Datasets:

dvtiendat
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hypergraph_operations_openthoughts_30k

row_index int64 0 29.4k	hypergraph dict	attempts int64 1 3	solution_approach dict
0	{ "problem": "Given real numbers \$ a, b, c \$ and a positive number \$ \\lambda \$ such that the polynomial \$ f(x) = x^3 + a x^2 + b x + c \$ has three real roots \$ x_1, x_2, x_3 \$, and the conditions \$ x_2 - x_1 = \\lambda \$ and \$ x_3 > \\frac{1}{2}(x_1 + x_2) \$ are satisfied, find the maximum va...	2	{ "hint": "Focus on the transformed polynomial and the relationship between the roots $ y_1, y_2, y_3 $ and the given conditions. Use the substitution $ x = y - \frac{a}{3} $ to simplify the polynomial and express the target expression in terms of $ y_3 $ and $ \frac{y_3}{\rho} $, where $ \rho = \frac{y_3}{\rho} $ is...
1	{ "problem": "Let \$ p = 2^{3009}, q = 3^{2006}, \$ and \$ r = 5^{1003} \$. Which of the following statements is true?\n(A) \$ p < q < r \$\n(B) \$ p < r < q \$\n(C) \$ q < p < r \$\n(D) \$ r < p < q \$\n(E) \$ q < r < p \$", "solution": "Given the values:\n\\[ p = 2^{3009}, \\quad q = 3^{2006}, \\qua...	1	{ "hint": "Compare the bases of the expressions after expressing all terms with the same exponent of 1003. Since the exponents are the same, the order of the terms depends on the order of their bases: 5, 8, and 9.", "tools": [ "exponent rules", "comparison of bases" ], "plan": [ "Express all terms w...
2	{ "problem": "Given that \$1 \\leq x, y, z \\leq 6\$, how many cases are there in which the product of natural numbers \$x, y, z\$ is divisible by 10?", "solution": "Given the constraints \$1 \\leq x, y, z \\leq 6\$, we are to find the number of natural number combinations \$(x, y, z)\$ such that their prod...	1	{ "hint": null, "tools": [], "plan": [], "confidence": "low", "status": "api_failed" }
3	{ "problem": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.", "solution": "\nGiven the problem, we are to find how many plums (sliv) balance with one pear (grusha...	1	{ "hint": "Focus on the relationship between apples, plums, and pears. Use the equations derived from the balances to substitute and solve for the weight of a pear in terms of plums.", "tools": [ "algebraic substitution", "simplification of equations" ], "plan": [ "Start with the two given equations...
4	{ "problem": "Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \$ n \$ is infinitely large.", "solution": "\n1. We begin by rewriting the given infinite product expression in a more manageable form. The given product is:\n\n \\[\n z = a \\sqrt...	1	{ "hint": "Focus on the exponent of $ a $ in the infinite product. Recognize that each term in the product can be written as $ a^{1/2^k} $, and combine the exponents using the property of exponents. The sum of the exponents forms an infinite geometric series with first term 1 and common ratio $ 1/2 $.", "tools": [ ...
5	{ "problem": "Give the value of \$0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\$. Only a numerical answer is expected.", "solution": "\nTo find the value of the series \$0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\$, we group the terms in pairs:\n1. Group terms in pairs:\n \\[\n (0 - 1) + (2 - 3) + (4 - 5) + \\l...	1	{ "hint": "Group the terms in pairs starting from the first two terms, then the next two terms, and so on, until the last term. Each pair except the last one simplifies to -1, and the last term is 50. Count the number of pairs and add the last term to the sum of the pairs.", "tools": [ "grouping", "arithmet...
6	{ "problem": "Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}\n$$\n\nDoes this equation have any other integer solutions?", "solution": "\nThe problem requires us to verify if \$x = 2\$ is a solution to the equation \$(x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3\$, and to de...	2	{ "hint": "Verify that $x = 2$ satisfies the equation by substituting it into the left-hand side and comparing it to the right-hand side. Then, determine if there are other integer solutions by factoring the equation and testing possible integer roots.", "tools": [ "algebraic substitution", "factoring", ...
