💡Alright, so I've got this problem here where I need to fill in the empty boxes with numbers. The rules are that each box in the middle of the marked lines should contain the sum of the numbers from its end boxes. Also, the sums of the numbers from the boxes on both diagonals need to be the same. Hmm, okay, let me try to visualize this.First, I think it's a grid, maybe a 3x3 grid since it mentions rows and diagonals. Let me sketch it out in my mind. There are three rows and three columns. Some of the boxes are already filled, and some are empty. The middle boxes in each row should be the sum of the numbers at the ends of that row. Also, the diagonals from top-left to bottom-right and top-right to bottom-left should have the same sum.Let me assume the grid looks something like this:```a b cd e fg h i```From the problem, I think some of these are already filled, and some are empty. For example, maybe a, c, g, and i are given, and b, d, f, h, and e are empty. Or maybe some other combination. Since the problem mentions "each box in the middle of the marked lines," I think that refers to the middle boxes in each row and column, so b, e, and f in the second row, and maybe d and h in the middle of the columns.Wait, actually, the problem says "each box in the middle of the marked lines contains the sum of the numbers from its end boxes." So, if there are lines connecting the end boxes, the middle box on each line is the sum of the end boxes. So, for example, if there's a line from a to c, then b should be a + c. Similarly, if there's a line from g to i, then h should be g + i. And if there's a vertical line from a to g, then d should be a + g, and similarly for the other vertical line.Also, the sums of the numbers on both diagonals should be the same. So, the diagonal from a to e to i should sum to the same value as the diagonal from c to e to g.Okay, let's try to assign variables to the unknown boxes. Suppose the given numbers are a, c, g, and i, and the rest are unknown. Then, b = a + c, d = a + g, f = c + i, h = g + i, and e is the middle box which might be related to both diagonals.Wait, but if e is part of both diagonals, then the sum of the main diagonal (a + e + i) should equal the sum of the other diagonal (c + e + g). So, a + e + i = c + e + g, which simplifies to a + i = c + g. That's an important equation.Also, from the lines, we have b = a + c, d = a + g, f = c + i, and h = g + i.So, if I can express all the variables in terms of a, c, g, and i, and then use the diagonal condition to find a relationship between them, I can solve for the unknowns.But wait, in the problem statement, it seems like some numbers are already given, and I need to fill in the rest. Maybe a, c, g, and i are given, and I need to find b, d, e, f, and h.Let me think of an example. Suppose a = 4, c = 8, g = 10, and i = something. Then b = 4 + 8 = 12. If g = 10, then d = 4 + 10 = 14. If i is unknown, then f = 8 + i, and h = 10 + i. Then, the diagonal sums: a + e + i = 4 + e + i, and c + e + g = 8 + e + 10 = 18 + e. So, 4 + e + i = 18 + e, which simplifies to 4 + i = 18, so i = 14.Wait, but if i = 14, then f = 8 + 14 = 22, and h = 10 + 14 = 24. Then, what about e? Is e determined by something else? Maybe e is the sum of the middle boxes or something. Wait, no, e is part of both diagonals, but we've already used the diagonal condition to find i.Hmm, maybe I need to think differently. Perhaps e is also determined by the vertical or horizontal lines. If there's a vertical line from b to f, then e should be b + f? Or maybe e is the sum of the middle boxes? Wait, the problem says "each box in the middle of the marked lines contains the sum of the numbers from its end boxes." So, if there's a vertical line from b to f, then e should be b + f. Similarly, if there's a horizontal line through e, maybe e is the sum of d and h? Or maybe e is the sum of the middle boxes from the rows or columns.Wait, I'm getting confused. Let me try to clarify. Each box in the middle of the marked lines is the sum of its end boxes. So, if there's a horizontal line from a to c, then b = a + c. If there's a vertical line from a to g, then d = a + g. Similarly, if there's a vertical line from c to i, then f = c + i. And if there's a horizontal line from g to i, then h = g + i.Now, if there's a diagonal line from a to i, then e = a + i. Similarly, if there's a diagonal line from c to g, then e = c + g. But wait, earlier we had a + i = c + g from the diagonal sums being equal. So, e = a + i = c + g, which is consistent.So, in this case, e is determined by either a + i or c + g, and they must be equal because of the diagonal sums condition.Okay, so let's try to put this together. Suppose a = 4, c = 8, g = 10, and i is unknown. Then, b = 4 + 8 = 12. d = 4 + 10 = 14. f = 8 + i. h = 10 + i. e = 4 + i (or e = 8 + 10 = 18). Wait, if e = 4 + i and e = 18, then 4 + i = 18, so i = 14.So, i = 14. Then, f = 8 + 14 = 22, h = 10 + 14 = 24, and e = 18.So, the grid would look like:```4 12 814 18 2210 24 14```Wait, but in this case, the diagonals: 4 + 18 + 14 = 36, and 8 + 18 + 10 = 36. So, that works.But wait, in the third row, the middle box h = 24, which should be the sum of g and i, which is 10 + 14 = 24. That's correct.Similarly, in the second row, the middle box e = 18, which is the sum of d and f, which is 14 + 22 = 36. Wait, that's not matching. Wait, no, e is supposed to be the sum of the end boxes of the vertical line, which would be d and f? Or is it the sum of the end boxes of the horizontal line?Wait, I think I made a mistake. If e is in the middle of a vertical line from d to f, then e should be d + f. But in our case, d = 14, f = 22, so e should be 14 + 22 = 36, but earlier we had e = 18. That's a contradiction.Hmm, so where did I go wrong? Let's go back.We have:- b = a + c = 4 + 8 = 12- d = a + g = 4 + 10 = 14- f = c + i = 8 + i- h = g + i = 10 + i- e = a + i = 4 + i (from the diagonal)- Also, from the other diagonal, e = c + g = 8 + 10 = 18So, 4 + i = 18 => i = 14Then, f = 8 + 14 = 22, h = 10 + 14 = 24Now, if e is supposed to be the sum of d and f, then e = 14 + 22 = 36, but we have e = 18. That's a problem.Wait, maybe e is not the sum of d and f, but rather, e is the sum of the end boxes of the vertical line through e. If the vertical line is from d to f, then e = d + f. But in our case, that would mean e = 14 + 22 = 36, which contradicts e = 18.Alternatively, maybe e is the sum of the end boxes of the horizontal line through e, which would be d and f. But again, that would be 14 + 22 = 36, not 18.Wait, perhaps I misunderstood the problem. Maybe the middle box is the sum of the end boxes of the line it's on, not necessarily the entire line. So, if e is in the middle of a vertical line from d to f, then e = d + f. But if e is also in the middle of a horizontal line from d to f, then e = d + f. But in our case, e is part of both diagonals, so e = a + i and e = c + g.But if e is also part of a vertical or horizontal line, then it should also be the sum of the end boxes of that line. So, if e is in the middle of a vertical line from d to f, then e = d + f. But we have e = 18, and d + f = 14 + 22 = 36, which is not equal to 18. So, that's a contradiction.Therefore, my initial assumption must be wrong. Maybe the lines are not all present. Maybe only some lines are marked, and e is not part of a vertical or horizontal line, only the diagonals.Wait, the problem says "each box in the middle of the marked lines contains the sum of the numbers from its end boxes." So, if e is in the middle of a marked line, then it should be the sum of the end boxes. But if e is only part of the diagonals, and not part of any vertical or horizontal line, then e is only determined by the diagonals.So, maybe in this problem, the only marked lines are the two diagonals, and the horizontal lines for the first and third rows, and the vertical lines for the first and third columns. So, e is only part of the diagonals, not part of any vertical or horizontal line. Therefore, e is determined by the diagonals, and the vertical and horizontal lines only affect the other middle boxes.In that case, e is determined by the diagonals, and the other middle boxes are determined by their respective lines.So, in our example:- b = a + c = 4 + 8 = 12- d = a + g = 4 + 10 = 14- f = c + i = 8 + i- h = g + i = 10 + i- e = a + i = 4 + i (from the main diagonal)- e = c + g = 8 + 10 = 18 (from the other diagonal)So, 4 + i = 18 => i = 14Then, f = 8 + 14 = 22, h = 10 + 14 = 24Now, the grid is:```4 12 814 18 2210 24 14```Now, checking the sums:- First row: 4 + 12 + 8 = 24- Second row: 14 + 18 + 22 = 54- Third row: 10 + 24 + 14 = 48- First column: 4 + 14 + 10 = 28- Second column: 12 + 18 + 24 = 54- Third column: 8 + 22 + 14 = 44- Main diagonal: 4 + 18 + 14 = 36- Other diagonal: 8 + 18 + 10 = 36So, the diagonals sum to 36, which is good. But the rows and columns don't necessarily have to sum to anything specific, unless specified. So, maybe this is acceptable.But wait, in