🧠:Alright, let's tackle this problem step by step. We need to find all integer pairs (x, y) that satisfy the following system of inequalities:1. (x^3 + y^2 - 3y + 1 < 0)2. (3x^3 - y^2 + 3y > 0)And then find the product of all such y values. Hmm, okay. Let me start by understanding each inequality and see how they interact.First, let's rewrite both inequalities to see if I can find a relationship between them. Maybe combining them would help.The first inequality is (x^3 + y^2 - 3y + 1 < 0). Let's denote this as Inequality (1).The second inequality is (3x^3 - y^2 + 3y > 0). Let's call this Inequality (2).If I add Inequality (1) and Inequality (2) together, maybe some terms will cancel out. Let's try that:Adding left sides: (x^3 + y^2 - 3y + 1 + 3x^3 - y^2 + 3y = 4x^3 + 1)Adding right sides: (0 + 0 = 0)So, combining them gives (4x^3 + 1 < 0 + 0)? Wait, no. Wait, the first inequality is less than 0, and the second is greater than 0. Adding them would result in:( (x^3 + y^2 - 3y + 1) + (3x^3 - y^2 + 3y) < 0 + 0 )Wait, no. Actually, the sum of the left sides is less than 0 (from the first inequality) plus greater than 0 (from the second inequality). But we can't directly add inequalities like that. Wait, actually, if we have two inequalities:Inequality 1: A < BInequality 2: C > DThen, A + C < B + D only if we know the relationships between A, B, C, D. But here, adding them might not be straightforward. Let me think again.Alternatively, maybe we can manipulate the inequalities to express them in terms of x^3 or y terms. Let me see.From Inequality (1): (x^3 < -y^2 + 3y - 1)From Inequality (2): (3x^3 > y^2 - 3y)So, if we substitute x^3 from Inequality (1) into Inequality (2):From Inequality (1), (x^3 < -y^2 + 3y - 1). Multiply both sides by 3 (since 3 is positive, the inequality direction remains):(3x^3 < -3y^2 + 9y - 3)But from Inequality (2), (3x^3 > y^2 - 3y). So combining these two:(y^2 - 3y < 3x^3 < -3y^2 + 9y - 3)Therefore, (y^2 - 3y < -3y^2 + 9y - 3)Let's solve this inequality:(y^2 - 3y < -3y^2 + 9y - 3)Bring all terms to the left side:(y^2 - 3y + 3y^2 - 9y + 3 < 0)Combine like terms:(4y^2 - 12y + 3 < 0)Now, this is a quadratic in y. Let's find its roots to determine where it's negative.Quadratic equation: (4y^2 -12y +3 = 0)Using quadratic formula: (y = [12 pm sqrt{144 - 48}]/8 = [12 pm sqrt{96}]/8 = [12 pm 4sqrt{6}]/8 = [3 pm sqrt{6}]/2)Approximate values of sqrt(6) ≈ 2.45, so roots are approximately:(3 + 2.45)/2 ≈ 5.45/2 ≈ 2.725and(3 - 2.45)/2 ≈ 0.55/2 ≈ 0.275So, the quadratic (4y^2 -12y +3) is a parabola opening upwards, so it's negative between its roots. Therefore, the inequality (4y^2 -12y +3 < 0) holds when y is between approximately 0.275 and 2.725. Since y is an integer, the possible integer values for y are 1 and 2.So y can be 1 or 2. Let's note that down: possible y values are 1 and 2.Now, we can check each y value (1 and 2) and find the corresponding x that satisfies both inequalities.First, let's check y = 1.Substitute y = 1 into Inequality (1):(x^3 + (1)^2 - 3(1) + 1 < 0)Simplify:(x^3 + 1 - 3 + 1 < 0)Which is:(x^3 -1 < 0) => (x^3 < 1)Since x is an integer, x^3 < 1 implies x <= 0 (since x=1 would be 1, which is not less than 1). So x <= 0.Now, check Inequality (2) with y = 1:(3x^3 - (1)^2 + 3(1) > 0)Simplify:(3x^3 -1 +3 > 0) => (3x^3 +2 > 0) => (3x^3 > -2)Divide both sides by 3 (positive, so inequality remains):(x^3 > -2/3)Since x is an integer, x^3 must be greater than approximately -0.666. So possible x values are x >= -0 since x is integer. Wait, x^3 > -2/3. Let's see:For integer x:If x = -1: (-1)^3 = -1 < -2/3? Yes, since -1 < -0.666. So x = -1 would give x^3 = -1 which is less than -2/3. So x^3 > -2/3 implies x >= 0.Wait, x is integer. Let's check:x = -1: x^3 = -1, which is less than -2/3, so 3x^3 + 2 = -3 + 2 = -1 > 0? No, -1 is not greater than 0. So x must be >= 0.But from Inequality (1), x <= 0. So combining these, x must be 0.So for y = 1, x must be 0.Check x = 0, y =1 in both inequalities:Inequality (1): 0 + 1 - 3 + 1 = -1 < 0: True.Inequality (2): 0 -1 +3 = 2 > 0: True.So (0, 1) is a solution.Now check y = 2.Substitute y =2 into Inequality (1):(x^3 + (2)^2 -3(2) +1 < 0)Simplify:(x^3 +4 -6 +1 <0) => (x^3 -1 <0) => (x^3 <1)Same as before: x <=0.Now check Inequality (2) for y=2:(3x^3 - (2)^2 +3(2) >0)Simplify:(3x^3 -4 +6 >0) => (3x^3 +2 >0)Same as when y=1: (3x^3 > -2), which implies x >=0 (since x is integer). So again, x must be 0.Check x=0, y=2:Inequality (1): 0 +4 -6 +1 = -1 <0: True.Inequality (2): 0 -4 +6 = 2 >0: True.So (0,2) is another solution.Wait, so both y=1 and y=2 have x=0. Are there other possible y values? Earlier, we concluded y must be 1 or 2 because the quadratic inequality 4y² -12y +3 <0 gives y between approx 0.275 and 2.725, so integer y=1,2. But let me confirm this because sometimes approximations can be misleading.Wait, let me compute the exact roots again. The quadratic was 4y² -12y +3. The discriminant is 144 - 48 = 96. So sqrt(96) is 4*sqrt(6), so the roots are (12 ±4sqrt(6))/8 = (3 ±sqrt(6))/2. The exact values are (3 - sqrt(6))/2 ≈ (3 -2.449)/2 ≈0.551/2≈0.2755 and (3 + sqrt(6))/2≈(3 +2.449)/2≈5.449/2≈2.7245. So indeed, between 0.2755 and 2.7245. So the integer y's are 1 and 2. So no other y's possible.Therefore, only y=1 and y=2. And for each, x must be 0.But let me just make sure there are no other possible y's if perhaps the quadratic inequality wasn't correctly derived. Let's retrace.We had from combining the two inequalities:From Inequality (1): x³ < -y² +3y -1From Inequality (2): 3x³ > y² -3ySo substituting x³ from (1) into (2):3x³ < 3*(-y² +3y -1) = -3y² +9y -3But from (2): 3x³ > y² -3yHence, y² -3y < 3x³ < -3y² +9y -3Therefore, y² -3y < -3y² +9y -3Bring all terms to left:y² -3y +3y² -9y +3 <0 =>4y² -12y +3 <0Yes, that seems correct. So the quadratic inequality is correct.Therefore, y must be 1 or 2.But just to be thorough, what if y is outside this range? Let's check y=0 and y=3, just in case.First, y=0:Substitute into Inequality (1): x³ +0 -0 +1 <0 =>x³ +1 <0 =>x³ < -1 =>x <= -2 (since x is integer: x=-2 gives (-2)^3=-8 < -1, x=-1 gives -1 < -1? No, -1 is not less than -1. So x <= -2.Now check Inequality (2) with y=0:3x³ -0 +0 >0 =>3x³ >0 =>x³ >0 =>x >=1But from Inequality (1), x <= -2. Contradiction. So no solutions for y=0.Similarly, y=3:Inequality (1): x³ +9 -9 +1 <0 =>x³ +1 <0 =>x <= -2Inequality (2):3x³ -9 +9 >0 =>3x³ >0 =>x >=1Again, contradiction. So no solutions.So y can only be 1 or 2. For these, x must be 0. Therefore, the pairs are (0,1) and (0,2). The product of all y's is 1*2=2.Wait, but hold on. Let me check x=0 for y=1 and y=2. We did that, but maybe we can check other x values? Wait, for y=1, x must be 0. For y=2, x must be 0. Let me confirm.Wait, for y=1:From Inequality (1): x³ <1. So x <=0.From Inequality (2):3x³ > -2. So x³ > -2/3. So x >=0.Thus, x=0. Similarly for y=2: same result. So x must be 0. So no other x values possible.Therefore, the only integer solutions are (0,1) and (0,2). The product of y values is 1*2=2. So the answer should be 2.But just to ensure there are no other possible x or y, let's consider that maybe x is positive or negative.Wait, for example, if y=1, x must be 0. What if x is 1? Let's test x=1, y=1:Inequality (1):1 +1 -3 +1 =0 <0? No, 0 is not less than 0. So invalid.x=-1, y=1:Inequality (1):-1 +1 -3 +1 =-2 <0: True.Inequality (2):3*(-1) -1 +3 =-3 -1 +3=-1 >0? No. So x=-1 doesn't work.Similarly, x=0 is the only possible x for y=1 and y=2.Similarly, for y=2, x=0. Any other x?x=1, y=2:Inequality (1):1 +4 -6 +1=0 <0? No.x=-1, y=2:Inequality (1):-1 +4 -6 +1=-2 <0: True.Inequality (2):3*(-1) -4 +6=-3 -4 +6=-1 >0? No.So again, only x=0 works for both y=1 and y=2.Therefore, the only pairs are (0,1) and (0,2). The product of the y's is 1*2=2.Final Answerboxed{2}

✅:To find all pairs of integers ((x, y)) that satisfy the system of inequalities:[begin{cases}x^3 + y^2 - 3y + 1 < 0 3x^3 - y^2 + 3y > 0end{cases}]we start by combining the inequalities. From the first inequality, we have (x^3 < -y^2 + 3y - 1). From the second inequality, we have (3x^3 > y^2 - 3y). Substituting (x^3) from the first inequality into the second, we get:[y^2 - 3y < 3x^3 < -3y^2 + 9y - 3]This simplifies to:[y^2 - 3y < -3y^2 + 9y - 3]Solving this quadratic inequality:[4y^2 - 12y + 3 < 0]The roots of the quadratic equation (4y^2 - 12y + 3 = 0) are approximately (0.275) and (2.725). Since (y) must be an integer, the possible values are (y = 1) and (y = 2).For (y = 1):- The first inequality becomes (x^3 - 1 < 0 implies x leq 0).- The second inequality becomes (3x^3 + 2 > 0 implies x geq 0).Thus, (x = 0) is the only solution, giving the pair ((0, 1)).For (y = 2):- The first inequality becomes (x^3 - 1 < 0 implies x leq 0).- The second inequality becomes (3x^3 + 2 > 0 implies x geq 0).Thus, (x = 0) is the only solution, giving the pair ((0, 2)).The product of all (y) values for the pairs ((0, 1)) and ((0, 2)) is (1 times 2 = 2).[boxed{2}]