7	{ "problem": "Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = 4x^{2} + 9x^{-2}$.", "solution": "To find the function \$ F(x) \$ whose graph passes through the point \$ M_0(3, -2) \$ and given that the derivative \$ F'(x) \$ is \$ 4x^2 + 9x^{-2} \$:\n\n1. **...	1	{ "hint": "To find the function $ F(x) $, start by integrating the given derivative $ F'(x) = 4x^2 + 9x^{-2} $ to obtain the general form of $ F(x) $. Then, use the point $ M_0(3, -2) $ to determine the constant of integration $ C $ by substituting $ x = 3 $ and $ F(x) = -2 $ into the integrated expression.", "tool...
8	{ "problem": "In an isosceles trapezoid with bases \$a = 21\$, \$b = 9\$ and height \$h = 8\$, find the radius of the circumscribed circle.", "solution": "\n1. Identify Given Data and Setup:\n - The given isosceles trapezoid \$ABCD\$ has bases \$AD\$ and \$BC\$ with lengths \$a = 21\$ and \\(b =...	1	{ "hint": "Focus on the diagonal AC and the sine of angle D to calculate the radius of the circumscribed circle using the formula $ R = \\frac{AC}{2 \\sin \\angle D} $.", "tools": [ "Pythagorean theorem", "sine of an angle in a right triangle", "circumradius formula for isosceles trapezoid" ], "plan...
9	{ "problem": "Given two linear functions \$ f(x) \$ and \$ g(x) \$ such that the graphs \$ y = f(x) \$ and \$ y = g(x) \$ are parallel lines that are not parallel to the coordinate axes. Find the minimum value of the function \$ 3(g(x))^2 + 2 f(x) \$, given that the minimum value of the function \\( 3(f(x))...	1	{ "hint": "Focus on the relationship between the two functions $ f(x) $ and $ g(x) $, which are parallel lines. Use the given minimum value of $ 3(f(x))^2 + 2g(x) $ to form an equation involving $ a $, $ b $, and $ c $. Then, use this to find the minimum value of $ 3(g(x))^2 + 2f(x) $ by leveraging the symmetry and p...
10	{ "problem": "Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \\%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratino arrive at the Field of...	1	{ "hint": "Focus on the relationship between Buratino's usual walking time and the time when he walks 25% faster. Use the given time difference of 1.5 hours to set up an equation and solve for the usual time it takes him to walk to the Field of Wonders.", "tools": [ "algebra", "time calculation" ], "pla...
11	{ "problem": "Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:\n\n$$\n\\begin{array}{ccccccc}\nX & 0 & 1 & 2 & \\cdots & k & \\cdots \\\\\nP & \\mathrm{e}^{-\\lambda} & \\lambda \\mathrm{e}^{-\\lambda} / 1! & \\lambda^{2} \\mathrm{e}^{-\\lambda} / 2! & \...	2	{ "hint": "Focus on calculating the expectation of $X^2$ by using the definition of expectation for the Poisson distribution and simplifying the resulting series. Then, use the known expectation $M(X) = \\lambda$ to compute the variance using the formula $D(X) = M(X^2) - [M(X)]^2$.", "tools": [ "Poisson distrib...
12	{ "problem": "Calculate the volume of the body bounded above by the surface \$z = xy^2\$ and below by the rectangle \$0 \\leq x \\leq 1\$, \$0 \\leq y \\leq 2\$.", "solution": "\n1. Identify the problem:\n To compute the volume of the solid bounded above by the surface \$z = x y^2\$ and below by the r...	2	{ "hint": "Focus on setting up the double integral over the given rectangular region with the function $z = xy^2$ as the upper bound and the xy-plane as the lower bound. Evaluate the inner integral with respect to $y$ first, treating $x$ as a constant, then evaluate the outer integral with respect to $x$.", "tools"...
13	{ "problem": "\nVasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the trian...	2	{ "hint": "Focus on the relationship between the angles at each vertex and the number of rotations required to return the triangle to its original position. Specifically, consider the angle at vertex 3 and how many times it must be rotated to complete a full 360-degree turn.", "tools": [ "angle relationships in...