the second row, the middle box e = 18, which is the sum of the end boxes of the diagonal, but not the sum of the end boxes of the row or column. Since e is only part of the diagonals, it's okay.So, in this case, the filled grid would be:```4 12 814 18 2210 24 14```But let me check if all the middle boxes are the sum of their end boxes:- b = 4 + 8 = 12 ✔️- d = 4 + 10 = 14 ✔️- f = 8 + 14 = 22 ✔️- h = 10 + 14 = 24 ✔️- e = 4 + 14 = 18 ✔️ (from main diagonal)- e = 8 + 10 = 18 ✔️ (from other diagonal)So, all conditions are satisfied.But wait, in the second row, the middle box e = 18, but the end boxes are 14 and 22. If there was a horizontal line from d to f, then e should be 14 + 22 = 36, but e is 18. So, that's a problem. Therefore, my assumption that the only marked lines are the diagonals and the first and third rows and columns might be incorrect.Perhaps, in reality, there are also horizontal and vertical lines through the second row and column, meaning that e should also be the sum of d and f, and the sum of b and h.So, let's re-examine.If e is the sum of d and f, then e = d + f = 14 + 22 = 36But earlier, we have e = 18 from the diagonals. So, 36 ≠ 18, which is a contradiction.Therefore, my initial assumption that the only marked lines are the diagonals and the first and third rows and columns must be wrong. Perhaps, the marked lines include all rows and columns, meaning that e is the sum of d and f, and also the sum of b and h.So, let's adjust our approach.Given:- b = a + c- d = a + g- f = c + i- h = g + i- e = d + f (from the vertical line)- e = b + h (from the horizontal line)- Also, the diagonals: a + e + i = c + e + gSo, from the diagonals: a + e + i = c + e + g => a + i = c + gFrom e = d + f and e = b + h, we have:d + f = b + hSubstituting d = a + g, f = c + i, b = a + c, h = g + i:(a + g) + (c + i) = (a + c) + (g + i)Simplify:a + g + c + i = a + c + g + iWhich is always true, so no new information.So, the key equations are:1. b = a + c2. d = a + g3. f = c + i4. h = g + i5. a + i = c + g6. e = d + f = b + hSo, with these equations, we can solve for the variables.Let's assign values to some variables to make it concrete. Suppose a = 4, c = 8, g = 10, and i is unknown.Then:b = 4 + 8 = 12d = 4 + 10 = 14f = 8 + ih = 10 + iFrom equation 5: 4 + i = 8 + 10 => 4 + i = 18 => i = 14So, i = 14Then:f = 8 + 14 = 22h = 10 + 14 = 24Now, e = d + f = 14 + 22 = 36Also, e = b + h = 12 + 24 = 36So, e = 36Now, let's check the diagonals:Main diagonal: a + e + i = 4 + 36 + 14 = 54Other diagonal: c + e + g = 8 + 36 + 10 = 54So, diagonals are equal.Now, let's check all the middle boxes:- b = a + c = 4 + 8 = 12 ✔️- d = a + g = 4 + 10 = 14 ✔️- f = c + i = 8 + 14 = 22 ✔️- h = g + i = 10 + 14 = 24 ✔️- e = d + f = 14 + 22 = 36 ✔️- e = b + h = 12 + 24 = 36 ✔️So, all conditions are satisfied.Therefore, the filled grid is:```4 12 814 36 2210 24 14```But wait, in the second row, the middle box e = 36, which is the sum of d and f, which is 14 + 22 = 36. That's correct.Similarly, in the second column, the middle box e = 36, which is the sum of b and h, which is 12 + 24 = 36. That's also correct.So, this grid satisfies all the given conditions.But let me double-check the diagonals:- Main diagonal: 4 + 36 + 14 = 54- Other diagonal: 8 + 36 + 10 = 54Yes, they are equal.Okay, so I think this is the correct solution.

💡Okay, so I have this function f(x) = mx² - mx - 1, and I need to solve two problems related to it. Let me take them one by one.First, I need to discuss the solution set of the inequality f(x) > 1 - 2x. Hmm, okay. So, I guess I should start by writing down the inequality:mx² - mx - 1 > 1 - 2x.Alright, let me rearrange this inequality to bring all terms to one side. Subtracting 1 and adding 2x to both sides gives:mx² - mx - 1 - 1 + 2x > 0.Simplifying that, I get:mx² + (-m + 2)x - 2 > 0.So, the inequality is mx² + (2 - m)x - 2 > 0. Now, this is a quadratic inequality. The solution set will depend on the value of m because the coefficient of x² is m, which can change the direction of the parabola or even make it a linear function if m is zero.Let me consider different cases for m.Case 1: m = 0.If m is zero, the quadratic term disappears, and the inequality becomes:0x² + (2 - 0)x - 2 > 0 => 2x - 2 > 0.Solving this, 2x - 2 > 0 => 2x > 2 => x > 1.So, when m = 0, the solution set is x > 1.Case 2: m ≠ 0.Now, if m is not zero, we have a quadratic inequality. To solve this, I can factor the quadratic expression if possible.Let me try factoring mx² + (2 - m)x - 