🔑:1. Given Problem: We are asked to find all pairs of integers (x, y) that satisfy the following system of inequalities: [ begin{cases} x^3 + y^2 - 3y + 1 < 0 3x^3 - y^2 + 3y > 0 end{cases} ]2. Step 1: Manipulating the Inequalities: To make it easier to solve the inequalities, we can multiply the first inequality by -3. This gives us: [ -3(x^3 + y^2 - 3y + 1) < 0 ] Simplifying, we get: [ -3x^3 - 3y^2 + 9y - 3 < 0 quad text{or} quad 3x^3 + 3y^2 - 9y + 3 > 0 ]3. Step 2: Adding the Inequalities: Adding the modified first inequality to the second inequality, we have: [ (3x^3 + 3y^2 - 9y + 3) + (3x^3 - y^2 + 3y) > 0 + 0 ] Combining like terms, we obtain: [ 6x^3 + 2y^2 - 6y + 3 > 0 ] Simplifying further by dividing everything by 3: [ 2x^3 + frac{2}{3}y^2 - 2y + 1 > 0 ]4. Step 3: Solving for Specific Values: Let's evaluate the specific values of y to see which pairs satisfy both inequalities. From the inequalities: [ x^3 + y^2 - 3y + 1 < 0 ] and [ 3x^3 - y^2 + 3y > 0 ] Let us test for integer values of y = 1 and y = 2: - For y = 1: [ x^3 + 1 - 3 cdot 1 + 1 < 0 quad Rightarrow quad x^3 - 1 < 0 quad Rightarrow quad x < 1 ] And [ 3x^3 - 1 + 3 cdot 1 > 0 quad Rightarrow quad 3x^3 + 2 > 0 quad Rightarrow quad x^3 > -frac{2}{3} quad Rightarrow quad x text{ is any integer (e.g., }0text{)} ] Trying x = 0: [ 0^3 + 1^2 - 3 cdot 1 + 1 = 0 < 0 text{ (true)} ] [ 3 cdot 0^3 - 1^2 + 3 cdot 1 = 2 > 0 text{ (true)} ] Thus, (0, 1) is a valid pair. - For y = 2: [ x^3 + 2^2 - 3 cdot 2 + 1 < 0 quad Rightarrow quad x^3 + 4 - 6 + 1 < 0 quad Rightarrow quad x^3 - 1 < 0 quad Rightarrow quad x < 1 ] And [ 3x^3 - 4 + 6 > 0 quad Rightarrow quad 3x^3 + 2 > 0 quad Rightarrow quad x^3 > -frac{2}{3} quad Rightarrow quad x text{ is any integer (e.g., }0text{)} ] Trying x = 0: [ 0^3 + 2^2 - 3 cdot 2 + 1 = -1 < 0 text{ (true)} ] [ 3 cdot 0^3 - 2^2 + 3 cdot 2 = 2 > 0 text{ (true)} ] Thus, (0, 2) is a valid pair.5. Conclusion: The integer pairs (x, y) that satisfy both inequalities are (0, 1) and (0, 2). Therefore, the product of all such y values is: [ 1 times 2 = 2 ][boxed{2}]

🧠:Okay, so I need to solve this math problem: 8 multiplied by 7 divided by 8 multiplied by 7. Let me write that down again to make sure I have it right. The expression is 8 × 7 ÷ 8 × 7. Hmm, order of operations is key here. Let me recall PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Since there are no parentheses or exponents here, I just need to handle the multiplication and division in order from left to right.Starting from the left, the first operation is 8 × 7. Let me compute that. 8 times 7 is 56. So now the expression becomes 56 ÷ 8 × 7. Next, moving to the right, the next operation is division: 56 divided by 8. That's 7. Now the expression is 7 × 7. Multiplying those gives 49. So the result is 49? Wait, but I want to check if I did that correctly because sometimes when there are multiple operations, people might misinterpret the order.Let me verify step by step again. Original expression: 8 × 7 ÷ 8 × 7. Left to right: First, 8 × 7 = 56. Then, 56 ÷ 8 = 7. Then, 7 × 7 = 49. Yeah, that seems right. But sometimes people might group the operations differently, especially if they think multiplication should be done before division, but actually in PEMDAS, multiplication and division are of equal precedence and are evaluated left to right. Similarly with addition and subtraction. So it's crucial to go left to right here.Alternatively, someone might misinterpret it as 8 × (7 ÷ 8) × 7. Let me compute that way to see if it's different. First, 7 ÷ 8 is 0.875. Then 8 × 0.875 is 7, and then 7 × 7 is 49. Wait, that's the same result. Huh. Interesting. So even if you do the division first here, you still end up with 49. But that might not always be the case. Let me test with different numbers to see if the order matters. Suppose I have 4 ÷ 2 × 3. Left to right: 4 ÷ 2 = 2, then 2 × 3 = 6. If someone does 4 ÷ (2 × 3) = 4 ÷ 6 ≈ 0.666. So in that case, the order changes the result. But in the original problem, maybe the operations are arranged such that it doesn't matter? Let me check.Wait, in the original problem, 8 × 7 ÷ 8 × 7, since the 8 and 7 are both multiplied and divided in a way that they might cancel out. Let me try another approach. Let's write it as fractions. So 8 × 7 divided by 8 × 7. So that's (8 × 7) / (8 × 7) if grouped that way, which would be 1. But that's not the case here. Wait, but the original expression is 8 × 7 ÷ 8 × 7. Without parentheses, the division is only by 8, not by (8 × 7). So it's 8 × 7 ÷ 8 × 7 = (8 × 7) ÷ 8 × 7. Which simplifies to (7) × 7 = 49. Because the 8 cancels out. Alternatively, if you rearrange the multiplication and division: 8 ÷ 8 × 7 × 7. Which is (8 ÷ 8) × 7 × 7 = 1 × 7 × 7 = 49. So regardless of the order, as long as you follow left to right for multiplication and division, you get the same result here. But that's only because the 8 cancels out. If the numbers were different, the result might change based on order. For example, 6 × 4 ÷ 2 × 3. Left to right: 6 ×4=24, 24÷2=12, 12×3=36. If you group differently: 6 × (4 ÷ 2) ×3=6×2×3=36. So same result. Wait, but in my previous example, 4 ÷ 2 × 3 vs 4 ÷ (2 ×3), that was different. So maybe when there's division followed by multiplication, the way you group matters. Let me check that again.If it's written as 4 ÷ 2 × 3, according to left to right, it's 4 ÷ 2 = 2, then 2 ×3=6. If someone does 4 ÷ (2 ×3), that's 4 ÷6=0.666. So the difference comes when you have division followed by multiplication. In the original problem, though, since the 8 and 8 cancel each other, and the 7s multiply, it's the same either way. Wait, but in the original problem, after canceling 8s, you have 7×7. But if we follow left to right, we have (8×7)/8×7. Let's compute that step by step. 8×7 is 56. Then divided by 8 is 7. Then times 7 is 49. Alternatively, if you think of it as (8/8) ×7×7, that's 1×49=49. So both ways, same result. Therefore, in this specific problem, even if someone groups differently, they end up with the same answer. But in other problems, changing the order might give different results, so it's important to follow left to right unless there are parentheses.But just to make sure, let's check with the actual expression. Let me parse the original expression: 8 × 7 ÷ 8 × 7. According to order of operations, since multiplication and division have the same precedence, we do them left to right. So first, 8 multiplied by 7: 56. Then, 56 divided by 8: 7. Then, 7 multiplied by 7: 49. So the answer is 49. Alternatively, if someone writes all as fractions: 8×7 divided by 8×7. But that could be interpreted as (8×7)/(8×7) which is 1, but that's incorrect because the original expression isn't written with parentheses. So the correct interpretation is ((8×7)/8)×7. So first, 8×7=56, divided by8=7, then multiplied by7=49. So yes, 49 is correct.Another way to see it is that the two 8s cancel each other: 8 ÷8=1, so the expression becomes 1×7×7=49. So that's another way to look at it, by rearranging the operations (since multiplication is commutative and associative). Wait, but division is not associative. However, in this case, since we have 8×7÷8×7, we can rearrange the multiplication and division steps. Let's see: 8 ÷8=1, then 1×7×7=49. But technically, according to strict left to right, you have to do 8×7 first. But in this case, rearranging gives the same result. So maybe because the operations are multiplication and division of the same numbers, the result is the same regardless of the order. But again, this isn't always the case. For example, 6 ÷3×2. Left to right: 6 ÷3=2, 2×2=4. If you rearrange: 6×2÷3=12÷3=4. Same result. Hmm. But in my previous example, 4 ÷2×3 vs 4 ÷(2×3), which are different. Wait, so maybe when you have division followed by multiplication, rearranging the order can sometimes preserve the result and sometimes not. It depends on the numbers. So in cases where the numbers can be factored out or canceled, rearranging might work. But as a general rule, to avoid confusion, we should follow left to right.But back to the original problem. It seems that both approaches give 49. Therefore, the answer is 49. I think that's solid. Let me just confirm with a calculator. 8×7=56, 56÷8=7, 7×7=49. Yes. So the answer is indeed 49.

✅:The expression to evaluate is (8 times 7 div 8 times 7). Following the order of operations (PEMDAS/BODMAS), multiplication and division are performed from left to right:1. First operation: (8 times 7 = 56). The expression becomes: (56 div 8 times 7).2. Second operation: (56 div 8 = 7). The expression simplifies to: (7 times 7).3. Final operation: (7 times 7 = 49).Alternative perspective: The (8) in the numerator and denominator cancels out: [frac{8 times 7 times 7}{8} = 7 times 7 = 49.]Final Answer: (boxed{49})

🔑:1. Evaluate the expression step by step following the left-to-right order:[ 8 times 7 div 8 times 7 ]2. First operation: Multiply 8 by 7:[ 8 times 7 = 56 ]3. Next operation: Divide 56 by 8:[ 56 div 8 = 7 ]4. Final operation: Multiply the result by 7:[ 7 times 7 = 49 ] Conclusion:The final answer is:[ boxed{49} ]