14	{ "problem": "If \$ n \$ is a positive integer, the notation \$ n! \$ (read \" \$ n \$ factorial\") represents the product of the integers from 1 to \$ n \$ inclusive. For example, \$ 5! = 1 \\times 2 \\times 3 \\times 4 \\times 5 = 120 \$. Which of the following is equal to a perfect square?\n(A) \\(\\frac...	2	{ "hint": "Focus on the expression $ (19!)^2 \times k $, where $ k $ is the result of dividing 20 by the denominator. Determine which value of $ k $ is a perfect square.", "tools": [ "factorial properties", "perfect square identification" ], "plan": [ "Express $ 20! $ as $ 20 \times 19! $ to simplif...
15	{ "problem": "What is the largest result we can find when we divide a 2-digit number by the sum of its digits?", "solution": "1. Let's define the problem more formally. We want to find the maximum value of the expression:\n\n\\[\n\\frac{10a + b}{a + b}\n\\]\n\nwhere \$a\$ and \$b\$ are digits of a two-digit num...	1	{ "hint": "Focus on the expression $\\frac{10a + b}{a + b}$ and consider how changing the digits $a$ and $b$ affects the value. Specifically, note that minimizing $\\frac{9b}{a + b}$ will maximize the overall expression. Try setting $b = 0$ to see if it leads to the maximum value.", "tools": [ "algebraic manipu...
16	{ "problem": "A person is practicing target shooting, starting at a distance of 100 meters from the target for the first shot. If the first shot misses, they move back 50 meters for the second shot and continue this pattern; each time they miss, they move back 50 meters and take another shot until they hit the target...	2	{ "hint": "Focus on the probability of hitting the target at each distance, noting that it is inversely proportional to the square of the distance. Use the given probability for the first shot to find the constant of proportionality. Then, calculate the probability of hitting the target on each subsequent shot as the...
17	{ "problem": "The force exerted by the airflow on a sail can be calculated using the formula:\n\n\\[ F = \\frac{C S \\rho (v_0 - v)^2}{2}, \\]\n\nwhere \$C\$ is the coefficient of aerodynamic force, \$S\$ is the area of the sail (\$S = 5 \\, \\text{m}^2\$), \$\\rho\$ is the air density, \$v_0\$ is the wind ...	1	{ "hint": "To find the speed of the sailboat when the instantaneous power of the wind reaches its maximum value, consider the relationship between power, force, and velocity. The power $N$ is given by the product of force $F$ and velocity $v$. Substitute the expression for $F$ into $N$ and then find the value of $v$ ...
18	{ "problem": "In the expansion of \$(a+b)^n\$, there are \$n+1\$ different terms. In the expansion of \$(a+b+c)^{10}\$, the number of different terms is:\n\n(A) 11 \n(B) 33 \n(C) 55 \n(D) 66 \n(E) 132 \n\n(Provided by the 9th American High School Mathematics Examination, 1958)", "solution": "\nWe need to...	2	{ "hint": "Focus on the number of non-negative integer solutions to the equation $i + j + k = 10$, which corresponds to the number of distinct terms in the expansion of $(a+b+c)^{10}$.", "tools": [ "binomial coefficient", "stars and bars method" ], "plan": [ "Identify the equation representing the d...
19	{ "problem": "Find any five consecutive natural numbers less than 100, whose product is divisible by 2014.", "solution": "\nTo find five consecutive natural numbers less than 100 whose product is divisible by 2014, we start by analyzing the prime factorization of 2014.\n\n#### Step 1: Prime Factorization of 2014\nW...	2	{ "hint": "Focus on the prime factor 53, as it is a prime number less than 100. Look for a sequence of five consecutive numbers that includes 53 and also contains a multiple of 19 and an even number to satisfy the factors of 2014.", "tools": [ "prime factorization", "consecutive number sequences" ], "pl...
