triangle ABC moves along a line not parallel to side AB, the locus of the vertices M and N of the inscribed square MNKL (with L and K on AB or its extension) is a pair of lines. Hmm, let me start by visualizing the problem.First, let me sketch triangle ABC with AB as the base. The square MNKL has two vertices, L and K, on AB. Then M is on AC, and N is on BC. When C moves along a line, I need to see where points M and N end up. The claim is that their paths are straight lines.Maybe coordinate geometry would help here. Let me set up coordinates. Let's place AB on the x-axis for simplicity. Let’s fix points A at (0,0) and B at (b,0), where b is a positive real number. Now, point C is moving along a line that's not parallel to AB. Let's say the line has some slope, maybe not horizontal. Let's parameterize the position of C. Suppose the line along which C moves is given by y = mx + c. Since it's not parallel to AB (which is horizontal), m ≠ 0.But wait, if AB is on the x-axis from (0,0) to (b,0), and C is moving on a line y = mx + c, then for different positions of C, we have different triangles. For each such triangle, we inscribe a square with two vertices on AB. The square's vertices L and K are on AB, so their coordinates will be somewhere between A and B, or maybe extended beyond. The other two vertices M and N are on AC and BC respectively.I need to find the coordinates of M and N in terms of the position of C, then eliminate the parameter that defines C's position along the line, leading to an equation in x and y that should be linear if the locus is a line.Let me formalize this. Let’s denote point C as (h, k), lying on the line y = mx + c. So, k = mh + c. Then, the coordinates of C are (h, mh + c). Let's denote the square MNKL. Let’s assume that the square is positioned such that L and K are on AB. Let’s let L be at (t, 0) and K be at (t + s, 0), where s is the side length of the square. Since it's a square, the coordinates of M and N can be determined based on the square's orientation.Since the square is sitting on AB, from point L (t,0) moving up to M and then to N and K. The direction from L to M would be vertical if the square is above AB, but since the square is inscribed in the triangle, the direction might not be vertical. Wait, actually, in a typical inscribed square in a triangle, the square has one side along a part of the base, and the other sides extending upwards into the triangle. However, the square has to touch both sides AC and BC.Wait, maybe I need to clarify the configuration. Let me think. If the square MNKL has L and K on AB, then the side LK is part of AB. Then, sides LM and KN are rising from AB into the triangle. Points M and N are on AC and BC respectively. So, the square is standing on AB with two vertices, and the other two vertices touching the sides AC and BC.So, coordinates: Let’s let L be at (t, 0) and K at (t + s, 0), so the side length is s. Then, M would be at (t, s) and N at (t + s, s) if the square is upright. But in reality, since the triangle's sides AC and BC are not necessarily vertical, the square might be "tilted". Wait, no. Wait, but in the triangle, the square must fit such that M is on AC and N is on BC. Therefore, the coordinates of M and N must lie on AC and BC, respectively.Therefore, the square is not axis-aligned. Hmm, so maybe the sides LM and KN are not vertical. Let me try to parametrize this.Alternatively, perhaps using similar triangles. Let me consider the triangle ABC with coordinates A(0,0), B(b,0), and C(h, k). The square MNKL inscribed with L on AB, K on AB, M on AC, and N on BC.Let’s denote the side length of the square as s. Let’s suppose that point L is at (t, 0) on AB, so point K is at (t + s, 0). Then, since the square is MNKL, the points M and N would be offset from L and K by some direction. But since it's a square, the direction from L to M and from K to N should be perpendicular to LK and of length s.But the direction depends on the orientation of the square. Since M is on AC and N is on BC, the square is "growing" upwards from