🧠:Okay, let me try to figure out how to find the height of this isosceles trapezoid. The problem says the diagonal is 10 cm and the area is 48 cm². Hmm. Alright, let's start by recalling some properties of trapezoids.First, an isosceles trapezoid has the legs (the non-parallel sides) equal in length, and the base angles are equal. The area of a trapezoid is given by the formula: Area = (1/2) × (sum of the two bases) × height. So, if I can find the sum of the two bases, maybe I can solve for the height. But wait, the problem doesn't give me the lengths of the bases. Instead, it gives me the diagonal. Hmm, how does the diagonal relate to the other parts of the trapezoid?Let me draw a mental picture. An isosceles trapezoid has two parallel sides, called the bases, and the other two sides are equal in length. The diagonal connects two non-adjacent vertices. If I draw one of the diagonals, it divides the trapezoid into two triangles. Since the trapezoid is isosceles, these two triangles should be congruent if the diagonal is drawn between the two upper and lower bases. Wait, actually, in an isosceles trapezoid, the diagonals are equal in length, so both diagonals are 10 cm each.Now, the area is 48 cm², so using the area formula: 48 = (1/2) × (a + b) × h, where a and b are the lengths of the two bases, and h is the height. So, if I can express a + b in terms of h, or find another relation involving a, b, and h using the diagonal length, I can solve for h.Let me think. The diagonal forms a triangle with the two bases. Let's say the trapezoid has bases of length a (top) and b (bottom), and the legs are each of length l. The height is h. When I draw the diagonal from the top left corner to the bottom right corner, this diagonal splits the trapezoid into two triangles: one is the triangle on the left, and the other is the triangle on the right. Wait, actually, in an isosceles trapezoid, the two triangles formed by a diagonal are congruent if the trapezoid is symmetrical. But maybe that's not the case here. Let me clarify.Alternatively, maybe I can use the Pythagorean theorem on one of the triangles formed by the diagonal. Let's take the diagonal of 10 cm. If I consider the triangle formed by the diagonal, the leg of the trapezoid, and the difference between the bases. Wait, no. The diagonal connects two vertices, so in the trapezoid, if we take the diagonal from the upper left to the lower right, then this diagonal, along with the leg and the lower base, forms a triangle. Wait, maybe not. Let me try to visualize this.Actually, the diagonal divides the trapezoid into two triangles. Each triangle has the diagonal as one side. Let's consider one of these triangles. The triangle will have sides of length 10 cm (the diagonal), the leg of the trapezoid (which we can call l), and the upper base a. Wait, no. Wait, actually, if the diagonal is from the top left to the bottom right, then the triangle formed will have sides: diagonal (10 cm), the leg (l), and the lower base (b). But I need to check this.Wait, maybe it's better to assign coordinates to the trapezoid to make it easier. Let's place the trapezoid on a coordinate system. Let me set the lower base of length b on the x-axis from (0, 0) to (b, 0). Then, the upper base of length a is parallel to the lower base and is at height h, so its coordinates would be from ((b - a)/2, h) to ((b + a)/2, h). This way, the trapezoid is symmetric about the vertical line through the midpoint of the bases.Now, the legs are the sides connecting (0, 0) to ((b - a)/2, h) and (b, 0) to ((b + a)/2, h). The length of each leg can be found using the distance formula: sqrt[ ((b - a)/2 - 0 )² + (h - 0)² ] = sqrt[ ((b - a)/2 )² + h² ].Similarly, the diagonal from (0, 0) to ((b + a)/2, h). Let's calculate the length of this diagonal. The coordinates are (0, 0) to ((b + a)/2, h). So the distance is sqrt[ ((b + a)/2 - 0 )² + (h - 0 )² ] = sqrt[ ((b + a)/2 )² + h² ].But the problem states that the diagonal is 10 cm. Therefore:sqrt[ ((b + a)/2 )² + h² ] = 10Squaring both sides:( (a + b)/2 )² + h² = 100So that's one equation involving a, b, and h.We also know the area is 48 cm², which gives:(1/2)(a + b)h = 48Let me write down both equations:1) ( (a + b)/2 )² + h² = 1002) (1/2)(a + b)h = 48Let me denote S = (a + b)/2. Then, equation 1 becomes:S² + h² = 100Equation 2 becomes:S * h = 48So we have S² + h² = 100 and S * h = 48. Now, we have two equations with two variables S and h. Let's solve for S and h.From equation 2: S = 48 / hSubstitute into equation 1:(48 / h)² + h² = 100Compute (48 / h)²:48² / h² + h² = 100Which is 2304 / h² + h² = 100Multiply both sides by h² to eliminate the denominator:2304 + h⁴ = 100 h²Bring all terms to one side:h⁴ - 100 h² + 2304 = 0Let me set y = h², so the equation becomes:y² - 100 y + 2304 = 0Now, solve this quadratic equation for y:y² - 100 y + 2304 = 0Using the quadratic formula:y = [100 ± sqrt(100² - 4 * 1 * 2304)] / 2Compute discriminant D:D = 10000 - 9216 = 784sqrt(784) = 28Therefore,y = [100 ± 28]/2Which gives two solutions:y = (100 + 28)/2 = 128/2 = 64y = (100 - 28)/2 = 72/2 = 36So y = 64 or y = 36. But y = h², so h² = 64 or h² = 36Therefore, h = sqrt(64) = 8 or h = sqrt(36) = 6But h must be positive, so possible heights are 8 cm or 6 cm. Wait, so two possible solutions? That seems odd. How can there be two possible heights? Let me check my steps again.Wait, in the equations, we derived that S = 48/h, and S = (a + b)/2. So S must be positive as well. So if h is 8, then S = 48/8 = 6. If h is 6, then S = 48/6 = 8.Therefore, there are two possibilities: either h = 8 cm and S = 6 cm, or h = 6 cm and S = 8 cm. Then, (a + b)/2 is 6 or 8. But does this correspond to real trapezoids?Wait, perhaps both solutions are valid. Depending on the lengths of the bases, you can have different heights. But since the problem doesn't specify any other constraints, both could be possible. But wait, the trapezoid is isosceles, so maybe both solutions are acceptable.But the problem asks for the height of the trapezoid. So are there two possible heights? That seems contradictory because usually, such problems expect a unique answer. Maybe I made a mistake somewhere.Wait, let's verify both solutions. Let's take h = 8 cm and S = 6 cm. Then (a + b)/2 = 6, so a + b = 12. Then from equation 1: S² + h² = 6² + 8² = 36 + 64 = 100, which matches the diagonal length. So that's valid.Similarly, h = 6 cm and S = 8 cm. Then a + b = 16. Then S² + h² = 8² + 6² = 64 + 36 = 100, which also matches. So both solutions are valid.So the height could be 6 cm or 8 cm. But the problem states "Find the height of the trapezoid". So why are there two answers? Hmm.Wait, perhaps there is a property of the isosceles trapezoid that I'm missing which would restrict this. Let me think. In an isosceles trapezoid, the legs are equal, and the bases are parallel. The height is the distance between the bases. The diagonals are equal. But is there a constraint that would make only one of these possible?Wait, if we have h = 8 cm and S = 6 cm (a + b = 12 cm), then the difference between the bases is related to the legs. The legs can be calculated using the formula for the legs of an isosceles trapezoid: leg = sqrt( h² + ((b - a)/2 )² )So if a + b = 12, then the difference (b - a) can be any value, but the legs must be positive real numbers. Similarly, if a + b = 16, then the legs would be sqrt( h² + ((b - a)/2 )² ) = sqrt(6² + ((b - a)/2 )² ). However, we don't have the length of the legs given in the problem. So both solutions are possible as long as the legs are real numbers. Since (b - a)/2 would need to be real, which is true as long as b > a or a > b, but since the trapezoid has bases of different lengths, the difference is non-zero. But in either case, ((b - a)/2 )² is non-negative, so the legs would have real lengths.Therefore, both solutions are possible. However, the problem asks for "the height", implying a unique answer. Therefore, perhaps I made a mistake in the equations.Wait, let's go back. The diagonal was calculated as sqrt( ((a + b)/2 )² + h² ). Wait, is that correct? Let me check the coordinate system again.Earlier, I placed the lower base from (0, 0) to (b, 0). The upper base is from ((b - a)/2, h) to ((b + a)/2, h). So the diagonal from (0, 0) to ((b + a)/2, h). Therefore, the horizontal distance is (b + a)/2 - 0 = (a + b)/2, and vertical distance is h. So the diagonal length is sqrt( ((a + b)/2 )² + h² ). That seems correct.Therefore, the equation is correct. So the two solutions are valid. Hmm.But the problem is in the context of an isosceles trapezoid. Maybe depending on whether the upper base is shorter or longer than the lower base, but in an isosceles trapezoid, the labels of the bases can be switched. So if a > b, then you can just call the other base the lower one. So both solutions are acceptable.But the problem says "the height", so maybe both answers are acceptable. However, in some textbooks, they might expect only the positive solution, but here we have two positive solutions. Wait, but both are positive. So maybe the answer is both 6 cm and 8 cm. But the problem is translated from Chinese, maybe in the original problem, there are some other constraints? Wait, no. The user provided the problem in English.Alternatively, maybe I missed something in the problem statement. Let me check again: "the diagonal of an isosceles trapezoid is 10 cm, and its area is 48 cm². Find the height of the trapezoid." There's no other information.Hmm. So according to my calculations, there are two possible heights: 6 cm and 8 cm. But why?Wait, let's consider specific examples. Suppose the trapezoid has a + b = 12 cm (so S = 6 cm) and h = 8 cm. Then, the legs would be sqrt( ((b - a)/2 )² + h² ). Let's assume that a is the upper base and b is the lower base. If a + b = 12, then (b - a)/2 is some value. Let's say a = x, so b = 12 - x. Then, the legs are sqrt( ((12 - x - x)/2 )² + 8² ) = sqrt( ((12 - 2x)/2 )² + 64 ) = sqrt( (6 - x)^2 + 64 ). But we don't know x. Similarly, for the other case where a + b = 16 cm and h = 6 cm, the legs would be sqrt( ((b - a)/2 )² + 36 ).But in both cases, the legs must be positive real numbers. However, without knowing the actual lengths of the bases, we can't determine which is which. Therefore, both configurations are possible. Thus, there are two possible heights.But the problem asks for "the height". In many math problems, especially in competitions or textbooks, sometimes problems are structured so that only one solution is valid, perhaps due to geometric constraints. Let me think again if there's a reason to discard one of the solutions.Wait, let's check the length of the legs. For h = 8 cm and S = 6 cm (a + b = 12), the legs would be sqrt( ((b - a)/2 )² + 8² ). The minimal possible length of the leg is when a = b, but since it's a trapezoid, a ≠ b. Wait, actually, in an isosceles trapezoid, a and b must be different, otherwise, it would be a rectangle. So (b - a)/2 cannot be zero. Therefore, the legs would be longer than 8 cm. But the problem doesn't mention the legs' length, so that's okay.Similarly, for h = 6 cm and S = 8 cm (a + b = 16), the legs would be sqrt( ((b - a)/2 )² + 6² ). Again, the legs must be longer than 6 cm. But since there's no constraint on the legs, both are possible.Therefore, both solutions are valid. So the problem might have two answers. However, since it's an isosceles trapezoid, maybe switching the roles of the bases doesn't create a different trapezoid. Therefore, the two solutions are essentially the same trapezoid but with the bases switched. But since the height is the same regardless of which base is on top, but in this case, the heights are different. Wait, no. If you switch the bases, the height remains the same. Wait, but in these two cases, the heights are different (6 and 8). So they are different trapezoids.Therefore, unless there's a constraint I'm missing, the problem allows for two possible heights. But maybe in the problem's context, considering that the diagonal is 10 cm, which is a fixed length, and the area is fixed, maybe there's a unique solution. Wait, but according to the equations, there are two solutions. So perhaps the answer is both 6 cm and 8 cm.But the problem says "Find the height of the trapezoid". Maybe I need to check if both are correct. Let me see. Let's take h = 8 cm. Then S = 6 cm, so a + b = 12 cm. Then the legs are sqrt( ((b - a)/2 )² + 8² ). Let's pick a = 4 cm and b = 8 cm (since a + b = 12). Then, (b - a)/2 = (8 - 4)/2 = 2 cm. So legs are sqrt(2² + 8²) = sqrt(4 + 64) = sqrt(68) ≈ 8.246 cm. Then the diagonal is sqrt( ((a + b)/2 )² + h² ) = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10 cm. That works.Now, take h = 6 cm. Then S = 8 cm, so a + b = 16 cm. Let's pick a = 6 cm and b = 10 cm. Then, (b - a)/2 = (10 - 6)/2 = 2 cm. The legs are sqrt(2² + 6²) = sqrt(4 + 36) = sqrt(40) ≈ 6.324 cm. Then the diagonal is sqrt( ((a + b)/2 )² + h² ) = sqrt(8² + 6²) = sqrt(64 + 36) = sqrt(100) = 10 cm. That also works.So both trapezoids exist. Therefore, the problem has two solutions. But perhaps the problem expects both, or maybe I missed a step where one of the solutions is invalid.Wait, but in an isosceles trapezoid, the legs must be of equal length, which they are in both cases. The bases are parallel, which they are. The diagonals are equal, which they are (both 10 cm). The area is 48 cm² in both cases. So both are valid isosceles trapezoids. Therefore, the answer should be that the height is either 6 cm or 8 cm.However, in many standard problems, unless specified otherwise, sometimes only the positive solution is considered, but here both are positive. Alternatively, maybe the problem assumes that the height is an integer, and both 6 and 8 are integers, so they are both acceptable. Therefore, the answer is 6 cm and 8 cm.But let me check the problem statement again. It says "Find the height of the trapezoid." Without any other constraints, but the answer might require both values. However, the user hasn't specified whether multiple answers are possible. In many cases, such problems might have a unique solution, so perhaps there is a mistake in my reasoning.Wait, let me check the area formula again. Area = (1/2)(a + b)h. So for h = 6, (a + b) = 16, area is (1/2)*16*6 = 48. Correct. For h = 8, (a + b) = 12, area is (1/2)*12*8 = 48. Correct.And the diagonal in both cases is 10. Correct. So both solutions are correct.Therefore, the answer should be both 6 cm and 8 cm. But maybe the problem expects the answer in a specific form. Since the user asked for the height, and in the problem statement, maybe both answers are acceptable.Alternatively, maybe the problem is from a source that expects a single answer, so I need to verify again.Wait, but the quadratic equation gave us two solutions. Unless there is a maximum possible height for the diagonal length. Let's see: for a given diagonal length of 10 cm, what is the maximum possible height?If we consider S² + h² = 100, then when S is minimized, h is maximized, and vice versa. Since S = (a + b)/2, and in a trapezoid, the sum of the bases must be positive, but there's no upper limit. Wait, but given that the area is fixed at 48, S and h are inversely related (S = 48/h). Therefore, as h increases, S decreases, and vice versa.But in this case, both solutions are within real numbers. Therefore, both are acceptable.Therefore, the height can be either 6 cm or 8 cm.But let me check some references or similar problems. For example, if you have a rectangle (which is a trapezoid with both bases equal), the diagonal would be sqrt(a² + h²), and the area is a*h. If given the diagonal and area, you could solve for a and h. In that case, there would be two solutions as well, but a rectangle with sides swapped. But in the case of a rectangle, swapping a and h just rotates the rectangle, so it's the same rectangle. But in our case, the trapezoid with h = 6 and S = 8 is different from the one with h = 8 and S = 6.Therefore, they are distinct trapezoids. Hence, both solutions are valid.Therefore, the answer is that the height is either 6 cm or 8 cm.But maybe the problem expects both answers. Let me confirm with calculation.Yes, as per the equations, both solutions satisfy the given conditions. Therefore, the height can be 6 cm or 8 cm.But the problem states "Find the height of the trapezoid." So, unless there's a convention to provide all possible answers, the answer should include both. However, in some educational contexts, maybe only the positive solution greater than zero is considered, but here both are positive. Therefore, the answer is 6 cm and 8 cm.But I need to check once more for any possible miscalculations.The quadratic in y was y² - 100y + 2304 = 0. Solutions y = [100 ± sqrt(10000 - 9216)] / 2 = [100 ± sqrt(784)] / 2 = [100 ± 28]/2. So y = 128/2 = 64 or 72/2 = 36. Therefore, h² = 64 or 36, so h = 8 or 6. Correct.So both are valid. Hence, the answer is 6 cm and 8 cm.But the problem might expect both answers. However, the way the problem is phrased, "Find the height of the trapezoid," suggests a single answer. Maybe the problem assumes that height is an integer and both are valid. Therefore, the answer is 6 cm and 8 cm. But I need to confirm.Alternatively, maybe I made a mistake in setting up the equations. Let me check:The diagonal length formula: sqrt( ((a + b)/2 )² + h² ) = 10. Squared: ((a + b)/2 )² + h² = 100. The area: (a + b)/2 * h = 48. Yes, that is correct.Therefore, substituting S = (a + b)/2, we have S² + h² = 100 and S*h = 48. Solving these gives S = 48/h, then substituting into the first equation, leading to h⁴ - 100 h² + 2304 = 0. Solutions h² = 64 or 36. So h = 8 or 6. Therefore, correct.Hence, the answer is both 6 cm and 8 cm.But let me check online if such a problem exists. Wait, I can't do that, but perhaps recall that in some problems with similar structure, two solutions exist. For example, in right triangles with given hypotenuse and area, there can be two solutions. So similarly here, since we're dealing with two variables and two equations, leading to two solutions.Therefore, the answer is 6 cm and 8 cm.But the problem might expect writing both answers. So I should present both.But the user might expect a single answer. Let me think. If the problem is from an exam, maybe only one answer is correct. But according to the mathematics, both are correct.Alternatively, maybe the problem has a typo, but assuming it's correct, both answers are valid.Therefore, the height of the trapezoid is either 6 cm or 8 cm.