20	{ "problem": "$A, B, C, D, E$ are seated in a train that consists of 5 carriages, with each carriage only holding one person. It is known that $D$ is seated in the last carriage, $A$ is immediately behind $E$, $B$ is in a carriage before $A$, and there is at least one person between $B$ and $C$. Who is seated in the ...	2	{ "hint": "Focus on the constraints involving positions of B, C, and the spacing between them. Since D is fixed in the last position and A is immediately before E, consider the possible placements of A and E. Then, determine where B and C can fit while maintaining the condition that there is at least one person betwe...

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row_index int64 0 29.4k	hypergraph dict	attempts int64 1 3	solution_approach dict
0	{ "problem": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satisfied, find the maximum va...	2	{ "hint": "Focus on the transformed polynomial and the relationship between the roots $ y_1, y_2, y_3 $ and the given conditions. Use the substitution $ x = y - \frac{a}{3} $ to simplify the polynomial and express the target expression in terms of $ y_3 $ and $ \frac{y_3}{\rho} $, where $ \rho = \frac{y_3}{\rho} $ is...
1	{ "problem": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)", "solution": "Given the values:\n\\[ p = 2^{3009}, \\quad q = 3^{2006}, \\qua...	1	{ "hint": "Compare the bases of the expressions after expressing all terms with the same exponent of 1003. Since the exponents are the same, the order of the terms depends on the order of their bases: 5, 8, and 9.", "tools": [ "exponent rules", "comparison of bases" ], "plan": [ "Express all terms w...
2	{ "problem": "Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?", "solution": "Given the constraints \\(1 \\leq x, y, z \\leq 6\\), we are to find the number of natural number combinations \\((x, y, z)\\) such that their prod...	1	{ "hint": null, "tools": [], "plan": [], "confidence": "low", "status": "api_failed" }
3	{ "problem": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.", "solution": "\nGiven the problem, we are to find how many plums (sliv) balance with one pear (grusha...	1	{ "hint": "Focus on the relationship between apples, plums, and pears. Use the equations derived from the balances to substitute and solve for the weight of a pear in terms of plums.", "tools": [ "algebraic substitution", "simplification of equations" ], "plan": [ "Start with the two given equations...
4	{ "problem": "Determine the value of\n\n$$\nz=a \\sqrt{a} \\sqrt[4]{a} \\sqrt[8]{a} \\ldots \\sqrt[2^{n}]{a} \\ldots\n$$\n\nif \\( n \\) is infinitely large.", "solution": "\n1. We begin by rewriting the given infinite product expression in a more manageable form. The given product is:\n\n \\[\n z = a \\sqrt...	1	{ "hint": "Focus on the exponent of $ a $ in the infinite product. Recognize that each term in the product can be written as $ a^{1/2^k} $, and combine the exponents using the property of exponents. The sum of the exponents forms an infinite geometric series with first term 1 and common ratio $ 1/2 $.", "tools": [ ...
5	{ "problem": "Give the value of \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\). Only a numerical answer is expected.", "solution": "\nTo find the value of the series \\(0 - 1 + 2 - 3 + 4 - 5 + \\ldots - 49 + 50\\), we group the terms in pairs:\n1. Group terms in pairs:\n \\[\n (0 - 1) + (2 - 3) + (4 - 5) + \\l...	1	{ "hint": "Group the terms in pairs starting from the first two terms, then the next two terms, and so on, until the last term. Each pair except the last one simplifies to -1, and the last term is 50. Count the number of pairs and add the last term to the sum of the pairs.", "tools": [ "grouping", "arithmet...
6	{ "problem": "Verify that 2 is a solution of the equation:\n\n$$\n(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}\n$$\n\nDoes this equation have any other integer solutions?", "solution": "\nThe problem requires us to verify if \\(x = 2\\) is a solution to the equation \\((x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3\\), and to de...	2	{ "hint": "Verify that $x = 2$ satisfies the equation by substituting it into the left-hand side and comparing it to the right-hand side. Then, determine if there are other integer solutions by factoring the equation and testing possible integer roots.", "tools": [ "algebraic substitution", "factoring", ...