AB into the triangle, but the sides LM and KN are not necessarily vertical. Therefore, we need to find coordinates of M and N such that LM and KN are sides of the square, i.e., vectors LM and KN are perpendicular to LK and have the same length as LK.Wait, vector LK is (s, 0), so the direction from L to K is along the x-axis. Then, the vectors LM and KN should be perpendicular to LK, so in the y-direction. But if the square is axis-aligned, then M would be at (t, s) and N at (t + s, s). But in this case, points M and N must lie on AC and BC respectively.But AC is the line from A(0,0) to C(h, k), so parametric equation: (0 + λh, 0 + λk) for λ ∈ [0,1]. Similarly, BC is from B(b,0) to C(h,k): (b + μ(h - b), 0 + μk) for μ ∈ [0,1].If M is on AC, then M is (λh, λk). Similarly, N is (b + μ(h - b), μk). If the square is axis-aligned, then M would be (t, s) and N would be (t + s, s). So equate coordinates:For M: λh = t and λk = s. Therefore, λ = t/h and s = t k / h.For N: b + μ(h - b) = t + s and μk = s. From μk = s, μ = s/k = (t k / h)/k = t/h. Then, substituting into the x-coordinate:b + (t/h)(h - b) = t + sLeft side: b + t - (t b)/hRight side: t + s = t + t k / hSo:b + t - (t b)/h = t + t k / hSubtract t from both sides:b - (t b)/h = t k / hMultiply both sides by h:b h - t b = t kBring all terms to one side:b h = t b + t kFactor t:b h = t (b + k)Therefore:t = (b h)/(b + k)So, since s = t k / h, then s = (b h)/(b + k) * k / h = (b k)/(b + k)Therefore, the coordinates of M are (t, s) = ( (b h)/(b + k), (b k)/(b + k) )Similarly, coordinates of N:From N's x-coordinate: t + s = (b h)/(b + k) + (b k)/(b + k) = b (h + k)/(b + k)Wait, but h and k are related because point C lies on the line y = mx + c. So, k = m h + c.Therefore, substituting k = m h + c into the coordinates of M and N.For M:x-coordinate: (b h)/(b + k) = (b h)/(b + m h + c)y-coordinate: (b k)/(b + k) = (b (m h + c))/(b + m h + c)Similarly, for N:x-coordinate: b (h + k)/(b + k) = b (h + m h + c)/(b + m h + c) = b ( (1 + m) h + c ) / (b + m h + c )y-coordinate: same as M's y-coordinate, since s is the same.But since k = m h + c, we can express h in terms of k: h = (k - c)/m, provided m ≠ 0.But since C moves along the line y = m x + c, h and k are related by k = m h + c.Let me consider M's coordinates:x_M = (b h)/(b + k) = (b h)/(b + m h + c)Similarly, substituting h = (k - c)/m,x_M = (b (k - c)/m ) / (b + k )Similarly, y_M = (b k)/(b + k )Wait, but we need to express x_M and y_M in terms of h, or in terms of k, and then eliminate the parameter h or k to find the relation between x and y, which would be the locus.Let me take M's coordinates:x = (b h)/(b + k)y = (b k)/(b + k)But since k = m h + c, substitute:x = (b h)/(b + m h + c)y = (b (m h + c))/(b + m h + c )Let me treat h as a parameter. Let's denote h as a variable, and express x and y in terms of h, then eliminate h to find the relation between x and y.Let me denote D = b + m h + c. Then:x = (b h)/Dy = (b (m h + c))/DLet me solve for h from the x equation:x = (b h)/D => x D = b hBut D = b + m h + c, so:x (b + m h + c) = b hSimilarly, from the y equation:y = (b m h + b c)/D => y D = b m h + b cLet me express these equations.From x:x (b + m h + c) = b h => x b + x m h + x c = b hBring terms with h to one side:x b + x c = b h - x m h = h (b - x m)Thus:h = (x b + x c)/(b - x m)Similarly, from y:y D = b m h + b cBut D = b + m h + c, so:y (b + m h + c) = b m h + b cSubstitute h from above:h = (x b + x c)/(b - x m)So:y [ b + m ( (x b + x c)/(b - x m) ) + c ] = b m ( (x b + x c)/(b - x m) ) + b cThis looks complicated, but maybe we can substitute h into the y equation and see if we can eliminate h.Alternatively, maybe express y in terms of x.From x:h = (x (b + c)) / (b - x m )Wait, wait, x (b + c)? Wait, from x:h = (x b + x c)/(b - x m) = x (b + c)/(b - x m)So h = x (b + c)/(b - x m)Then, substitute h into the expression for y.From y:y = [b m h + b c]/D = [b m h + b c]/[b + m h + c]Substitute h:y = [b m (x (b + c)/(b - x m)) + b c] / [b + m (x (b + c)/(b - x m)) + c ]Multiply numerator and denominator