✅:The height of the isosceles trapezoid can be found by solving the system of equations derived from the given diagonal and area. 1. Area Formula: [ text{Area} = frac{1}{2} times (a + b) times h = 48 implies (a + b)h = 96 ]2. Diagonal Formula: Using the Pythagorean theorem for the diagonal: [ sqrt{left(frac{a + b}{2}right)^2 + h^2} = 10 implies left(frac{a + b}{2}right)^2 + h^2 = 100 ]Let ( S = frac{a + b}{2} ). Substituting ( S = frac{48}{h} ) into the diagonal equation:[left(frac{48}{h}right)^2 + h^2 = 100 implies frac{2304}{h^2} + h^2 = 100]Multiply through by ( h^2 ):[2304 + h^4 = 100h^2 implies h^4 - 100h^2 + 2304 = 0]Let ( y = h^2 ), then:[y^2 - 100y + 2304 = 0 implies y = frac{100 pm sqrt{784}}{2} = 64 text{ or } 36]Thus, ( h = sqrt{64} = 8 , text{cm} ) or ( h = sqrt{36} = 6 , text{cm} ).Both solutions satisfy the original constraints. Therefore, the height of the trapezoid is:[boxed{6 text{ cm}} quad text{and} quad boxed{8 text{ cm}}]

🔑:1. Understand the problem: We are given an isosceles trapezoid (ABCD) with the following properties: - Diagonal (AC = 10 text{ cm}) - Area of the trapezoid ( = 48 text{ cm}^2 ) - (AB = CD)2. Identify variables: - Let (CE = h) be the height of the trapezoid. - Let (AE = x) and (AD + BC = 2y) (because (AE) is the average width of the trapezoid).3. Set up the equation for the area: The height (h) and the average of the bases provide the height and the effective base for calculating the area: [ xy = 48 text{ cm}^2 ]4. Use the Pythagorean theorem in (triangle AEC): Since ( angle AEC = 90^circ ), [ AE^2 + CE^2 = AC^2 implies x^2 + h^2 = 100 ]5. Solve the system of equations: We have two equations: [ begin{aligned} &1. quad x^2 + h^2 = 100 &2. quad xh = 48 end{aligned} ]6. Express (h) from the second equation: [ h = frac{48}{x} ]7. Substitute (h) into the first equation: [ x^2 + left(frac{48}{x}right)^2 = 100 ] Multiply through by (x^2) to clear the denominator: [ x^4 + frac{48^2}{x^2} = 100x^2 ] Simplify: [ x^4 + 2304 = 100x^2 ] Let (y = x^2): [ y^2 - 100y + 2304 = 0 ] Solve this quadratic equation using the quadratic formula (y = frac{-b pm sqrt{b^2 - 4ac}}{2a}): [ y = frac{100 pm sqrt{10000 - 9216}}{2} ] [ y = frac{100 pm sqrt{784}}{2} ] [ y = frac{100 pm 28}{2} ] [ y = 64 quad text{or} quad y = 36 ] Therefore, [ x^2 = 64 implies x = 8 quad text{or} quad x^2 = 36 implies x = 6 ]8. Find (h) using (h = frac{48}{x}): [ text{If } x = 8, text{ then } h = frac{48}{8} = 6 text{ cm} ] [ text{If } x = 6, text{ then } h = frac{48}{6} = 8 text{ cm} ]9. Conclusion: The height of the trapezoid can be (6 text{ cm}) or (8 text{ cm}).[boxed{6 text{ cm}, 8 text{ cm}}]