2.Looking for two numbers that multiply to (m * -2) = -2m and add up to (2 - m). Hmm, that might be tricky. Alternatively, I can use the quadratic formula to find the roots.The quadratic equation is mx² + (2 - m)x - 2 = 0.Using the quadratic formula, x = [-b ± √(b² - 4ac)] / (2a), where a = m, b = (2 - m), and c = -2.Calculating the discriminant:D = b² - 4ac = (2 - m)² - 4 * m * (-2) = (4 - 4m + m²) + 8m = 4 + 4m + m².So, D = m² + 4m + 4 = (m + 2)².That's a perfect square, which is good because it means the roots are real and equal if D = 0, but since D is a square, it's always non-negative.So, the roots are:x = [-(2 - m) ± √(m + 2)²] / (2m) = [m - 2 ± |m + 2|] / (2m).Wait, since √(m + 2)² is |m + 2|, which is equal to m + 2 if m + 2 ≥ 0, i.e., m ≥ -2, and -(m + 2) if m < -2.So, let's consider two subcases based on the value of m.Subcase 2a: m > 0.If m is positive, the quadratic opens upwards. The roots are:x = [m - 2 + (m + 2)] / (2m) and x = [m - 2 - (m + 2)] / (2m).Simplifying the first root:[m - 2 + m + 2] / (2m) = (2m) / (2m) = 1.Simplifying the second root:[m - 2 - m - 2] / (2m) = (-4) / (2m) = -2/m.So, the roots are x = 1 and x = -2/m.Since m > 0, -2/m is negative. So, the quadratic crosses the x-axis at x = -2/m and x = 1. Since it opens upwards, the inequality mx² + (2 - m)x - 2 > 0 is satisfied when x < -2/m or x > 1.Therefore, for m > 0, the solution set is x < -2/m or x > 1.Subcase 2b: m < 0.If m is negative, the quadratic opens downwards. Let's find the roots again:x = [m - 2 + |m + 2|] / (2m) and x = [m - 2 - |m + 2|] / (2m).Again, considering m < 0, let's see when m + 2 is positive or negative.If m + 2 ≥ 0, i.e., m ≥ -2, then |m + 2| = m + 2.If m + 2 < 0, i.e., m < -2, then |m + 2| = -(m + 2).So, Subsubcase 2b1: -2 ≤ m < 0.Then, |m + 2| = m + 2.So, the roots are:x = [m - 2 + m + 2] / (2m) = (2m) / (2m) = 1.x = [m - 2 - (m + 2)] / (2m) = (m - 2 - m - 2) / (2m) = (-4) / (2m) = -2/m.Since m is negative, -2/m is positive. So, the roots are x = 1 and x = -2/m, with -2/m > 1 because m is negative.Since the quadratic opens downward, the inequality mx² + (2 - m)x - 2 > 0 is satisfied between the roots. So, 1 < x < -2/m.Subsubcase 2b2: m < -2.Here, |m + 2| = -(m + 2).So, the roots are:x = [m - 2 + (-m - 2)] / (2m) = (m - 2 - m - 2) / (2m) = (-4) / (2m) = -2/m.x = [m - 2 - (-m - 2)] / (2m) = (m - 2 + m + 2) / (2m) = (2m) / (2m) = 1.So, the roots are x = 1 and x = -2/m. But since m < -2, -2/m is positive and less than 1 because m is negative and less than -2, so -2/m is between 0 and 1.Therefore, the quadratic opens downward, and the inequality is satisfied between the roots. So, -2/m < x < 1.Subsubcase 2b3: m = -2.At m = -2, the discriminant D = (m + 2)² = 0, so there's only one root.x = [m - 2 ± 0] / (2m) = (m - 2) / (2m).Plugging m = -2:x = (-2 - 2) / (2*(-2)) = (-4)/(-4) = 1.So, the quadratic becomes (-2)x² + (2 - (-2))x - 2 = -2x² + 4x - 2.Which factors as -2(x² - 2x + 1) = -2(x - 1)².So, the inequality is -2(x - 1)² > 0.But since (x - 1)² is always non-negative, multiplying by -2 makes it non-positive. So, -2(x - 1)² is always ≤ 0, and it's equal to zero at x = 1.Therefore, the inequality -2(x - 1)² > 0 has no solution. So, the solution set is empty.So, summarizing Case 2:- If m > 0: x < -2/m or x > 1.- If -2 < m < 0: 1 < x < -2/m.- If m < -2: -2/m < x < 1.- If m = -2: No solution.Alright, that was the first part. Now, moving on to the second problem.If f(x) < -m + 4 holds for any x ∈ [1, 3], find the range of the parameter m.So, f(x) = mx² - mx - 1 < -m + 4 for all x in [1, 3].Let me rewrite this inequality:mx² - mx - 1 < -m + 4.Bring all terms to one side:mx² - mx - 1 + m - 4 < 0.Simplify:mx² - mx + m - 5 < 0.Factor m from the first three terms:m(x² - x + 1) - 5 < 0.So, m(x² - x + 1) < 5.Therefore, m < 5 / (x² - x + 1).But this must hold for all x in [1, 3]. So, m must be less than the minimum value of 5 / (x² - x + 1) over x ∈ [1, 3].Wait, actually, since the inequality is m < 5 / (x² - x + 1) for all x in [1, 3], m must be less than the minimum of 5 / (x² - x + 1) on [1, 3].Alternatively, m must be less than or equal to the infimum of 5 / (x² - x + 1) on [1, 3].So, to find the range of m, I need to find the minimum value of 5 / (x² - x + 1) on [1, 3].First, let's analyze the function g(x) = x² - x + 1 on [1, 3].Compute its derivative to find extrema.g'(x) = 2x - 1.Set g'(x) = 0: 2x - 1 = 0 => x = 1/2.But 1/2 is not in [1, 3], so the extrema on [1, 3] occur at the endpoints.Compute g(1) = 1 - 1 + 1 = 1.g(3) = 9 - 3 + 1 = 7.So, on [1, 3], g(x) increases from 1 to 7.Therefore, the minimum of g(x) on [1, 3] is 1, and the maximum is 7.Thus, 1 ≤ g(x) ≤ 7 on [1, 3].Therefore, 5 / g(x) is between 5/7 and 5.So, 5/7 ≤ 5 / g(x) ≤ 5.But since m must be less than 5 / g(x) for all x in [1, 3], m must be less than the minimum of 5 / g(x), which is 5/7.Therefore, m < 5/7.Wait, but let me double-check.If m < 5 / g(x) for all x in [1, 3], then m must be less than the smallest value of 5 / g(x) on [1, 3].Since g(x) is minimized at x = 1 with g(1) = 1, 5 / g(x) is maximized at x = 1 with 5/1 = 5, and minimized at x = 3 with 5/7.Wait, no. Wait, if g(x) is increasing, then 5 / g(x) is decreasing.So, at x = 1, 5 / g(x) = 5 / 1 = 5.At x = 3, 5 / g(x) = 5 / 7.So, 5 / g(x) decreases from 5 to 5/7 as x increases from 1 to 3.Therefore, the minimum value of 5 / g(x) on [1, 3] is 5/7, and the maximum is 5.Thus, for m to be less than 5 / g(x) for all x in [1, 3], m must be less than or equal to the minimum of 5 / g(x), which is 5/7.But since the inequality is strict (m < 5 / g(x)), m must be strictly less than 5/7.Therefore, the range of m is m < 5/7.Wait, but let me think again.If m is less than 5 / g(x) for all x in [1, 3], then m must be less than the infimum of 5 / g(x) over [1, 3].Since 5 / g(x) attains its minimum at x = 3, which is 5/7, m must be less than 5/7.Yes, that makes sense.So, the range of m is m < 5/7.But just to be thorough, let me test m = 5/7.If m = 5/7, then f(x) = (5/7)x² - (5/7)x - 1.We need to check if f(x) < -m + 4 for all x in [1, 3].Compute f(x) + m - 4:(5/7)x² - (5/7)x - 1 + 5/7 - 4.Simplify:(5/7)x² - (5/7)x - 1 + 5/7 - 4 = (5/7)x² - (5/7)x - (1 - 5/7 + 4).Compute constants:1 - 5/7 = 2/7, so -2/7 + 4 = -2/7 + 28/7 = 26/7.Thus, f(x) + m - 4 = (5/7)x² - (5/7)x - 26/7.We need this to be < 0 for all x in [1, 3].But at x = 1:(5/7)(1) - (5/7)(1) - 26/7 = 0 - 26/7 = -26/7 < 0.At x = 3:(5/7)(9) - (5/7)(3) - 26/7 = (45/7 - 15/7) - 26/7 = (30/7) - 26/7 = 4/7 > 0.So, at x = 3, f(x) + m - 4 = 4/7 > 0, which violates the inequality.Therefore, m cannot be equal to 5/7. Hence, m must be strictly less than 5/7.Therefore, the range of m is m < 5/7.So, summarizing:1. The solution set of f(x) > 1 - 2x depends on m: - If m = 0: x > 1. - If m > 0: x < -2/m or x > 1. - If -2 < m < 0: 1 < x < -2/m. - If m < -2: -2/m < x < 1. - If m = -2: No solution.2. The range of m such that f(x) < -m + 4 for all x in [1, 3] is m < 5/7.I think that's it. Let me just make sure I didn't make any calculation mistakes.In the first part, when factoring, I considered different cases for m and found the roots correctly. The quadratic formula was applied properly, and the discriminant was calculated correctly as a perfect square, leading to the roots 1 and -2/m. Then, depending on the sign of m, I determined the intervals where the quadratic is positive or negative.In the second part, I correctly transformed the inequality into m < 5 / (x² - x + 1) and analyzed the function g(x) = x² - x + 1 on [1, 3]. Since g(x) is increasing on this interval, its minimum is at x = 1 and maximum at x = 3. Therefore, 5 / g(x) is decreasing, with maximum at x = 1 and minimum at x = 3. Thus, m must be less than the minimum value of 5 / g(x), which is 5/7.Yes, everything seems consistent. I think I'm confident with these results.

💡Okay, so I have this problem about an ellipse and then I need to find the equation of a hyperbola. Let me try to break it down step by step.First, the ellipse is given by the equation (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a > b > 0). The problem says that the two foci and one endpoint of the minor axis form an equilateral triangle. Hmm, okay, so I need to visualize this. The ellipse has two foci on the major axis, which is the x-axis in this case because the major term is under (x^2). The minor axis is along the y-axis, so one endpoint of the minor axis would be at ((0, b)).Now, the two foci are located at ((pm c, 0)), where (c) is the distance from the center to each focus. For an ellipse, we know that (c^2 = a^2 - b^2). So, (c = sqrt{a^2 - b^2}).The problem states that the two foci and one endpoint of the minor axis form an equilateral triangle. So, the three points are ((c, 0)), ((-c, 0)), and ((0, b)). In an equilateral triangle, all sides are equal. So, the distance between each pair of these points should be the same.Let me calculate the distances:1. Distance between ((c, 0)) and ((-c, 0)): This is straightforward. It's just the distance along the x-axis, which