by (b - x m) to eliminate denominators:Numerator: b m x (b + c) + b c (b - x m)Denominator: [b (b - x m) + m x (b + c) + c (b - x m)]Compute numerator:= b m x (b + c) + b c (b) - b c x m= b m x (b + c) + b² c - b c x mFactor terms:= b m x b + b m x c + b² c - b c x m= b² m x + b m x c + b² c - b c x mNotice that b m x c and - b c x m cancel each other:= b² m x + b² cFactor out b²:= b² (m x + c)Denominator:= b (b - x m) + m x (b + c) + c (b - x m)Expand each term:= b² - b x m + m x b + m x c + c b - c x mSimplify:= b² - b x m + b x m + m x c + b c - c x mAgain, -b x m and +b x m cancel:= b² + m x c + b c - c x m= b² + b cSo denominator = b² + b c = b (b + c)Therefore, y = [b² (m x + c)] / [b (b + c)] = [b (m x + c)] / (b + c)Thus, y = (b m x + b c)/(b + c)Which simplifies to:y = (b m / (b + c)) x + (b c)/(b + c)This is a linear equation in x and y, which is a straight line with slope (b m)/(b + c) and y-intercept (b c)/(b + c). Therefore, the locus of point M is the line y = (b m)/(b + c) x + (b c)/(b + c)Similarly, let's find the locus of point N.From earlier, the coordinates of N are:x_N = b (h + k)/(b + k)But since k = m h + c,x_N = b (h + m h + c)/(b + m h + c) = b ( (1 + m) h + c ) / (b + m h + c )Similarly, y_N = s = (b k)/(b + k) = (b (m h + c))/(b + m h + c )Wait, so y-coordinate of N is same as y-coordinate of M, which makes sense since both are at height s above AB.But let's check. Wait, in the square, points M and N are both at the top side of the square, so their y-coordinates should be equal. So y_M = y_N = s.But according to the previous calculations, both M and N have the same y-coordinate, which is (b k)/(b + k). So when we derived the locus for M, it turned out to be a straight line. Let's see if N's locus is another straight line.Let me compute x_N:x_N = b ( (1 + m) h + c ) / (b + m h + c )Again, express h in terms of x. But perhaps we can use similar steps as with M.Let me express x_N in terms of h:x_N = b [ (1 + m) h + c ] / (b + m h + c )Let me denote D = b + m h + c, same as before.Then:x_N = b [ (1 + m) h + c ] / DBut we can express h from the previous result: h = x (b + c)/(b - x m )Wait, no, that h was from the equation for x_M. Here, for x_N, we need another relation. Alternatively, since h is the same parameter (as C is moving, h changes), perhaps express x_N in terms of h and then relate to y.But y_N = (b k)/(b + k) = (b (m h + c))/(b + m h + c )Same as y_M. Therefore, for point N, since its y-coordinate is the same as that of M, and given that the locus of M is a straight line, maybe the locus of N is another straight line. Let me try to derive it.Express x_N:x_N = [ b ( (1 + m) h + c ) ] / D, where D = b + m h + cExpress h in terms of x_N and D.Alternatively, let me use the same approach as for M. Let me solve for h in terms of x_N.From x_N = [ b ( (1 + m) h + c ) ] / DMultiply both sides by D:x_N D = b ( (1 + m) h + c )But D = b + m h + c, so substitute:x_N (b + m h + c ) = b ( (1 + m) h + c )Expand left side: x_N b + x_N m h + x_N cRight side: b (1 + m) h + b cBring all terms to left side:x_N b + x_N m h + x_N c - b (1 + m) h - b c = 0Factor terms with h:h [ x_N m - b (1 + m) ] + (x_N b + x_N c - b c ) = 0Solve for h:h [ x_N m - b - b m ] = - (x_N b + x_N c - b c )h = [ - (x_N b + x_N c - b c ) ] / [ x_N m - b (1 + m ) ]= [ - x_N (b + c ) + b c ] / [ x_N m - b (1 + m ) ]= [ b c - x_N (b + c ) ] / [ x_N m - b - b m ]Now, from the expression for y_N:y_N = (b (m h + c )) / DBut D = b + m h + cSo y_N = [ b m h + b c ] / [ b + m h + c ]Let me substitute h from above into this expression.First, h = [ b c - x_N (b + c ) ] / [ x_N m - b (1 + m ) ]Let me denote numerator of h as Num = b c - x_N (b + c )Denominator of h as Den = x_N m - b (1 + m )So h = Num / DenThen,y_N = [ b m (Num / Den ) + b c ] / [ b + m (Num / Den ) + c ]Multiply numerator and denominator by Den to eliminate fractions:Numerator: [ b m Num + b c Den ] / DenDenominator: [ b Den + m Num + c Den ] / DenThus,y_N = [ b m Num + b c Den ] / [ b Den + m Num + c Den ]Substitute Num and Den:Num = b c - x_N (b + c )Den = x_N m - b (1 + m )Compute numerator:b m (b c - x_N (b + c )) + b c (x_N m - b (1 + m ))= b m b c - b m x_N (b + c ) + b c x_N m - b c b (1 + m )Simplify term by term:First term: b² m cSecond term: -b m x_N (b + c )Third term: + b c x_N mFourth term: -b² c (1 + m )Combine second and third terms:- b m x_N (b + c ) + b c x_N m = -b m x_N (b + c - c ) = -b m x_N b = -b² m x_NThus, numerator becomes:b² m c - b² m x_N - b² c (1 + m )Factor b²:= b² [ m c - m x_N - c (1 + m ) ]= b² [ m c - m x_N - c - c m ]Simplify inside the brackets:m c - c m cancels, leaving -m x_N - cThus, numerator = b² ( -m x_N - c )Denominator:b Den + m Num + c Den= b (x_N m - b (1 + m )) + m (b c - x_N (b + c )) + c (x_N m - b (1 + m ))Expand each term:First term: b x_N m - b² (1 + m )Second term: m b c - m x_N (b + c )Third term: c x_N m - c b (1 + m )Combine all terms:= b x_N m - b² - b² m + m b c - m x_N b - m x_N c + c x_N m - c b - c b mCombine like terms:Terms with x_N m:b x_N m - m x_N b - m x_N c + c x_N m= (b x_N m - b x_N m ) + ( -m x_N c + c x_N m ) = 0 + 0 = 0Terms with constants:- b² - b² m + m b c - c b - c b m= -b² (1 + m ) + m b c - c b (1 + m )Factor:= -b² (1 + m ) - c b (1 + m ) + m b c= -(1 + m )(b² + c b ) + m b cHmm, seems messy. Let me compute each term:- b² - b² m: common factor -b² (1 + m )+ m b c: + m b c- c b - c b m: -c b (1 + m )So altogether:- b² (1 + m ) - c b (1 + m ) + m b cFactor out -(1 + m ) from first two terms:= -(1 + m )(b² + c b ) + m b cThis is as simplified as it gets. Let me write the entire denominator:Denominator = -(1 + m )(b² + c b ) + m b c= - (1 + m ) b (b + c ) + m b c= -b (b + c ) - m b (b + c ) + m b c= -b (b + c ) - m b (b + c - c )= -b (b + c ) - m b (b )= -b (b + c ) - m b²= -b² - b c - m b²Factor:= -b² (1 + m ) - b cSo denominator = -b² (1 + m ) - b cThus, denominator = -b ( b (1 + m ) + c )Therefore, putting numerator and denominator together:y_N = [ -b² (m x_N + c ) ] / [ -b ( b (1 + m ) + c ) ]Simplify:The negatives cancel, and b² / b = b:y_N = [ b (m x_N + c ) ] / [ b (1 + m ) + c ]Cancel b:Wait, denominator is b (1 + m ) + c, numerator is b (m x_N + c )Wait, no, wait:Wait, numerator was b² (-m x_N - c ), denominator was -b ( b (1 + m ) + c )Thus,y_N = [ -b² (m x_N + c ) ] / [ -b ( b (1 + m ) + c ) ]The negatives cancel:= [ b² (m x_N + c ) ] / [ b ( b (1 + m ) + c ) ]Cancel one b:= [ b (m x_N + c ) ] / [ b (1 + m ) + c ]Therefore,y_N = [ b m x_N + b c ] / [ b (1 + m ) + c ]This is a linear equation in x_N and y_N. Let me write it as:y_N [ b (1 + m ) + c ] = b m x_N + b cBring all terms to left:y_N [ b (1 + m ) + c ] - b m x_N - b c = 0But this can be rearranged as:b m x_N - [ b (1 + m ) + c ] y_N + b c = 0But perhaps express y_N in terms of x_N:y_N = [ b m x_N + b c ] / [ b (1 + m ) + c ]Which is a linear equation, hence the locus of N is also a straight line.Therefore, both points M and N trace straight lines as C moves along the line y = m x + c. Hence, the locus is a pair of lines.Wait, but the problem statement says "a pair of lines", which would be the two lines traced by M and N. So, since both M and N move along their own respective straight lines, the locus is indeed a pair of lines. Therefore, this proves the statement.I should check if there are any special cases or if my approach missed something. For example, if the line along which C moves is parallel to AB, but the problem states it's not parallel, so m ≠ 0. Also, I assumed coordinates with A at (0,0) and B at (b,0). Since the problem doesn't specify coordinates, this should be general because we can always place AB on the x-axis through affine transformation, and the line of C is arbitrary not parallel to AB, so our coordinate choice is valid.Another check: When C approaches the line AB, the square becomes smaller. When C is far away, the square becomes larger. The equations we derived for the loci of M and N are linear, so regardless of the position of C, M and N move along these lines.Therefore, the conclusion is that the loci are two straight lines, one for M and one for N. Hence, the proof is complete.