🧠:Okay, so I need to find all pairs of natural numbers m and n (excluding zero) such that the polynomial f(x) = 1 + x^n + x^{2n} + ... + x^{mn} is divisible by g(x) = 1 + x + x^2 + ... + x^m. Hmm, let's start by understanding what the problem is asking. Divisibility here means that when we divide f(x) by g(x), there's no remainder. So, f(x) must be a multiple of g(x). First, maybe I should write expressions for both f(x) and g(x). Let's recall that these are both geometric series. For g(x), which is 1 + x + x² + ... + x^m, the sum is (x^{m+1} - 1)/(x - 1) when x ≠ 1. Similarly, f(x) is 1 + x^n + x^{2n} + ... + x^{mn}, which can be written as (x^{n(m+1)} - 1)/(x^n - 1) assuming x ≠ 1 and x^n ≠ 1. So, f(x) is a geometric series with ratio x^n and m+1 terms. So, the problem reduces to finding m and n such that g(x) divides f(x). That would mean that f(x) is a multiple of g(x), so f(x) = g(x) * h(x) for some polynomial h(x) with integer coefficients. Alternatively, since both are polynomials with leading coefficient 1, h(x) must also have integer coefficients.Another approach could be to use the roots of g(x). If g(x) divides f(x), then every root of g(x) must be a root of f(x). So, the roots of g(x) are the (m+1)-th roots of unity except x = 1. Because g(x) = (x^{m+1} - 1)/(x - 1), so its roots are the primitive (m+1)-th roots of unity. So, if ω is a root of g(x), then ω^{m+1} = 1 and ω ≠ 1. Then, for f(x) to have ω as a root, we need f(ω) = 0. Therefore, the condition is that for every primitive (m+1)-th root of unity ω, f(ω) = 0. Let's compute f(ω). Since ω^n + ω^{2n} + ... + ω^{mn} + 1 = 0. Wait, f(ω) = 1 + ω^n + ω^{2n} + ... + ω^{mn}. For this to be zero, we must have that the sum of these terms equals zero. But since ω is a root of g(x), which is 1 + x + ... + x^m, we know that 1 + ω + ω² + ... + ω^m = 0. So maybe there is a relationship between the exponents in f(x) and the exponents in g(x). Perhaps when n divides (m+1), or something like that?Alternatively, maybe the exponents in f(x) need to cover the same residues modulo (m+1) as the exponents in g(x). Let's think in terms of exponents. If we can relate n and m such that when we take exponents n, 2n, ..., mn modulo (m+1), they cover certain residues. Hmm, not sure yet.Wait, let's note that since ω^{m+1} = 1, then ω^k = ω^{k mod (m+1)}. So, f(ω) = 1 + ω^n + ω^{2n} + ... + ω^{mn} = sum_{k=0}^{m} ω^{kn}. So, this is a geometric series with ratio ω^n. So, sum_{k=0}^{m} ω^{kn} = (1 - ω^{n(m+1)})/(1 - ω^n), provided ω^n ≠ 1. But since ω is a primitive (m+1)-th root of unity, ω^n = 1 if and only if (m+1) divides n. But n is a positive integer, so unless m+1 divides n, ω^n ≠ 1. Therefore, if (m+1) does not divide n, then f(ω) = (1 - ω^{n(m+1)})/(1 - ω^n). But ω^{m+1} = 1, so ω^{n(m+1)} = (ω^{m+1})^n = 1^n = 1. Therefore, numerator becomes 1 - 1 = 0, so f(ω) = 0. Wait, but denominator is 1 - ω^n. If ω^n ≠ 1, then denominator is non-zero. Therefore, f(ω) = 0. But if ω^n = 1, which is when (m+1) divides n, then the sum f(ω) would be 1 + 1 + ... +1 (m+1 times), since each term ω^{kn} = (ω^n)^k = 1^k = 1. Then, f(ω) = m+1, which is not zero. Therefore, in this case, if (m+1) divides n, then f(ω) = m+1 ≠ 0, so ω is not a root of f(x), hence g(x) does not divide f(x). Conversely, if (m+1) does not divide n, then f(ω) = 0 for all roots ω of g(x), hence g(x) divides f(x). Wait, so the conclusion is that g(x) divides f(x) if and only if (m+1) does not divide n. But hold on, this seems contradictory. Let me check again.Wait, if (m+1) divides n, then n = k(m+1) for some k. Then, ω^n = ω^{k(m+1)} = (ω^{m+1})^k = 1^k = 1. Therefore, f(ω) = 1 + 1 + ... +1 (m+1 terms) = m+1. So, f(ω) = m+1 ≠ 0. Therefore, if (m+1) divides n, then f(ω) ≠ 0, so g(x) does not divide f(x). On the other hand, if (m+1) does not divide n, then ω^n ≠ 1, so f(ω) = (1 - ω^{n(m+1)})/(1 - ω^n) = (1 - 1)/(1 - ω^n) = 0. Therefore, in this case, f(ω) = 0 for all roots ω of g(x), so g(x) divides f(x). Therefore, the conclusion is that g(x) divides f(x) if and only if (m+1) does not divide n. Therefore, the pairs (m, n) are those where m+1 does not divide n. But wait, the problem states m, n ∈ N*, so m and n are positive integers. So, the answer is all positive integers m, n such that m + 1 does not divide n. Hmm, but let me check with specific examples to see if this holds.Let's take m = 1. Then g(x) = 1 + x. So, m+1 = 2. So, we need 2 not dividing n. Therefore, n must be odd. Let's check. For m = 1, f(x) = 1 + x^n + x^{2n} + ... + x^{1*n} = 1 + x^n. Because mn = 1*n = n, so it's 1 + x^n. Then, g(x) = 1 + x. So, 1 + x divides 1 + x^n? Yes, if and only if x = -1 is a root of 1 + x^n. Plugging x = -1, we get 1 + (-1)^n. So, this is 0 when (-1)^n = -1, which is when n is odd. Therefore, when n is odd, 1 + x divides 1 + x^n. So, in this case, when m =1, the condition is that n is odd, which is equivalent to m+1=2 not dividing n. So, that works.Another example: m=2. Then, g(x)=1 + x + x². m+1=3. So, we need 3 not dividing n. Let's check. If n=1, then f(x)=1 + x + x², which is equal to g(x). So, of course, it's divisible. Since 3 does not divide 1. If n=2, then f(x)=1 + x² + x⁴. Let's check if 1 + x + x² divides this. Let's divide 1 + x² + x⁴ by 1 + x + x². Alternatively, check if roots of g(x) are roots of f(x). The roots of g(x) are the primitive 3rd roots of unity, ω and ω², where ω³=1. Let's compute f(ω). f(ω)=1 + ω² + ω⁴. But ω³=1, so ω⁴=ω. Therefore, f(ω)=1 + ω² + ω = 1 + (ω + ω²) = 1 + (-1) = 0. Similarly, f(ω²)=0. So, yes, when n=2 and m=2, m+1=3 does not divide n=2, so 3 does not divide 2, so it works. If n=3, m=2, then m+1=3 divides n=3, so according to the condition, it shouldn't work. Let's check f(x)=1 + x³ + x⁶. Divided by g(x)=1 + x + x². Let's check f(ω)=1 + ω³ + ω⁶ = 1 + 1 + 1 = 3 ≠ 0. So, f(x) is not divisible by g(x). Correct. So, n=3, m=2 doesn't work. So, the condition seems to hold.Another example: m=3. Then, m+1=4. So, we need 4 not dividing n. Let's take n=1. Then, f(x)=1 + x + x² + x³ = g(x). So, divisible. n=2. Then, f(x)=1 + x² + x⁴ + x⁶. Let's check divisibility. Let's plug in the roots of g(x), which are the 4th roots of unity except 1, so i, -1, -i. But wait, g(x)=1 + x + x² + x³. The roots are the primitive 4th roots of unity? Wait, g(x)=(x⁴ -1)/(x -1), so roots are the 4th roots of unity except 1. So, they are -1, i, -i. Let's check f(-1)=1 + (-1)^2 + (-1)^4 + (-1)^6 = 1 +1 +1 +1=4≠0. So, f(-1)≠0. But according to our previous logic, since m+1=4 does not divide n=2, then g(x) should divide f(x). But here, f(-1)=4≠0, which contradicts. Wait, that's a problem. What's going on here.Wait, when m=3, n=2. Then, m+1=4. Since 4 does not divide 2, according to our previous conclusion, g(x) should divide f(x). But in reality, when we check f(-1)=4≠0. Therefore, our conclusion must be wrong. So, there's a mistake in the reasoning. Let me check where.Earlier, we considered that if (m+1) does not divide n, then ω^n ≠1 for any primitive (m+1)-th root of unity. Then, f(ω)=0. But in the case m=3, n=2, m+1=4. The roots of g(x) are the primitive 4th roots of unity: i and -i, and also -1? Wait, no. Wait, g(x)=1 + x + x² + x³. The roots are the 4th roots of unity except 1. So, they are e^(2πik/4) for k=1,2,3. So, that's i, -1, -i. So, ω = i, ω=-1, ω=-i. Let's check f(ω) for ω=i. f(i)=1 + (i)^2 + (i)^4 + (i)^6 = 1 + (-1) + 1 + (-1)=0. Similarly, f(-i)=1 + (-i)^2 + (-i)^4 + (-i)^6=1 + (-1)+1 + (-1)=0. But f(-1)=1 + (-1)^2 + (-1)^4 + (-1)^6=1 +1 +1 +1=4≠0. So, even though 4 does not divide n=2, there exists a root ω=-1 of g(x) such that f(ω)≠0. Therefore, our previous conclusion was incorrect. So, why did this happen?Ah! Because when m+1 divides n(m+1). Wait, no. Wait, the problem was that even though (m+1) does not divide n, ω^n might still be equal to 1 for some roots ω of g(x). In this case, ω=-1 is a root of g(x)=1 + x + x² + x³, and n=2. So, ω^n = (-1)^2=1. Therefore, even though 4 does not divide n=2, there is a root ω=-1 of g(x) where ω^n=1. Hence, f(ω)=sum_{k=0}^{m} 1^{k}=m+1=4≠0. Therefore, even if (m+1) doesn't divide n, but if there exists a root ω of g(x) such that ω^n=1, then f(ω)=m+1≠0, hence g(x) does not divide f(x). So, our previous reasoning was flawed because we assumed that if (m+1) does not divide n, then ω^n ≠1 for all roots ω of g(x). But actually, even if (m+1) does not divide n, there might be some roots ω where ω^n=1. For example, when m+1=4 and n=2, then ω=-1 is a root of g(x), and ω^2=1. So, the previous conclusion was incorrect. Therefore, we need a different approach.So, perhaps the correct condition is that for all primitive (m+1)-th roots of unity ω, ω^n ≠1. That is, the order of ω, which is m+1, does not divide n. But ω is a primitive (m+1)-th root of unity, so its order is m+1. Therefore, ω^n=1 if and only if m+1 divides n. Therefore, if m+1 divides n, then for all roots ω of g(x), ω^n=1, which would make f(ω)=m+1≠0. If m+1 does not divide n, then for all primitive (m+1)-th roots of unity ω, ω^n ≠1, hence f(ω)=0. However, in the case where m+1 is composite, some roots of g(x) may have orders that divide m+1 but are less than m+1. For example, when m+1=4, the roots include -1, which has order 2. So, even if m+1=4 does not divide n=2, the root -1 has order 2, which divides n=2. Therefore, ω^n=1 for ω=-1. Therefore, the condition is actually stronger: for every divisor d of m+1 with d >1, d does not divide n. Wait, but that's equivalent to saying that gcd(m+1, n) =1. Because if gcd(m+1, n)=1, then none of the divisors of m+1 (other than 1) divide n. Conversely, if gcd(m+1, n)=d>1, then d divides n and d divides m+1. Therefore, the correct condition is that gcd(m+1, n)=1. Let's verify this.Take m=3 (m+1=4), n=2. gcd(4,2)=2≠1. Therefore, g(x) does not divide f(x). Which matches our previous example where f(-1)=4≠0. If we take m=2 (m+1=3), n=2. gcd(3,2)=1. Then, according to this, g(x) divides f(x). Which was true in our earlier example. Similarly, m=1 (m+1=2), n=1. gcd(2,1)=1. So, g(x) divides f(x). Which is true because 1 + x divides 1 + x. Wait, but m=1, n=1: f(x)=1 + x, which is equal to g(x). So, yes, divisible. If m=1, n=2. gcd(2,2)=2≠1. So, not divisible. Which is correct, since 1 + x does not divide 1 + x².Another example: m=4, m+1=5. If n=2, gcd(5,2)=1. So, g(x)=1 +x +x² +x³ +x⁴. f(x)=1 +x² +x⁴ +x⁶ +x⁸. Let's check if roots of g(x) satisfy f(ω)=0. The roots of g(x) are primitive 5th roots of unity, ω, ω², ω³, ω⁴. So, ω^5=1. Then, f(ω)=1 + ω² + ω⁴ + ω⁶ + ω⁸. But ω⁶=ω^(5+1)=ω, and ω⁸=ω^(5+3)=ω³. So, f(ω)=1 + ω² + ω⁴ + ω + ω³. But 1 + ω + ω² + ω³ + ω⁴=0, since it's the sum of all 5th roots of unity except 1. Therefore, f(ω)=0. Similarly for the other roots. Therefore, gcd(5,2)=1 implies divisibility. If n=5, gcd(5,5)=5≠1, so not divisible. So, f(ω)=1 + ω^5 + ... + ω^25=1 +1 +...+1=5≠0. Correct.Therefore, the correct condition seems to be that gcd(m+1, n)=1. Therefore, the answer is all pairs (m, n) where m and n are positive integers with gcd(m+1, n)=1. But let's check another case. Take m=4, n=3. gcd(5,3)=1. Then, f(x)=1 +x³ +x⁶ +x⁹ +x^{12}. Check at ω, a primitive 5th root of unity. f(ω)=1 + ω³ + ω⁶ + ω⁹ + ω^{12}. Simplify exponents modulo 5. 6 mod5=1, 9 mod5=4, 12 mod5=2. So, f(ω)=1 + ω³ + ω + ω⁴ + ω². Which is 1 + ω + ω² + ω³ + ω⁴=0. Correct. Therefore, gcd(m+1,n)=1 is the condition.But wait, earlier when m=3, n=2, gcd(4,2)=2≠1, so it doesn't divide, which matches. When m=2, n=1, gcd(3,1)=1, which works. So, this seems to hold. Therefore, the conclusion is that g(x) divides f(x) if and only if gcd(m+1, n)=1.Wait, but let's check m=3, n=1. gcd(4,1)=1. Then, f(x)=1 + x + x² +x³, which is exactly g(x), so divisible. Correct. If m=3, n=3. gcd(4,3)=1. So, f(x)=1 +x³ +x⁶ +x⁹. Check divisibility. At ω=i, f(i)=1 + (-i) + (-1) + i=0. At ω=-1, f(-1)=1 + (-1) +1 + (-1)=0. At ω=-i, same as ω=i. Wait, but g(x)=1 +x +x² +x³. The roots are the 4th roots of unity except 1: i, -1, -i. For ω=i: f(ω)=1 + i³ + i⁶ + i⁹=1 + (-i) + (-1) + i=0. For ω=-1: f(-1)=1 + (-1)³ + (-1)^6 + (-1)^9=1 -1 +1 -1=0. For ω=-i: same as ω=i. So, all roots satisfy f(ω)=0. Hence, divisible. But gcd(4,3)=1. So, it works. But when m=3, n=2, gcd(4,2)=2≠1, and we saw f(-1)=4≠0. So, this holds.Therefore, the correct condition is gcd(m+1, n)=1. Hence, all pairs (m, n) where m and n are positive integers with gcd(m+1, n)=1.But wait, let's confirm with m=5, n=2. gcd(6,2)=2≠1. So, g(x)=1 +x +x² +x³ +x⁴ +x⁵. f(x)=1 +x² +x⁴ +x⁶ +x⁸ +x^{10}. Let's check ω, a primitive 6th root of unity. The roots of g(x) are e^(2πik/6) for k=1,2,3,4,5. So, ω=e^(πi/3)=cos(π/3)+i sin(π/3)= (1/2)+i(√3/2). Let's compute f(ω). ω²= e^(2πi/3), ω⁴=e^(4πi/3), ω⁶=1, ω⁸=ω², ω^{10}=ω⁴. So, f(ω)=1 + ω² + ω⁴ +1 + ω² + ω⁴=2 + 2ω² + 2ω⁴. But 1 + ω + ω² + ω³ + ω⁴ + ω⁵=0. So, ω + ω² + ω³ + ω⁴ + ω⁵=-1. But ω³= e^(πi)= -1. So, ω + ω² + (-1) + ω⁴ + ω⁵= -1. Therefore, ω + ω² + ω⁴ + ω⁵=0. But ω⁵=ω^{-1}. Therefore, maybe this is getting too complicated. Alternatively, compute f(ω)=2 + 2(ω² + ω⁴). But ω² + ω⁴= e^(2πi/3) + e^(4πi/3) = -1. Because in the sum 1 + ω + ω² + ... + ω⁵=0, subtract 1 + ω³=1 -1=0. Therefore, ω + ω² + ω⁴ + ω⁵=0. Since ω³=-1, so ω⁴=ω³ω=-ω, ω⁵=ω³ω²=-ω². So, ω + ω² + (-ω) + (-ω²)=0. So, indeed. Therefore, ω² + ω⁴=ω² - ω. But this may not be helpful. Alternatively, since f(ω)=2 + 2(ω² + ω⁴). If we can compute ω² + ω⁴. Let's take ω=e^(πi/3). Then, ω²=e^(2πi/3), ω⁴=e^(4πi/3). The sum of these two is e^(2πi/3) + e^(4πi/3) = -1/2 + i√3/2 + (-1/2 -i√3/2) = -1. Therefore, f(ω)=2 + 2*(-1)=0. Wait, so f(ω)=0. Similarly, check for another root. Let's take ω=e^(2πi/3). But this is a primitive 3rd root of unity, but m+1=6, so ω=e^(2πi/6)=e^(πi/3). Wait, confused. Anyway, if m=5, n=2, and gcd(6,2)=2≠1, but in this case, f(ω)=0. Wait, this contradicts our previous conclusion. So, what's happening?Wait, let's compute f(x) for m=5, n=2. f(x)=1 +x² +x⁴ +x⁶ +x⁸ +x^{10}. Let's divide this by g(x)=1 +x +x² +x³ +x⁴ +x⁵. Let me perform polynomial division.Alternatively, note that x^6 -1 = (x-1)g(x). So, x^6=1 + (x-1)g(x). Therefore, x^6 ≡1 mod g(x). Similarly, x^{6k}≡1^k=1 mod g(x). So, in f(x)=1 +x² +x⁴ +x⁶ +x⁸ +x^{10} ≡1 +x² +x⁴ +1 +x² +x⁴ mod g(x). Because x^6≡1, x^8≡x², x^{10}≡x⁴. Therefore, f(x)≡ (1+1) + (x² +x²) + (x⁴ +x⁴) =2 + 2x² + 2x⁴ mod g(x). Therefore, f(x)≡2(1 +x² +x⁴). But 1 +x² +x⁴ is not congruent to 0 mod g(x). Wait, but what is 1 +x² +x⁴ in terms of g(x). Let's compute 1 +x² +x⁴ = (x⁶ -1)/(x² -1). But since x⁶ -1=(x-1)g(x), so 1 +x² +x⁴= (x-1)g(x)/(x² -1)= (x-1)g(x)/[(x-1)(x+1)] = g(x)/(x+1). Therefore, 1 +x² +x⁴ = g(x)/(x+1). Therefore, unless x+1 divides g(x), which it doesn't (since g(-1)=1 -1 +1 -1 +1 -1=0 for m=5? Wait, m=5, g(x)=1 +x +x² +x³ +x⁴ +x⁵. g(-1)=1 -1 +1 -1 +1 -1=0. So, x+1 divides g(x). Indeed, g(x)= (x⁶ -1)/(x -1), so roots are 6th roots of unity except 1. So, x+1 divides g(x) because -1 is a root. Therefore, g(x)= (x+1)(x^4 +x^2 +1). Wait, but that's not important here. The point is, f(x)=2(1 +x² +x⁴)=2(g(x)/(x+1)). So, unless 2(g(x)/(x+1)) is divisible by g(x), which would require that (x+1) divides 2. But (x+1) is a polynomial of degree 1, and 2 is a constant. So, unless 2=0, which it isn't, this is not possible. Therefore, f(x) ≡ 2(g(x)/(x+1)) mod g(x), which is not zero. Therefore, f(x) is not divisible by g(x). But earlier, when we checked at ω=e^(πi/3), we found f(ω)=0. But according to polynomial division, f(x) is not divisible by g(x). This is a contradiction. What's wrong here?Wait, when we computed f(ω)=0, but polynomial division suggests that the remainder is non-zero. This inconsistency arises because when we reduce f(x) modulo g(x), the remainder is 2(1 +x² +x⁴). However, if we evaluate this remainder at a root ω of g(x), which is a 6th root of unity, we get 2(1 + ω² + ω⁴). But for ω=e^(πi/3), 1 + ω² + ω⁴=1 + e^(2πi/3) + e^(4πi/3)=0. Therefore, even though the remainder is a non-zero polynomial, it evaluates to zero at all roots of g(x). This implies that the remainder polynomial is divisible by g(x). But the remainder is of lower degree than g(x) (degree 4 vs degree 5). The only way this can happen is if the remainder is the zero polynomial. Therefore, there must be an error in the calculation.Wait, let's re-examine. For m=5, n=2. f(x)=1 +x² +x⁴ +x⁶ +x⁸ +x^{10}. Since x^6≡1 mod g(x), then x^6=1, x^8=x², x^{10}=x⁴. Therefore, f(x)≡1 +x² +x⁴ +1 +x² +x⁴=2 +2x² +2x⁴ mod g(x). So, the remainder is 2(1 +x² +x⁴). However, 1 +x² +x⁴ is indeed divisible by x+1 when m=5? Wait, m=5, g(x)=1 +x +x² +x³ +x⁴ +x⁵. We know that x+1 divides g(x) because g(-1)=0. Therefore, since x+1 divides g(x), and g(x) divides x^6 -1, which factors as (x-1)(x+1)(x² +x +1)(x² -x +1). Therefore, the roots of g(x) are the 6th roots of unity except 1. So, the roots include -1, e^(πi/3), e^(5πi/3). Now, the remainder polynomial is 2(1 +x² +x⁴). Let's check its value at x=-1: 2(1 +1 +1)=6≠0. At x=e^(πi/3): 2(1 + e^(2πi/3) + e^(4πi/3))=2(0)=0. Similarly at e^(5πi/3). Therefore, the remainder is a polynomial that is zero at some roots of g(x) but not all. Specifically, it's zero at the primitive 6th roots of unity but not at x=-1. Therefore, the remainder is not the zero polynomial, which means that g(x) does not divide f(x). However, when we tested ω=e^(πi/3), we got f(ω)=0, but for ω=-1, f(-1)=6≠0. Therefore, f(x) is not divisible by g(x). Therefore, the earlier calculation where we thought f(ω)=0 was incorrect because we only checked the primitive roots (order 6) but not the roots with lower order (like -1, order 2). Therefore, even though gcd(m+1,n)=1 (in this case, gcd(6,2)=2≠1, so it's excluded), but even when gcd(m+1,n)=d>1, there are some roots where f(ω)=0 and others where it's not. Therefore, the correct condition must be that for all roots ω of g(x), f(ω)=0. Which happens if and only if ω^n ≠1 for all roots ω of g(x). But ω is a root of g(x), which are the (m+1)-th roots of unity except 1. So, ω has order dividing m+1. Therefore, ω^n=1 if and only if the order of ω divides n. So, for f(ω)=0, we need that for all ω in the set of roots of g(x), the order of ω does not divide n. Therefore, the condition is that the multiplicative order of each root ω of g(x) does not divide n. Since the roots of g(x) are all the primitive (m+1)-th roots of unity and the others with orders dividing m+1 but less than m+1 (in case m+1 is not prime). Therefore, the necessary and sufficient condition is that n is not divisible by any divisor d of m+1 where d >1. Which is equivalent to gcd(m+1, n)=1. Because if gcd(m+1,n)=d>1, then d divides both m+1 and n, so there exists a root ω of g(x) with order d (since d divides m+1), and ω^n=1. Conversely, if gcd(m+1,n)=1, then no divisor d>1 of m+1 divides n, hence for all roots ω of g(x), ω^n ≠1, so f(ω)=0. Therefore, the correct condition is that gcd(m+1,n)=1. However, in our example with m=5, n=2, gcd(6,2)=2≠1, so the condition is not met, hence f(x) is not divisible by g(x), which matches our previous result. Similarly, when m=3, n=1, gcd(4,1)=1, so divisible. When m=3, n=2, gcd(4,2)=2≠1, not divisible. So, this seems to hold. Therefore, the answer is that all pairs (m, n) of positive integers where gcd(m+1, n)=1.But wait, let's check another example where m+1 is composite. Let's take m=5, n=5. Then, gcd(6,5)=1. Therefore, should be divisible. Let's see. f(x)=1 +x⁵ +x^{10} +x^{15} +x^{20} +x^{25}. Divided by g(x)=1 +x +x² +x³ +x⁴ +x⁵. Let's check at x=ω, a primitive 6th root of unity. For ω=-1, f(-1)=1 + (-1)^5 + (-1)^{10} + (-1)^{15} + (-1)^{20} + (-1)^{25}=1 -1 +1 -1 +1 -1=0. For ω=e^(πi/3), f(ω)=1 + ω^5 + ω^{10} + ω^{15} + ω^{20} + ω^{25}. Since ω^6=1, ω^5=ω^{-1}, ω^{10}=ω^{4}, ω^{15}=ω^{3}= -1, ω^{20}=ω^{2}, ω^{25}=ω^{1}. So, f(ω)=1 + ω^{-1} + ω^{4} + (-1) + ω^{2} + ω. But ω^{-1}=ω^5, and ω^4 + ω^2 + ω + ω^5= sum of all primitive 6th roots except 1? Wait, sum of all 6th roots of unity is 0. So, 1 + sum_{k=1}^5 ω^k=0. Therefore, sum_{k=1}^5 ω^k= -1. But in f(ω)=1 + ω^5 + ω^4 + (-1) + ω^2 + ω= (1 -1) + (ω + ω² + ω⁴ + ω⁵)=0 + (-1 - ω³). But ω³= e^(πi)= -1. So, -1 -(-1)=0. Therefore, f(ω)=0. Similarly for other roots. Therefore, when gcd(m+1,n)=1, even if m+1 is composite, it works. Therefore, the correct condition is indeed gcd(m+1,n)=1.Therefore, the solution is all pairs of positive integers m, n with gcd(m+1, n)=1.Final AnswerThe natural numbers ( m ) and ( n ) must satisfy ( gcd(m + 1, n) = 1 ). Thus, the solution is (boxed{gcd(m + 1, n) = 1}).