is (2c).2. Distance between ((c, 0)) and ((0, b)): Using the distance formula, this is (sqrt{(c - 0)^2 + (0 - b)^2} = sqrt{c^2 + b^2}).3. Distance between ((-c, 0)) and ((0, b)): Similarly, this is (sqrt{(-c - 0)^2 + (0 - b)^2} = sqrt{c^2 + b^2}).So, we have two sides of length (sqrt{c^2 + b^2}) and one side of length (2c). Since it's an equilateral triangle, all sides must be equal. Therefore, (2c = sqrt{c^2 + b^2}).Let me square both sides to eliminate the square root:[(2c)^2 = (sqrt{c^2 + b^2})^2 4c^2 = c^2 + b^2 4c^2 - c^2 = b^2 3c^2 = b^2 b = csqrt{3}]Okay, so (b = csqrt{3}). That's one relationship between (b) and (c).Next, the problem says that the maximum distance from a focus to a point on the ellipse is (3sqrt{3}). I need to figure out what this maximum distance is.In an ellipse, the maximum distance from a focus to a point on the ellipse occurs at the farthest vertex from that focus. Since the foci are on the x-axis, the farthest point would be the vertex on the opposite side of the ellipse. The vertices on the major axis are at ((pm a, 0)). So, the distance from a focus at ((c, 0)) to the vertex at ((-a, 0)) is (a + c). Similarly, the distance from ((-c, 0)) to ((a, 0)) is also (a + c).Therefore, the maximum distance is (a + c = 3sqrt{3}).So, we have another equation: (a + c = 3sqrt{3}).Now, let me summarize what I have so far:1. (b = csqrt{3})2. (a + c = 3sqrt{3})3. For an ellipse, (c^2 = a^2 - b^2)I can substitute (b) from the first equation into the third equation.So, substituting (b = csqrt{3}) into (c^2 = a^2 - b^2):[c^2 = a^2 - (csqrt{3})^2 c^2 = a^2 - 3c^2 c^2 + 3c^2 = a^2 4c^2 = a^2 a = 2c]So, (a = 2c). Now, substitute this into the second equation (a + c = 3sqrt{3}):[2c + c = 3sqrt{3} 3c = 3sqrt{3} c = sqrt{3}]So, (c = sqrt{3}). Then, (a = 2c = 2sqrt{3}), and (b = csqrt{3} = sqrt{3} times sqrt{3} = 3).So, now I have (a = 2sqrt{3}), (b = 3), and (c = sqrt{3}).But wait, the problem asks for the standard equation of a hyperbola with real semi-axis (a) and imaginary semi-axis (b), where the foci lie on the y-axis.Hmm, okay, so for a hyperbola, the standard form when the foci are on the y-axis is (frac{y^2}{A^2} - frac{x^2}{B^2} = 1), where (A) is the real semi-axis and (B) is the imaginary semi-axis.But in the problem, it says "real semi-axis (a)" and "imaginary semi-axis (b)". So, does that mean (A = a) and (B = b)?Wait, but in hyperbola terminology, the real axis is the transverse axis, and the imaginary axis is the conjugate axis. So, if the foci are on the y-axis, the transverse axis is the y-axis, so the real semi-axis is along the y-axis, which would correspond to (A), and the imaginary semi-axis is along the x-axis, which would correspond to (B).But the problem says "real semi-axis (a)" and "imaginary semi-axis (b)", so I think that means (A = a) and (B = b).Therefore, the standard equation would be (frac{y^2}{a^2} - frac{x^2}{b^2} = 1).But wait, in our case, (a = 2sqrt{3}), so (a^2 = (2sqrt{3})^2 = 4 times 3 = 12). And (b = 3), so (b^2 = 9).Therefore, the equation of the hyperbola is (frac{y^2}{12} - frac{x^2}{9} = 1).Let me just double-check my steps to make sure I didn't make a mistake.1. The ellipse equation is given, and the two foci and one endpoint of the minor axis form an equilateral triangle. I found that (b = csqrt{3}).2. The maximum distance from a focus to a point on the ellipse is (3sqrt{3}), which is (a + c). So, (a + c = 3sqrt{3}).3. Using the relationship (c^2 = a^2 - b^2), I substituted (b = csqrt{3}) and found (a = 2c).4. Substituted (a = 2c) into (a + c = 3sqrt{3}) to find (c = sqrt{3}), hence (a = 2sqrt{3}) and (b = 3).5. Then, for the hyperbola with real semi-axis (a) and imaginary semi-axis (b), with foci on the y-axis, the equation is (frac{y^2}{a^2} - frac{x^2}{b^2} = 1), which becomes (frac{y^2}{12} - frac{x^2}{9} = 1).I think that's correct. Let me just verify the hyperbola part.For a hyperbola, the distance between the foci is (2C), where (C^2 = A^2 + B^2). In our case, the foci are on the y-axis, so (C) is the distance from the center to each focus along the y-axis.Given (A = a = 2sqrt{3}) and (B = b = 3), then (C^2 = (2sqrt{3})^2 + 3^2 = 12 + 9 = 21), so (C = sqrt{21}). Therefore, the foci are at ((0, pm sqrt{21})). But the problem didn't specify the foci's positions, just that they lie on the y-axis, so I think that's fine.Therefore, the equation I found should be correct.**Final Answer**The standard equation of the hyperbola is boxed{dfrac{y^2}{12} - dfrac{x^2}{9} = 1}.