✅:To prove that the locus of vertices ( M ) and ( N ) of the inscribed square ( MNKL ) in triangle ( ABC ) (as described) forms a pair of lines when vertex ( C ) moves along a line not parallel to ( AB ), we proceed as follows: Coordinate Setup:1. Fix Coordinates: Place ( A ) at ( (0,0) ) and ( B ) at ( (b,0) ). Let ( C ) move along the line ( y = mx + c ), where ( m neq 0 ) (since the line is not parallel to ( AB )).2. Parametrize ( C ): Let ( C ) have coordinates ( (h, mh + c) ). Square Inscription:3. Square Vertices: - ( L ) and ( K ) lie on ( AB ): ( L(t,0) ), ( K(t+s,0) ). - ( M ) and ( N ) lie on ( AC ) and ( BC ), respectively. The side length ( s ) is determined by the square's position. Deriving Coordinates of ( M ) and ( N ):4. Coordinates of ( M ): - ( M ) lies on ( AC ): ( M(lambda h, lambda(mh + c)) ). - From the square's geometry, ( lambda h = t ) and ( s = lambda(mh + c) ). Solving gives ( t = frac{bh}{b + mh + c} ) and ( s = frac{b(mh + c)}{b + mh + c} ). - Thus, ( M ) has coordinates ( left(frac{bh}{b + mh + c}, frac{b(mh + c)}{b + mh + c}right) ).5. Coordinates of ( N ): - ( N ) lies on ( BC ): Using similar substitutions, ( N ) has coordinates ( left(frac{b((1 + m)h + c)}{b + mh + c}, frac{b(mh + c)}{b + mh + c}right) ). Locus Analysis:6. Eliminate Parameter ( h ): - For ( M ), express ( h ) in terms of ( x ): ( h = frac{x(b + c)}{b - mx} ). Substitute into ( y )-coordinate to get ( y = frac{bmx + bc}{b + c} ), a linear equation. - For ( N ), similarly eliminate ( h ) to derive ( y = frac{bmx + bc}{b(1 + m) + c} ), another linear equation. Conclusion:7. Linear Loci: - The coordinates of ( M ) and ( N ) satisfy linear equations in ( x ) and ( y ). Hence, as ( C ) moves along ( y = mx + c ), the loci of ( M ) and ( N ) are straight lines.Final Answer: The locus of vertices ( M ) and ( N ) is indeed a pair of lines, as required. Therefore, (boxed{text{The loci of } M text{ and } N text{ are each a straight line.}}).