7	{ "problem": "Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = 4x^{2} + 9x^{-2}$.", "solution": "To find the function \\( F(x) \\) whose graph passes through the point \\( M_0(3, -2) \\) and given that the derivative \\( F'(x) \\) is \\( 4x^2 + 9x^{-2} \\):\n\n1. **...	1	{ "hint": "To find the function $ F(x) $, start by integrating the given derivative $ F'(x) = 4x^2 + 9x^{-2} $ to obtain the general form of $ F(x) $. Then, use the point $ M_0(3, -2) $ to determine the constant of integration $ C $ by substituting $ x = 3 $ and $ F(x) = -2 $ into the integrated expression.", "tool...
8	{ "problem": "In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), find the radius of the circumscribed circle.", "solution": "\n1. Identify Given Data and Setup:\n - The given isosceles trapezoid \\(ABCD\\) has bases \\(AD\\) and \\(BC\\) with lengths \\(a = 21\\) and \\(b =...	1	{ "hint": "Focus on the diagonal AC and the sine of angle D to calculate the radius of the circumscribed circle using the formula $ R = \\frac{AC}{2 \\sin \\angle D} $.", "tools": [ "Pythagorean theorem", "sine of an angle in a right triangle", "circumradius formula for isosceles trapezoid" ], "plan...
9	{ "problem": "Given two linear functions \\( f(x) \\) and \\( g(x) \\) such that the graphs \\( y = f(x) \\) and \\( y = g(x) \\) are parallel lines that are not parallel to the coordinate axes. Find the minimum value of the function \\( 3(g(x))^2 + 2 f(x) \\), given that the minimum value of the function \\( 3(f(x))...	1	{ "hint": "Focus on the relationship between the two functions $ f(x) $ and $ g(x) $, which are parallel lines. Use the given minimum value of $ 3(f(x))^2 + 2g(x) $ to form an equation involving $ a $, $ b $, and $ c $. Then, use this to find the minimum value of $ 3(g(x))^2 + 2f(x) $ by leveraging the symmetry and p...
10	{ "problem": "Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \\%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratino arrive at the Field of...	1	{ "hint": "Focus on the relationship between Buratino's usual walking time and the time when he walks 25% faster. Use the given time difference of 1.5 hours to set up an equation and solve for the usual time it takes him to walk to the Field of Wonders.", "tools": [ "algebra", "time calculation" ], "pla...
11	{ "problem": "Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:\n\n$$\n\\begin{array}{ccccccc}\nX & 0 & 1 & 2 & \\cdots & k & \\cdots \\\\\nP & \\mathrm{e}^{-\\lambda} & \\lambda \\mathrm{e}^{-\\lambda} / 1! & \\lambda^{2} \\mathrm{e}^{-\\lambda} / 2! & \...	2	{ "hint": "Focus on calculating the expectation of $X^2$ by using the definition of expectation for the Poisson distribution and simplifying the resulting series. Then, use the known expectation $M(X) = \\lambda$ to compute the variance using the formula $D(X) = M(X^2) - [M(X)]^2$.", "tools": [ "Poisson distrib...
12	{ "problem": "Calculate the volume of the body bounded above by the surface \\(z = xy^2\\) and below by the rectangle \\(0 \\leq x \\leq 1\\), \\(0 \\leq y \\leq 2\\).", "solution": "\n1. Identify the problem:\n To compute the volume of the solid bounded above by the surface \\(z = x y^2\\) and below by the r...	2	{ "hint": "Focus on setting up the double integral over the given rectangular region with the function $z = xy^2$ as the upper bound and the xy-plane as the lower bound. Evaluate the inner integral with respect to $y$ first, treating $x$ as a constant, then evaluate the outer integral with respect to $x$.", "tools"...
13	{ "problem": "\nVasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the trian...	2	{ "hint": "Focus on the relationship between the angles at each vertex and the number of rotations required to return the triangle to its original position. Specifically, consider the angle at vertex 3 and how many times it must be rotated to complete a full 360-degree turn.", "tools": [ "angle relationships in...