✅:To determine all pairs ( m, n in mathbf{N}^{*} ) such that the polynomial ( f(x) = 1 + x^n + x^{2n} + cdots + x^{mn} ) is divisible by ( g(x) = 1 + x + x^2 + cdots + x^m ), we start by analyzing the roots of these polynomials. 1. Roots of ( g(x) ): - ( g(x) ) is a geometric series sum: ( g(x) = frac{x^{m+1} - 1}{x - 1} ) for ( x neq 1 ). - The roots of ( g(x) ) are the ((m+1))-th roots of unity excluding 1, i.e., ( omega ) where ( omega^{m+1} = 1 ) and ( omega neq 1 ).2. Roots of ( f(x) ): - ( f(x) ) is also a geometric series sum: ( f(x) = frac{x^{n(m+1)} - 1}{x^n - 1} ) for ( x neq 1 ) and ( x^n neq 1 ). - For ( f(x) ) to be divisible by ( g(x) ), every root ( omega ) of ( g(x) ) must be a root of ( f(x) ).3. Condition for Divisibility: - For each root ( omega ) of ( g(x) ), ( f(omega) = 0 ). This requires ( 1 + omega^n + omega^{2n} + cdots + omega^{mn} = 0 ). - Since ( omega ) is a primitive ((m+1))-th root of unity, ( omega^n neq 1 ) if and only if ( m+1 ) does not divide ( n ). - However, we need to ensure that ( omega^n neq 1 ) for all roots ( omega ) of ( g(x) ). This happens if and only if the greatest common divisor (gcd) of ( m+1 ) and ( n ) is 1, i.e., ( gcd(m+1, n) = 1 ).4. Verification: - If ( gcd(m+1, n) = 1 ), then no divisor of ( m+1 ) (other than 1) divides ( n ), ensuring ( omega^n neq 1 ) for all roots ( omega ). - If ( gcd(m+1, n) neq 1 ), there exists a root ( omega ) such that ( omega^n = 1 ), leading to ( f(omega) neq 0 ).Thus, the solution is that ( m ) and ( n ) must satisfy ( gcd(m + 1, n) = 1 ).[boxed{gcd(m + 1, n) = 1}]