🔑:1. Let us consider the triangle ABC with vertices A, B, and C. A square MNKL is inscribed in the triangle such that the vertices L and K lie on side AB (or its extension), while the vertices M and N lie on sides AC and BC respectively. 2. Assume that vertex C of the triangle moves along a line that is not parallel to side AB. We need to prove that the geometric locus of vertices M and N forms a pair of straight lines. 3. First, observe the segments AC and BC divided by points M and N respectively. To facilitate understanding, denote the height of the triangle from vertex C to base AB as CH_{3}. When the vertex C moves, the height CH_{3} adjusts accordingly. 4. According to Menelaus' theorem and its converse, we can establish ratios on sides AC and BC divided by M and N. Specifically, the points M and N divide the segments AC and BC in a ratio proportional to the segments AB and CH_{3}:[frac{AM_i}{M_iC_i} = frac{AB}{C_iH_{3}}, quad frac{BN_i}{N_iC_i} = frac{AB}{C_iH_{3}}]5. Using the converse of Menelaus' theorem on the triangle ABC with the cutting transversals passing through M, M_i, and S:[frac{AM}{MC} cdot frac{CN}{NB} cdot frac{BS}{SA} = 1 ]This implies points M, M_i, and S are collinear.6. Similarly, for points N, N_i, and S:[frac{BM}{MC} cdot frac{CN}{NA} cdot frac{AS}{SB} = 1 ]So, points N, N_i, and S are collinear. 7. Given these collinearities, we infer that the geometric loci of points M and N are the straight lines MS and NS respectively.8. Therefore, we conclude that as the vertex C of the triangle ABC moves along a line not parallel to side AB, the geometric loci of vertices M and N of the inscribed square form two straight lines.[boxed{M S text{ and } N S}]

🧠:Okay, let's tackle this problem step by step. So, the question is about Anton's performance in a school test with 5 sections, each having the same number of questions. Anton got 20 correct answers, and his correct percentage is between 60% and 70%. We need to find the total number of questions in the test.First, let me parse the problem again. There are 5 sections, each with the same number of questions. So, if each section has 'n' questions, the total number of questions in the test would be 5n. Anton answered 20 correctly. His percentage correct is more than 60% but less than 70%. So, 60% < (20 / total questions) * 100 < 70%. Wait, actually, hold on. If the percentage is the percentage of correct answers out of total questions, then it's (20 / total) * 100. But the problem says his correct answer percentage is more than 60% but less than 70%. Wait, but that would mean 60% < (20 / total) * 100 < 70%. So, solving for total would give the possible total number of questions. But since the total must be 5n, where n is the number of questions per section, which is an integer. So, we need to find 5n such that 60 < (20 / 5n) * 100 < 70. Hmm.Wait, let me write that down properly. Let total questions be T = 5n. Then, Anton's percentage is (20 / T) * 100. The percentage is between 60% and 70%, so:60 < (20 / T) * 100 < 70.Let me solve this inequality for T.First, divide all parts by 100 to get:0.6 < 20 / T < 0.7Then, take reciprocals, remembering to reverse the inequalities when we do that:1/0.7 < T/20 < 1/0.6Calculating 1/0.7 ≈ 1.42857 and 1/0.6 ≈ 1.66667.So,1.42857 < T/20 < 1.66667Multiply all parts by 20:1.42857 * 20 < T < 1.66667 * 20Calculating:1.42857 * 20 ≈ 28.57141.66667 * 20 ≈ 33.3333So, T must be between approximately 28.5714 and 33.3333. Since T is the total number of questions, which must be a multiple of 5 (because there are 5 sections with equal number of questions), T can be 30 or 35. Wait, 30 is within the range, but 35 would be 35, which is higher than 33.3333. Wait, so 30 is 30, which is between 28.57 and 33.33. But 35 is outside the upper bound. So, T must be 30? Wait, but let's check.Wait, 5 sections, each with n questions. So possible T values are 5n, n being integer. So possible T values are 5,10,15,20,25,30,35,... But from the inequality, T is between roughly 28.57 and 33.33. So the only multiple of 5 in that interval is 30. Therefore, T=30. Therefore, total questions is 30. Let's check that. If T=30, then Anton's percentage is (20/30)*100 ≈ 66.666...%, which is 66.67%, which is between 60% and 70%, so that works. If T were 25, then 20/25=80%, which is over 70%, so no. If T=35, 20/35≈57.14%, which is below 60%. So only 30 fits. So the answer is 30.Wait, but hold on. Let me verify again. The problem says "more than 60% but less than 70%". So 66.666...% is exactly two-thirds, which is between 60 and 70. So that works. Therefore, total questions is 30. So each section has 6 questions (since 30 divided by 5 is 6). So Anton got 20 out of 30 right, which is 66.666...%, which is correct.But let me just think again—is there any other possible T? Because sometimes, depending on rounding, maybe another multiple of 5 is possible? Let's see. For example, if T=35, as I thought earlier, 20/35≈57.14%, which is below 60%. If T=25, 20/25=80%, which is above 70%. So no, the only possible T is 30. Therefore, the answer should be 30.Wait, but let me check another way. Let's use the inequalities strictly. The percentage is more than 60% and less than 70%. So:60 < (20 / T) * 100 < 70Divide all parts by 100:0.6 < 20 / T < 0.7Multiply all parts by T (assuming T positive, which it is):0.6T < 20 < 0.7TSo, 0.6T < 20 and 20 < 0.7TFirst inequality: 0.6T < 20 ⇒ T < 20 / 0.6 ≈ 33.333...Second inequality: 20 < 0.7T ⇒ T > 20 / 0.7 ≈ 28.571...So T must be between approximately 28.571 and 33.333. Since T is a multiple of 5, the only integer T in that range is 30. Therefore, total questions is 30. So each section has 6 questions. That seems to check out.So, I think that's the answer. Let me just confirm once more. If there are 5 sections, each with 6 questions, total is 5*6=30. Anton answered 20. 20/30 is 2/3 ≈66.67%, which is between 60% and 70%. So yes, that's correct.I don't think there are any other possibilities here. The key was realizing that the total number of questions must be a multiple of 5, and then finding the multiple that falls within the calculated range. 30 is the only one that works. So the answer is 30.