14	{ "problem": "If \\( n \\) is a positive integer, the notation \\( n! \\) (read \" \\( n \\) factorial\") represents the product of the integers from 1 to \\( n \\) inclusive. For example, \\( 5! = 1 \\times 2 \\times 3 \\times 4 \\times 5 = 120 \\). Which of the following is equal to a perfect square?\n(A) \\(\\frac...	2	{ "hint": "Focus on the expression $ (19!)^2 \times k $, where $ k $ is the result of dividing 20 by the denominator. Determine which value of $ k $ is a perfect square.", "tools": [ "factorial properties", "perfect square identification" ], "plan": [ "Express $ 20! $ as $ 20 \times 19! $ to simplif...
15	{ "problem": "What is the largest result we can find when we divide a 2-digit number by the sum of its digits?", "solution": "1. Let's define the problem more formally. We want to find the maximum value of the expression:\n\n\\[\n\\frac{10a + b}{a + b}\n\\]\n\nwhere \\(a\\) and \\(b\\) are digits of a two-digit num...	1	{ "hint": "Focus on the expression $\\frac{10a + b}{a + b}$ and consider how changing the digits $a$ and $b$ affects the value. Specifically, note that minimizing $\\frac{9b}{a + b}$ will maximize the overall expression. Try setting $b = 0$ to see if it leads to the maximum value.", "tools": [ "algebraic manipu...
16	{ "problem": "A person is practicing target shooting, starting at a distance of 100 meters from the target for the first shot. If the first shot misses, they move back 50 meters for the second shot and continue this pattern; each time they miss, they move back 50 meters and take another shot until they hit the target...	2	{ "hint": "Focus on the probability of hitting the target at each distance, noting that it is inversely proportional to the square of the distance. Use the given probability for the first shot to find the constant of proportionality. Then, calculate the probability of hitting the target on each subsequent shot as the...
17	{ "problem": "The force exerted by the airflow on a sail can be calculated using the formula:\n\n\\[ F = \\frac{C S \\rho (v_0 - v)^2}{2}, \\]\n\nwhere \\(C\\) is the coefficient of aerodynamic force, \\(S\\) is the area of the sail (\\(S = 5 \\, \\text{m}^2\\)), \\(\\rho\\) is the air density, \\(v_0\\) is the wind ...	1	{ "hint": "To find the speed of the sailboat when the instantaneous power of the wind reaches its maximum value, consider the relationship between power, force, and velocity. The power $N$ is given by the product of force $F$ and velocity $v$. Substitute the expression for $F$ into $N$ and then find the value of $v$ ...
18	{ "problem": "In the expansion of \\((a+b)^n\\), there are \\(n+1\\) different terms. In the expansion of \\((a+b+c)^{10}\\), the number of different terms is:\n\n(A) 11 \n(B) 33 \n(C) 55 \n(D) 66 \n(E) 132 \n\n(Provided by the 9th American High School Mathematics Examination, 1958)", "solution": "\nWe need to...	2	{ "hint": "Focus on the number of non-negative integer solutions to the equation $i + j + k = 10$, which corresponds to the number of distinct terms in the expansion of $(a+b+c)^{10}$.", "tools": [ "binomial coefficient", "stars and bars method" ], "plan": [ "Identify the equation representing the d...
19	{ "problem": "Find any five consecutive natural numbers less than 100, whose product is divisible by 2014.", "solution": "\nTo find five consecutive natural numbers less than 100 whose product is divisible by 2014, we start by analyzing the prime factorization of 2014.\n\n#### Step 1: Prime Factorization of 2014\nW...	2	{ "hint": "Focus on the prime factor 53, as it is a prime number less than 100. Look for a sequence of five consecutive numbers that includes 53 and also contains a multiple of 19 and an even number to satisfy the factors of 2014.", "tools": [ "prime factorization", "consecutive number sequences" ], "pl...
20	{ "problem": "$A, B, C, D, E$ are seated in a train that consists of 5 carriages, with each carriage only holding one person. It is known that $D$ is seated in the last carriage, $A$ is immediately behind $E$, $B$ is in a carriage before $A$, and there is at least one person between $B$ and $C$. Who is seated in the ...	2	{ "hint": "Focus on the constraints involving positions of B, C, and the spacing between them. Since D is fixed in the last position and A is immediately before E, consider the possible placements of A and E. Then, determine where B and C can fit while maintaining the condition that there is at least one person betwe...