💡Okay, so I need to find the coefficient of (x^7) in the expansion of ((1 + 2x - x^2)^4). Hmm, that seems a bit tricky, but I think I can figure it out step by step.First, I remember that when you have a polynomial raised to a power, you can expand it using the multinomial theorem. The multinomial theorem generalizes the binomial theorem to polynomials with more than two terms. In this case, our polynomial has three terms: 1, (2x), and (-x^2).The multinomial theorem says that:[(a + b + c)^n = sum_{k_1 + k_2 + k_3 = n} frac{n!}{k_1! k_2! k_3!} a^{k_1} b^{k_2} c^{k_3}]So in our case, (a = 1), (b = 2x), and (c = -x^2), and (n = 4). We need to find the terms where the power of (x) is 7.Let me denote the exponents of (a), (b), and (c) as (k_1), (k_2), and (k_3) respectively. Since (a = 1), any power of (a) is just 1, so we don't have to worry about that. The important part is the exponents of (x) from (b) and (c).Each term in the expansion will look like:[frac{4!}{k_1! k_2! k_3!} cdot 1^{k_1} cdot (2x)^{k_2} cdot (-x^2)^{k_3}]Simplifying this, we get:[frac{4!}{k_1! k_2! k_3!} cdot 2^{k_2} cdot (-1)^{k_3} cdot x^{k_2 + 2k_3}]We need the exponent of (x) to be 7, so:[k_2 + 2k_3 = 7]Also, since the total number of terms is 4, we have:[k_1 + k_2 + k_3 = 4]So now we have a system of equations:1. (k_2 + 2k_3 = 7)2. (k_1 + k_2 + k_3 = 4)I need to find non-negative integers (k_1), (k_2), and (k_3) that satisfy both equations.Let me try to solve these equations. From the second equation, (k_1 = 4 - k_2 - k_3). Since (k_1) must be a non-negative integer, (4 - k_2 - k_3 geq 0).From the first equation, (k_2 = 7 - 2k_3). Substituting this into the inequality:[4 - (7 - 2k_3) - k_3 geq 0]Simplify:[4 - 7 + 2k_3 - k_3 geq 0][-3 + k_3 geq 0][k_3 geq 3]Since (k_3) must be an integer, the smallest possible value is 3.Let me check (k_3 = 3):- Then (k_2 = 7 - 2*3 = 1)- Then (k_1 = 4 - 1 - 3 = 0)So one solution is (k_1 = 0), (k_2 = 1), (k_3 = 3).Let me check if there are any other solutions. If (k_3 = 4), then (k_2 = 7 - 2*4 = -1), which is negative, so that's not possible. Similarly, higher values of (k_3) will make (k_2) even more negative. So the only solution is (k_1 = 0), (k_2 = 1), (k_3 = 3).Now, let's compute the coefficient for this term.The multinomial coefficient is:[frac{4!}{0! 1! 3!} = frac{24}{1 * 1 * 6} = 4]Then, the coefficient from (b) is (2^{k_2} = 2^1 = 2), and from (c) is ((-1)^{k_3} = (-1)^3 = -1).Multiplying these together:[4 * 2 * (-1) = -8]So the coefficient of (x^7) is (-8).Wait, let me double-check in case I missed another term. Are there any other combinations of (k_2) and (k_3) that could give (x^7)?Earlier, I saw that (k_3) must be at least 3, and (k_3 = 3) gives a valid solution. (k_3 = 4) gives a negative (k_2), which isn't allowed. So there are no other solutions. Therefore, the only term contributing to (x^7) is the one we found with coefficient (-8).So, I think the answer is (-8), which corresponds to option A.**Final Answer**boxed{A}

💡Okay, so I have this problem here about a right rectangular prism, and it says that three of its faces have areas of 56, 63, and 72 square units. I need to find the volume of the prism and express the answer to the nearest whole number. Hmm, let me think about how to approach this.First, I remember that a right rectangular prism has three dimensions: length, width, and height. Let's call them ( a ), ( b ), and ( c ) respectively. The areas of the three adjacent faces would then be ( ab ), ( bc ), and ( ac ). So, according to the problem, these areas are 56, 63, and 72. But I'm not sure which area corresponds to which pair of dimensions. Maybe it doesn't matter because multiplication is commutative, so the order might not affect the final result. I'll keep that in mind.The goal is to find the volume, which is ( abc ). I know that if I can find the values of ( a ), ( b ), and ( c ), I can multiply them together to get the volume. But since I only have the areas of the faces, I need a way to relate these areas to the volume.I recall that if I multiply the three areas together, I get ( (ab)(bc)(ac) ), which simplifies to ( a^2b^2c^2 ). That's because each dimension is multiplied twice: once for each face it appears on. So, ( (ab)(bc)(ac) = a^2b^2c^2 ). Let me write that down:[(ab)(bc)(ac) = 56 times 63 times 72]So, ( a^2b^2c^2 = 56 times 63 times 72 ). To find ( abc ), which is the volume, I need to take the square root of both sides. That makes sense because ( (abc)^2 = a^2b^2c^2 ), so taking the square root gives me ( abc ).Alright, let's compute ( 56 times 63 times 72 ). Hmm, that's a bit of a large multiplication. Maybe I can break it down step by step.First, multiply 56 and 63:[56 times 63]Let me calculate that. 56 times 60 is 3360, and 56 times 3 is 168. Adding those together gives 3360 + 168 = 3528. So, 56 times 63 is 3528.Now, I need to multiply that result by 72:[3528 times 72]This is a larger multiplication. Maybe I can break it down further. Let's do 3528 times 70 and 3528 times 2, then add them together.First, 3528 times 70:3528 × 70 = 3528 × 7 × 103528 × 7: Let's compute that.3528 × 7:7 × 3000 = 210007 × 500 = 35007 × 28 = 196Adding those together: 21000 + 3500 = 24500, then 24500 + 196 = 24696So, 3528 × 70 = 24696 × 10 = 246960Now, 3528 × 2 = 7056Adding 246960 and 7056:246960 + 7056 = 254,016So, 56 × 63 × 72 = 254,016Therefore, ( a^2b^2c^2 = 254,016 )To find ( abc ), take the square root of 254,016.Let me compute that. Hmm, square root of 254,016.I know that 500 squared is 250,000, so sqrt(250,000) = 500.So, sqrt(254,016) is a bit more than 500.Let me see how much more.Compute 504 squared:504 × 504Let me compute 500 × 500 = 250,000Then, 500 × 4 = 20004 × 500 = 20004 × 4 = 16So, (500 + 4)^2 = 500^2 + 2×500×4 + 4^2 = 250,000 + 4000 + 16 = 254,016Wow, that's exactly the number we have.So, sqrt(254,016) = 504Therefore, ( abc = 504 )So, the volume of the prism is 504 cubic units.