✅:The test has a total of 30 questions. Here's the breakdown:1. Define Variables: Let each section have ( n ) questions. Since there are 5 sections, the total number of questions is ( T = 5n ).2. Set Up Inequality: Anton answered 20 questions correctly, and his percentage is between 60% and 70%: [ 60 < frac{20}{T} times 100 < 70 ]3. Solve for ( T ): - Convert percentages to decimals: [ 0.6 < frac{20}{T} < 0.7 ] - Take reciprocals (reversing inequalities): [ frac{1}{0.7} < frac{T}{20} < frac{1}{0.6} ] - Calculate bounds: [ 28.57 < T < 33.33 ]4. Determine Valid ( T ): - ( T ) must be a multiple of 5 (since ( T = 5n )). - The only multiple of 5 in the range ( 28.57 < T < 33.33 ) is ( boxed{30} ).5. Verification: - For ( T = 30 ), the percentage is ( frac{20}{30} times 100 approx 66.67% ), which satisfies ( 60% < 66.67% < 70% ).Answer: There were 30 questions in total.

🔑:Given:- There are 5 sections in the school test.- Each section contains an equal number of questions.- Anton answered 20 questions correctly.- The percentage of his correct answers is more than 60% but less than 70%.We need to determine the total number of questions in the test, denoted as ( x ).1. Set Up the Inequality: Let's translate the information about the percentage into a mathematical inequality. [ 60% < frac{20}{x} < 70% ] Converting percentages to fractions: [ frac{60}{100} < frac{20}{x} < frac{70}{100} ] Simplifying these fractions: [ 0.6 < frac{20}{x} < 0.7 ] 2. Solve for ( x ): To isolate ( x ), we can solve the parts of the inequality separately. For the lower bound: [ 0.6 < frac{20}{x} Rightarrow 0.6x < 20 Rightarrow x > frac{20}{0.6} ] Calculating ( frac{20}{0.6} ): [ frac{20}{0.6} = frac{20 times 10}{0.6 times 10} = frac{200}{6} = frac{100}{3} approx 33.33 ] Therefore, the lower bound is: [ x > 33.33 ] For the upper bound: [ frac{20}{x} < 0.7 Rightarrow 20 < 0.7x Rightarrow x < frac{20}{0.7} ] Calculating ( frac{20}{0.7} ): [ frac{20}{0.7} = frac{20 times 10}{0.7 times 10} = frac{200}{7} approx 28.57 ] Therefore, the upper bound is: [ x < 28.57 ]3. Combine the Results and Find Possible Values for ( x ): Combining the above results, we get: [ 28.57 < x < 33.33 ] 4. Determine ( x ) Considering the Number of Sections: Given that there are 5 sections and each section must have an equal number of questions, ( x ) must be a multiple of 5. Therefore, the possible value of ( x ) within this range is: [ x = 30 ] Conclusion:The total number of questions in the test is:[boxed{30}]

🧠:Okay, let's see. I need to find the expected value of the product of two independent random variables X and Y. The given expected values are E(X) = 4 and E(Y) = 5. Hmm, right. So, the question is asking for E(XY). First, I remember that for independent random variables, there's a property related to the expectation of their product. Wait, isn't it that if two variables are independent, the expected value of their product is equal to the product of their expected values? So, E(XY) = E(X) * E(Y) if X and Y are independent. Is that correct?Let me double-check. I know that for independent variables, the covariance is zero, which means that Cov(X,Y) = E(XY) - E(X)E(Y) = 0. Therefore, rearranging that gives E(XY) = E(X)E(Y). Yeah, that seems right. So, if X and Y are independent, their covariance is zero, which implies the expectation of the product is the product of expectations.But wait, is independence a stronger condition than just having zero covariance? Because I recall that independence implies zero covariance, but zero covariance doesn't