🔑:1. Let f(x) = 1 + x^n + x^{2n} + cdots + x^{mn} and g(x) = 1 + x + x^2 + cdots + x^m.2. Note that g(x) is the sum of a geometric series and can be expressed as: [ g(x) = frac{x^{m+1} - 1}{x - 1} ]3. Since x^{m+1} - 1 = (x - 1)g(x), we can write: [ g(x) = left(x - varepsilon_1right)left(x - varepsilon_2right) cdots left(x - varepsilon_mright) ]4. Here, each (varepsilon_k) (1 leq k leq m) is an (m+1)-th root of unity, given by: [ varepsilon_k = cos frac{2kpi}{m+1} + isin frac{2kpi}{m+1} ]5. Clearly, when k neq j, (varepsilon_k neq varepsilon_j), meaning that the roots of (g(x)) are distinct.6. For (g(x)) to divide (f(x)), it is necessary and sufficient that (f(varepsilon_k) = 0) for all k = 1, 2, cdots, m. That is: [ f(varepsilon_k) = 1 + varepsilon_k^n + varepsilon_k^{2n} + cdots + varepsilon_k^{mn} = 0 ]7. Since (f(x)) can also be expressed using roots in terms of (n): [ f(x) = frac{x^{n(m+1)} - 1}{x^n - 1} ]8. Therefore, the required condition for (g(x) mid f(x)) also translates to: [ varepsilon_k^{n(m+1)} - 1 = 0 quad text{and} quad varepsilon_k^n neq 1 ] for all k = 1, 2, cdots, m.9. Since (varepsilon_k^{n(m+1)} - 1 = 0) is true for any (n in mathbb{N}^*), the sufficient condition for (g(x) mid f(x)) is: [ varepsilon_k^n = cos frac{2knpi}{m+1} + isin frac{2knpi}{m+1} neq 1 ] 10. It follows that (n) and (m+1) must be coprime (i.e., (gcd(n, m+1)=1)). Conclusion:All pairs of integers (m, n) such that (m in mathbb{N}^*) and (n in mathbb{N}^*) making (g(x) mid f(x)) are given by:[boxed{{(m, n) mid m in mathbb{N}^*, n in mathbb{N}^*, n text{ and } m+1 text{ are coprime}}}]

🧠:Okay, so I need to solve this equation: ( a^{2} + b^{2} + c^{2} + d^{2} - ab - bc - cd - d + frac{2}{5} = 0 ). Hmm, let's see. First, I should probably try to simplify this equation by completing the square for each variable. Since there are four variables, maybe I can group them or handle them step by step. Let me look at the terms involving a and b first. The equation has ( a^2 - ab + b^2 ). Wait, that looks like part of a quadratic form. For two variables, the expression ( a^2 - ab + b^2 ) can be rewritten as ( frac{3}{4}(a - b)^2 + frac{1}{4}(a + b)^2 ), but I'm not sure if that's helpful here. Alternatively, maybe I can complete the square for a in terms of b. Let's try that. The terms with a are ( a^2 - ab ). Completing the square for a:( a^2 - ab = a^2 - ab + frac{b^2}{4} - frac{b^2}{4} = left(a - frac{b}{2}right)^2 - frac{b^2}{4} ).So substituting back into the equation:( left(a - frac{b}{2}right)^2 - frac{b^2}{4} + b^2 + c^2 + d^2 - bc - cd - d + frac{2}{5} = 0 ).Simplify the b terms: ( -frac{b^2}{4} + b^2 = frac{3b^2}{4} ). So now the equation becomes:( left(a - frac{b}{2}right)^2 + frac{3b^2}{4} + c^2 + d^2 - bc - cd - d + frac{2}{5} = 0 ).Next, let's look at the terms involving b and c. We have ( frac{3b^2}{4} - bc + c^2 ). Maybe we can complete the square for b here. Let's factor out the coefficient of ( b^2 ):( frac{3}{4}b^2 - bc + c^2 = frac{3}{4}left( b^2 - frac{4c}{3}b right) + c^2 ).Completing the square inside the parentheses:( b^2 - frac{4c}{3}b = left(b - frac{2c}{3}right)^2 - frac{4c^2}{9} ).Substituting back:( frac{3}{4}left( left(b - frac{2c}{3}right)^2 - frac{4c^2}{9} right) + c^2 = frac{3}{4}left(b - frac{2c}{3}right)^2 - frac{3}{4} cdot frac{4c^2}{9} + c^2 ).Simplify the c² terms:( - frac{3}{4} cdot frac{4c^2}{9} = - frac{c^2}{3} ).So total becomes:( frac{3}{4}left(b - frac{2c}{3}right)^2 - frac{c^2}{3} + c^2 = frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2c^2}{3} ).Now substituting this back into the main equation:( left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2c^2}{3} + d^2 - cd - d + frac{2}{5} = 0 ).Moving on to the terms with c and d. The remaining terms involving c and d are ( frac{2c^2}{3} - cd + d^2 - d ). Let's try to complete the square for c here. Let's look at ( frac{2c^2}{3} - cd ). Maybe factor out the coefficient of c²:( frac{2}{3}c^2 - cd = frac{2}{3}left( c^2 - frac{3d}{2}c right) ).Completing the square inside the parentheses:( c^2 - frac{3d}{2}c = left(c - frac{3d}{4}right)^2 - frac{9d^2}{16} ).Substituting back:( frac{2}{3}left( left(c - frac{3d}{4}right)^2 - frac{9d^2}{16} right) = frac{2}{3}left(c - frac{3d}{4}right)^2 - frac{2}{3} cdot frac{9d^2}{16} = frac{2}{3}left(c - frac{3d}{4}right)^2 - frac{3d^2}{8} ).So now, the terms involving c and d become:( frac{2}{3}left(c - frac{3d}{4}right)^2 - frac{3d^2}{8} + d^2 - d ).Simplify the d² terms:( - frac{3d^2}{8} + d^2 = frac{5d^2}{8} ).So now, the equation is:( left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5d^2}{8} - d + frac{2}{5} = 0 ).Now, we need to handle the remaining d terms: ( frac{5d^2}{8} - d ). Let's complete the square for d here.Factor out the coefficient of d²:( frac{5}{8}d^2 - d = frac{5}{8}left( d^2 - frac{8}{5}d right) ).Completing the square inside:( d^2 - frac{8}{5}d = left(d - frac{4}{5}right)^2 - frac{16}{25} ).Substituting back:( frac{5}{8}left( left(d - frac{4}{5}right)^2 - frac{16}{25} right) = frac{5}{8}left(d - frac{4}{5}right)^2 - frac{5}{8} cdot frac{16}{25} = frac{5}{8}left(d - frac{4}{5}right)^2 - frac{2}{5} ).So now, substituting this back into the equation:( left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5}{8}left(d - frac{4}{5}right)^2 - frac{2}{5} + frac{2}{5} = 0 ).The constants at the end cancel each other out: ( - frac{2}{5} + frac{2}{5} = 0 ). So the equation simplifies to:( left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5}{8}left(d - frac{4}{5}right)^2 = 0 ).Since each term on the left-hand side is a square multiplied by a positive coefficient, the sum of these terms can only be zero if each individual squared term is zero. Therefore, we set each squared term equal to zero:1. ( a - frac{b}{2} = 0 ) ⇒ ( a = frac{b}{2} )2. ( b - frac{2c}{3} = 0 ) ⇒ ( b = frac{2c}{3} )3. ( c - frac{3d}{4} = 0 ) ⇒ ( c = frac{3d}{4} )4. ( d - frac{4}{5} = 0 ) ⇒ ( d = frac{4}{5} )Now, we can substitute each equation into the next to find the values of a, b, c, and d.Starting with the fourth equation: ( d = frac{4}{5} ).Substitute d into the third equation: ( c = frac{3}{4} cdot frac{4}{5} = frac{3}{5} ).Substitute c into the second equation: ( b = frac{2}{3} cdot frac{3}{5} = frac{2}{5} ).Substitute b into the first equation: ( a = frac{1}{2} cdot frac{2}{5} = frac{1}{5} ).So, the solution is ( a = frac{1}{5} ), ( b = frac{2}{5} ), ( c = frac{3}{5} ), ( d = frac{4}{5} ).Let me verify this solution by plugging the values back into the original equation:Compute each term:- ( a^2 = left(frac{1}{5}right)^2 = frac{1}{25} )- ( b^2 = left(frac{2}{5}right)^2 = frac{4}{25} )- ( c^2 = left(frac{3}{5}right)^2 = frac{9}{25} )- ( d^2 = left(frac{4}{5}right)^2 = frac{16}{25} )- ( -ab = -frac{1}{5} cdot frac{2}{5} = -frac{2}{25} )- ( -bc = -frac{2}{5} cdot frac{3}{5} = -frac{6}{25} )- ( -cd = -frac{3}{5} cdot frac{4}{5} = -frac{12}{25} )- ( -d = -frac{4}{5} )- ( frac{2}{5} )Now sum all these:( frac{1}{25} + frac{4}{25} + frac{9}{25} + frac{16}{25} - frac{2}{25} - frac{6}{25} - frac{12}{25} - frac{4}{5} + frac{2}{5} ).First, combine the fractions with denominator 25:Numerator: 1 + 4 + 9 + 16 - 2 - 6 - 12 = (1+4=5; 5+9=14; 14+16=30; 30-2=28; 28-6=22; 22-12=10). So total is 10/25.Then the terms with denominator 5: -4/5 + 2/5 = (-4 + 2)/5 = -2/5.So total equation becomes:10/25 - 2/5 = 10/25 - 10/25 = 0. Perfect, it checks out.Therefore, the only real solution is ( a = frac{1}{5} ), ( b = frac{2}{5} ), ( c = frac{3}{5} ), ( d = frac{4}{5} ).Final AnswerThe solution is (boxed{a = dfrac{1}{5}}), (boxed{b = dfrac{2}{5}}), (boxed{c = dfrac{3}{5}}), and (boxed{d = dfrac{4}{5}}).

✅:To solve the equation ( a^{2} + b^{2} + c^{2} + d^{2} - ab - bc - cd - d + frac{2}{5} = 0 ), we can complete the square for each variable step by step.1. Complete the square for (a) and (b): [ a^2 - ab + b^2 = left(a - frac{b}{2}right)^2 - frac{b^2}{4} + b^2 = left(a - frac{b}{2}right)^2 + frac{3b^2}{4} ] Substituting back, the equation becomes: [ left(a - frac{b}{2}right)^2 + frac{3b^2}{4} + c^2 + d^2 - bc - cd - d + frac{2}{5} = 0 ]2. Complete the square for (b) and (c): [ frac{3}{4}b^2 - bc + c^2 = frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2c^2}{3} ] Substituting back, the equation becomes: [ left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2c^2}{3} + d^2 - cd - d + frac{2}{5} = 0 ]3. Complete the square for (c) and (d): [ frac{2}{3}c^2 - cd + d^2 = frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5d^2}{8} ] Substituting back, the equation becomes: [ left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5d^2}{8} - d + frac{2}{5} = 0 ]4. Complete the square for (d): [ frac{5}{8}d^2 - d = frac{5}{8}left(d - frac{4}{5}right)^2 - frac{2}{5} ] Substituting back, the equation becomes: [ left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5}{8}left(d - frac{4}{5}right)^2 = 0 ]Since each term is a square multiplied by a positive coefficient, each term must be zero:1. ( a - frac{b}{2} = 0 ) ⇒ ( a = frac{b}{2} )2. ( b - frac{2c}{3} = 0 ) ⇒ ( b = frac{2c}{3} )3. ( c - frac{3d}{4} = 0 ) ⇒ ( c = frac{3d}{4} )4. ( d - frac{4}{5} = 0 ) ⇒ ( d = frac{4}{5} )Substituting these equations step by step:- ( d = frac{4}{5} )- ( c = frac{3}{4} cdot frac{4}{5} = frac{3}{5} )- ( b = frac{2}{3} cdot frac{3}{5} = frac{2}{5} )- ( a = frac{1}{2} cdot frac{2}{5} = frac{1}{5} )Verification confirms the solution satisfies the original equation. Final AnswerThe solution is (boxed{a = dfrac{1}{5}}), (boxed{b = dfrac{2}{5}}), (boxed{c = dfrac{3}{5}}), and (boxed{d = dfrac{4}{5}}).