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, 56 × 63: 56 × 60 = 3360, 56 × 3 = 168, total 3528. That seems correct.Then, 3528 × 72: I broke it into 3528 × 70 and 3528 × 2.3528 × 70: 3528 × 7 = 24,696, then ×10 is 246,960. Correct.3528 × 2 = 7,056. Correct.Adding 246,960 + 7,056: 246,960 + 7,000 = 253,960, then +56 = 254,016. Correct.Then, sqrt(254,016): I thought of 504 squared, which is 254,016. Correct.So, yes, the volume is 504 cubic units.I think that's the answer. It all checks out. I didn't make any calculation errors that I can see.**Final Answer**The volume of the prism is boxed{504} cubic units.

💡Okay, so I have this problem about a convex 1000-sided polygon with 500 points inside it, and I need to figure out how many triangles are formed when I triangulate the polygon. All the points, both the 1000 on the polygon and the 500 inside, are vertices of the triangles, and there are no other vertices. Hmm, that sounds a bit complicated, but let me try to break it down.First, I remember that triangulation means dividing a polygon into triangles such that no two triangles overlap and all the original vertices are used. For a convex polygon, the number of triangles formed by triangulation is always (n - 2), where n is the number of sides. So, for a convex polygon with 1000 sides, the number of triangles would be 1000 - 2 = 998 triangles. But wait, that's just for the polygon itself without any interior points.Now, in this problem, there are 500 additional points inside the polygon. These points are not on the edges or vertices of the polygon, and no three of them are collinear. So, each of these points will affect the triangulation. I think each interior point will add more triangles because the triangulation has to account for these points.I recall that when you have a polygon with interior points, the number of triangles increases. There's a formula for this, but I can't remember exactly what it is. Let me think. I think it's related to Euler's formula for planar graphs, which is V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. In this case, the faces would be the triangles plus the outer face.So, let's try applying Euler's formula here. The total number of vertices V is 1000 (from the polygon) plus 500 (interior points), so V = 1500. The number of faces F would be the number of triangles plus 1 (the outer face). Let's denote the number of triangles as T, so F = T + 1.Now, we need to find the number of edges E. Each triangle has 3 edges, but each edge is shared by two triangles, except for the edges on the boundary of the polygon, which are only part of one triangle. So, if we let B be the number of boundary edges, which is 1000, then the total number of edges can be expressed as E = (3T + B)/2. Wait, let me make sure that's correct. Each triangle contributes 3 edges, but each interior edge is shared by two triangles, so the total number of edges is (3T + B)/2.So, plugging into Euler's formula: V - E + F = 2. Substituting the values we have:1500 - [(3T + 1000)/2] + (T + 1) = 2Let me simplify this equation step by step.First, expand the terms:1500 - (3T + 1000)/2 + T + 1 = 2Combine like terms:1500 + 1 - (3T + 1000)/2 + T = 21501 - (3T + 1000)/2 + T = 2Now, let's get rid of the fraction by multiplying the entire equation by 2:2*1501 - (3T + 1000) + 2T = 2*23002 - 3T - 1000 + 2T = 4Combine like terms:(3002 - 1000) + (-3T + 2T) = 42002 - T = 4Now, solve for T:-T = 4 - 2002-T = -1998Multiply both sides by -1:T = 1998So, according to this, the number of triangles formed by the triangulation is 1998.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from Euler's formula:V - E + F = 2V = 1500F = T + 1E = (3T + B)/2, where B = 1000So, plugging in:1500 - [(3T + 1000)/2] + (T + 1) = 2Yes, that's correct.Then, simplifying:1500 + 1 - (3T + 1000)/2 + T = 21501 - (3T + 1000)/2 + T = 2Multiply by 2:3002 - 3T - 1000 + 2T = 42002 - T = 4T = 1998Yes, that seems consistent. So, I think the number of triangles is indeed 1998.Alternatively, I remember another way to think about this. For a convex polygon with n vertices and k interior points, the number of triangles in a triangulation is n + 2k - 2. Let me check if that formula applies here.In this case, n = 1000 and k = 500.So, plugging into the formula:Number of triangles = 1000 + 2*500 - 2 = 1000 + 1000 - 2 = 1998Yes, that's the same result as before. So, that confirms it.Therefore, the number of triangles formed by the triangulation is 1998.**Final Answer**The number of triangles formed by the triangulation is boxed{1998}.