necessarily imply independence. However, in this problem, they do state that X and Y are independent, so we can safely use that property.So, applying the formula here, since X and Y are independent, then E(XY) = E(X) * E(Y) = 4 * 5 = 20. That seems straightforward. But let me make sure there's no catch here. Wait, maybe I should verify this with an example. Suppose I take two simple independent random variables and compute their expectations. Let's say X can be 3 or 5 each with probability 0.5, so E(X) = (3 + 5)/2 = 4. Let Y be 4 or 6 each with probability 0.5, so E(Y) = (4 + 6)/2 = 5. Now, since they are independent, the joint probabilities would be the product of individual probabilities. So, each combination (3,4), (3,6), (5,4), (5,6) has a probability of 0.25. Calculating E(XY): (3*4 + 3*6 + 5*4 + 5*6)/4 = (12 + 18 + 20 + 30)/4 = (80)/4 = 20. Which is indeed 4*5. So that example checks out. Alternatively, if X and Y were not independent, this might not hold. For example, suppose X and Y are perfectly correlated. Let's take X as before: 3 or 5 with probability 0.5 each. Let Y be 4 when X=3 and 6 when X=5. Then E(Y) would still be (4 + 6)/2 = 5. But E(XY) would be (3*4 + 5*6)/2 = (12 + 30)/2 = 42/2 = 21, which is different from 20. So, in that case, even though E(X) = 4 and E(Y) = 5, the expectation of the product is 21 because they are dependent. But in our problem, since X and Y are independent, the covariance is zero, so the formula holds. Therefore, the answer should be 20. Let me check if there's another approach. Maybe using the definition of expectation. The expected value of XY is the double integral over all x and y of xy times the joint probability density function f(x,y). If X and Y are independent, the joint density is the product of the marginal densities: f(x,y) = f_X(x)f_Y(y). Therefore, E(XY) = ∫∫ xy f_X(x)f_Y(y) dx dy = (∫x f_X(x) dx)(∫y f_Y(y) dy) = E(X)E(Y). So, this mathematical derivation also confirms that when variables are independent, the expectation of the product is the product of expectations. Therefore, even from the definition, the result holds. So, regardless of the specific distributions of X and Y, as long as they are independent, the expected value of their product is simply 4 * 5 = 20. I think that's solid. No other factors are mentioned in the problem, like variances or specific distributions, so we don't need to consider anything else. The key point here is the independence of the variables. As long as that's given, the multiplicative property applies. So, final answer should be 20.

✅:The expected value of the product of two independent random variables ( X ) and ( Y ) is the product of their individual expected values. Given ( E(X) = 4 ) and ( E(Y) = 5 ), we use the property of independence:[E(XY) = E(X) cdot E(Y) = 4 cdot 5 = 20]Final Answer: boxed{20}

🔑:Given:- M(X) = 4- M(Y) = 5We need to find the mathematical expectation M(X cdot Y) of the product of two independent random variables X and Y.1. Utilize the property of expectation for independent random variables: When X and Y are independent, the expectation of their product is the product of their expectations. This can be written as: [ M(X cdot Y) = M(X) cdot M(Y) ] This property is given by the equation (2.4.15). 2. Substitute the given values into the formula: [ M(X cdot Y) = M(X) cdot M(Y) ] Here, we substitute M(X) = 4 and M(Y) = 5: [ M(X cdot Y) = 4 cdot 5 ]3. Perform the multiplication: [ M(X cdot Y) = 20 ]4. Conclusion:Therefore, the mathematical expectation of the product of X and Y is:[boxed{20}]