🔑:1. Given equation:[a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + frac{2}{5} = 0]2. Rewrite the equation in the form of squares by completing the square for each variable. Attempt to match each part to a perfect square setup:[a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + frac{2}{5} = left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5}{8}left(d - frac{4}{5}right)^2]3. Now, to justify the expansion and ensure no terms are omitted:[(a - frac{b}{2})^2 = a^2 - ab + frac{b^2}{4}]and similarly for others:[frac{3}{4}left(b - frac{2c}{3}right)^2 = frac{3}{4}(b^2 - frac{4bc}{3} + frac{4c^2}{9}) = b^2 - frac{4bc}{3} + frac{4c^2}{9}][frac{2}{3}left(c - frac{3d}{4}right)^2 = frac{2}{3}(c^2 - frac{3cd}{2} + frac{9d^2}{16}) = frac{2}{3}(c^2 - frac{3cd}{2} + frac{9d^2}{16})]After expanding, combining terms, and comparing it to the original equation, it is observed that all individual squared terms must be zero for equality to hold because:[left(a - frac{b}{2}right)^2 + frac{3}{4}left(b - frac{2c}{3}right)^2 + frac{2}{3}left(c - frac{3d}{4}right)^2 + frac{5}{8}left(d - frac{4}{5}right)^2 = 0 implies text{each squared term must be zero}]4. This necessitates:[a - frac{b}{2} = 0 implies a = frac{b}{2}][b - frac{2c}{3} = 0 implies b = frac{2c}{3}][c - frac{3d}{4} = 0 implies c = frac{3d}{4}][d - frac{4}{5} = 0 implies d = frac{4}{5}]5. Solve from the last equation upwards:[d = frac{4}{5}][c = frac{3 cdot frac{4}{5}}{4} = frac{3 cdot 4}{4 cdot 5} = frac{3}{5}][b = frac{2 cdot frac{3}{5}}{3} = frac{2 cdot 3}{3 cdot 5} = frac{2}{5}][a = frac{frac{2}{5}}{2} = frac{2}{5} cdot frac{1}{2} = frac{1}{5}] Conclusion:[boxed{a = frac{1}{5}, b = frac{2}{5}, c = frac{3}{5}, d = frac{4}{5}}]

🧠:Okay, let's try to figure out how many nights Xin Xin's family stayed at the seaside. Hmm, they arrived during the day on May 30th, right? And May has 31 days, so May 30th is followed by May 31st. Then they left during the day on June 4th. The question is asking for the number of nights they spent there.Wait, nights are different from days. If you check into a hotel on a certain day, the first night would be that night, right? So let's break it down day by day. Let's list the dates they were there.They arrived on May 30th during the day. So they checked in and probably stayed the night of May 30th. Then May 31st is the next day, and they would stay the night of May 31st. Then June 1st, night of June 1st. June 2nd, night of June 2nd. June 3rd, night of June 3rd. Then they left during the day on June 4th. So they didn't stay the night of June 4th because they left during the day.Let me count the nights: May 30th night, May 31st night, June 1st night, June 2nd night, June 3rd night. That's 5 nights? But wait, the options don't have 5. Wait, the options are A.3, B.4, C.5, D.6, E.7. So C is 5. But wait, maybe I'm overcounting. Let me check again.Wait, arrival on May 30th (daytime). So the first night is May 30th to 31st. Then May 31st to June 1st, that's the second night. June 1st to 2nd, third. June 2nd to 3rd, fourth. June 3rd to 4th, fifth. But they left on June 4th during the day, so they didn't stay the night of June 4th. So that's five nights? But the options include C.5. So why is the answer not C? Wait, maybe the problem is that check-out is on June 4th during the day. So depending on hotel check-out times, maybe the night before is counted. Let me think.Wait, typically, when you check out on a certain day, you don't pay for that night. For example, if you check in on the 1st and check out on the 2nd, you stay one night (the night of the 1st). So if they checked in on May 30th and checked out on June 4th, how many nights is that?From May 30 to May 31: night 1May 31 to June 1: night 2June 1 to 2: night 3June 2 to 3: night 4June 3 to 4: night 5Wait, but they checked out on June 4th during the day, so they didn't stay the night of June 4th. So that's 5 nights. But the options have C as 5. But maybe the answer is 5. But wait, maybe there's a mistake here.Wait, let's count the days they were there. From May 30 (arrival) to June 4 (departure). The duration in days is June 4 minus May 30. Since May has 31 days, May 30 to May 31 is 1 day, then June 1 to June 4 is 4 days. So total days is 1 + 4 = 5 days. But the number of nights is usually one less than the number of days if you're leaving on the same day as arrival. Wait, no. Wait, maybe the confusion is between days stayed and nights stayed.For example, if you arrive on day 1 and leave on day 2, you stayed one night. So the number of nights is the number of days minus one. Wait, but in this case, arrival is May 30, departure is June 4. Let's calculate the days from May 30 to June 4 inclusive? Wait, no. If you check in on May 30 and check out on June 4, how many days is that?May 30, 31, June 1, 2, 3, 4. That's 6 days. But since they check out on the 4th during the day, they don't stay the 4th night. So the number of nights is 5. But that contradicts the initial calculation. Wait, let's see: May 30 (check-in day) to June 4 (check-out day) is 5 nights. Because each full day after check-in adds a night. Wait, perhaps the confusion is in how hotels count nights. Usually, each night corresponds to the day you check in. So if you check in on May 30, the first night is May 30, then May 31, etc.But let's see: May 30 check-in. Nights spent would be:Night of May 30 (after check-in)Night of May 31Night of June 1Night of June 2Night of June 3Then check out on June 4, so no night on June 4. That's 5 nights. So answer C.5. But the options given include C.5. So why is this confusing? Maybe the answer is 5, but let me check again.Wait, perhaps the problem is that the family arrived on May 30 (daytime) and left on June 4 (daytime). So the total number of nights is from May 30 to June 3. Let's count:May 30-31: 1May 31-June 1: 2June 1-2:3June 2-3:4June 3-4:5Wait, but they left during the day on June 4, so they didn't stay the night of June 3-4. Wait, no, if they check out on June 4 during the day, then they did stay the night before, which is June 3-4. Because they would have slept there on June 3rd night and checked out the next day. So that's the fifth night. Therefore, total of 5 nights. But the options have C as 5. So why the answer is not C? Wait, maybe the answer is B.4. Wait, perhaps I'm overcounting.Wait, let's think differently. Let's list each day they stayed and count the nights.Check-in on May 30 (day). They stay the night of May 30 (so night 1).May 31: day, night (night 2).June 1: day, night (night 3).June 2: day, night (night 4).June 3: day, night (night 5).June 4: check out during the day, so no night.Wait, so that's 5 nights. So the answer should be C.5. But maybe the options have a different answer. Wait, but in the problem statement, the options are A.3, B.4, C.5, D.6, E.7. So C is 5, which is what I'm getting. But I need to make sure.Alternatively, maybe the problem is considering the departure day as not counting. Let's calculate the duration from May 30 to June 4.May 30 to May 31: 1 nightMay 31 to June 1: 1 nightJune 1 to June 2: 1 nightJune 2 to June 3: 1 nightJune 3 to June 4: 1 nightWait, that's 5 nights. But maybe the answer is 5. But let's see another approach. The number of nights is equal to the number of days they stayed minus 1 if they left on the same day. Wait, no. For example, if you arrive in the morning of day 1 and leave in the evening of day 2, that's 1 night. But if you leave in the morning of day 2, you only stayed 1 night (day 1 night). So maybe the formula is: nights = departure day - arrival day.But dates are tricky. Let's convert the dates to numbers. May 30 is day 30, May has 31 days, so June 4 is day 4 of June. So from May 30 to June 4:Total days from May 30 to June 4 inclusive is (31 - 30 +1) + 4 = 2 +4=6 days? Wait, no. Let me compute the days between May 30 and June 4, including both check-in and check-out days.From May 30 to June 4: May 30, 31, June 1,2,3,4. That's 6 days. But the number of nights would be 6 -1 =5? Because each full day is a night. So 6 days, 5 nights. So answer C.5. Therefore, the correct answer should be C.5.But maybe the problem is presented in a way that the answer is B.4. Where am I going wrong?Wait, maybe the confusion is that when you check out on June 4 during the day, you don't count that day as a night. Let's list each night:- Check-in May 30: night of May 30 (night 1)- Night of May 31 (night 2)- Night of June 1 (night 3)- Night of June 2 (night 4)- Night of June 3 (night 5)Then check out on June 4, so no night there. That's 5 nights. So answer C.5. But maybe the options are different. Wait, the problem says "arrived during the day on May 30" and "left during the day on June 4". So if they arrived on May 30, they would spend the night of May 30, 31, June 1, 2, 3. That's 5 nights. Therefore, the answer should be C.5.But maybe the test expects the answer to be B.4. Maybe a miscalculation. Wait, let's think again. Suppose they check in on May 30, check out on June 4. The number of nights is June 4 minus May 30 = 4 nights? Wait, no. Let's subtract the dates: June 4 is day 4 of June, May 30 is day 30 of May. Since May has 31 days, the total days between May 30 and June 4 is (31-30)+4=1+4=5 days. But again, the number of nights is days minus 1, so 4 nights. Wait, now I'm confused.Wait, let's do this step by step:From May 30 to May 31: 1 day, 1 nightMay 31 to June 1: 1 day, 1 nightJune 1 to June 2: 1 day, 1 nightJune 2 to June 3: 1 day, 1 nightJune 3 to June 4: 1 day, but they left during the day, so no night here.Wait, so total days from May 30 to June 4 (excluding June 4 night) is 5 days, but nights are 4. Wait, no. Each full day they stayed includes a night. Wait, if they arrived on May 30, then:Night 1: May 30-31Night 2: May 31-June 1Night 3: June 1-2Night 4: June 2-3Night 5: June 3-4But they check out on June 4 during the day, so they do stay the night of June 3-4, right? Because they have to sleep somewhere on June 3rd night, then check out the next day. So that's 5 nights. Therefore, answer C.5.I think the correct answer is C.5. Maybe the options are correct. But I need to check again. Let's count the nights one by one:1. May 30 night2. May 31 night3. June 1 night4. June 2 night5. June 3 nightThen leave on June 4 day. So that's 5 nights. So answer is C.5. But maybe there's a cultural difference or a different way hotels count nights. For example, in some countries, if you check in late, they might count the next day. But the problem states they arrived during the day, so they would definitely spend that night.Alternatively, maybe the problem is considering that the departure day is not counted, so from May 30 to June 4 is 5 days, hence 4 nights. But that doesn't make sense because each day after arrival adds a night. For example, arriving on day 1 and leaving on day 2 is 1 night. Day 1 to day 3 is 2 nights, etc.So from May 30 to June 4 is 5 days (May 30,31, June 1,2,3,4), but wait, that's 6 days. Wait, May 30 to June 4 inclusive is 6 days. So nights would be 6-1=5. Hence, answer C.5. Yes, that's correct.Therefore, the answer should be C.5.

✅:To determine how many nights Xin Xin's family spent at the seaside, we analyze their stay from arrival on May 30 to departure on June 4:1. Check-in: Arrived during the day on May 30. - First night: May 30–31.2. Subsequent nights: - Second night: May 31–June 1. - Third night: June 1–2. - Fourth night: June 2–3. - Fifth night: June 3–4.3. Check-out: Left during the day on June 4, so no night spent on June 4. Total nights = 5 (May 30–31, May 31–June 1, June 1–2, June 2–3, June 3–4). Answer: C. 5.

🔑:To determine how many nights Xin Xin's family stayed by the seaside, we need to understand the duration of their stay in detail.1. Determine the number of days stayed: - Xin Xin's family arrived on May 30th and they left on June 4th. - In May, there are 2 days remaining after May 30th, including May 30th itself (i.e., May 30th and May 31st). - In June, they stayed for 4 days (June 1st to June 4th, inclusive).2. Calculate the total number of nights: - To find the number of nights, we subtract one from the total number of days stayed. - The total number of days is from May 30th to June 4th inclusive. - Counting the days gives: [ text{May 30, 31} text{ (2 days)} + text{June 1, 2, 3, 4} text{ (4 days)} = 6 text{ days} ] - Hence, the number of nights is: [ text{Total nights} = 6 text{ days} - 1 = 5 text{ nights} ]3. Conclusion: The total number of nights Xin Xin's family stayed by the seaside is:[ boxed{C}]