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"content": "Three spheres touch each other externally; the radius of the greatest one is equal to the sum of the radii of the other two. All three spheres touch also the faces of a cube internally, whose edges have the length of 42 cm.\nFind the diameters of the spheres.",
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"con... | Three spheres touch each other externally; the radius of the greatest one is equal to the sum of the radii of the other two. All three spheres touch also the faces of a cube internally, whose edges have the length of 42 cm.
Find the diameters of the spheres. | Three spheres touch each other externally; the radius of the greatest one is equal to the sum of the radii of the other two. All three spheres touch also the faces of a cube internally, whose edges have the length of 42 cm.
Find the diameters of the spheres. | ai2-adapt-dev/openmath-2-math | 0.997884 | Okay, let me try to figure out this problem. So, there are three spheres that touch each other externally, and the radius of the largest one is equal to the sum of the radii of the other two. All three spheres also touch the faces of a cube internally, and the cube has edges of 42 cm. We need to find the diameters of t... | Given three spheres that touch each other externally, with the radius of the largest sphere equal to the sum of the radii of the other two, and all three spheres touching the faces of a cube with edge length 42 cm, we need to find the diameters of the spheres.
1. **Understanding the Cube and Spheres**:
- The cube h... | 6,379 | <think>
Okay, let me try to figure out this problem. So, there are three spheres that touch each other externally, and the radius of the largest one is equal to the sum of the radii of the other two. All three spheres also touch the faces of a cube internally, and the cube has edges of 42 cm. We need to find the diamet... | [
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"value": "Three spheres touch each other externally; the radius of the greatest one is equal to the sum of the radii of the other two. All three spheres touch also the faces of a cube internally, whose edges have the length of 42 cm.\nFind the diameters of the spheres."
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"from... |
[{"content":"Let $g(x)$ be a quadratic function such that $g(g(x))=9g(x)+5x+3$. Given that $g(0)=-3$(...TRUNCATED) | "Let $g(x)$ be a quadratic function such that $g(g(x))=9g(x)+5x+3$. Given that $g(0)=-3$ and $g(1)=1(...TRUNCATED) | "Let $g(x)$ be a quadratic function such that $g(g(x))=9g(x)+5x+3$. Given that $g(0)=-3$ and $g(1)=1(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.997952 | "Okay, so I need to find the coefficient of the x² term in the quadratic function g(x). The problem(...TRUNCATED) | "Given the quadratic function \\( g(x) = ax^2 + bx + c \\) with the conditions \\( g(0) = -3 \\) and(...TRUNCATED) | 16,521 | "<think>\nOkay, so I need to find the coefficient of the x² term in the quadratic function g(x). Th(...TRUNCATED) | [{"from":"human","value":"Let $g(x)$ be a quadratic function such that $g(g(x))=9g(x)+5x+3$. Given t(...TRUNCATED) |
[{"content":"Let $a$ and $b$ be two positive integers such that their product is 240. Determine the (...TRUNCATED) | "Let $a$ and $b$ be two positive integers such that their product is 240. Determine the sum of all p(...TRUNCATED) | "Let $a$ and $b$ be two positive integers such that their product is 240. Determine the sum of all p(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.997641 | "Okay, so I need to figure out the sum of all possible values of the sum of two positive integers a (...TRUNCATED) | "Given two positive integers \\(a\\) and \\(b\\) such that their product is 240, we need to determin(...TRUNCATED) | 1,624 | "<think>\nOkay, so I need to figure out the sum of all possible values of the sum of two positive in(...TRUNCATED) | [{"from":"human","value":"Let $a$ and $b$ be two positive integers such that their product is 240. D(...TRUNCATED) |
[{"content":"Let $C_n$ be the product of the first $n$ terms of the arithmetic series $1+2+3+\\dotsb(...TRUNCATED) | "Let $C_n$ be the product of the first $n$ terms of the arithmetic series $1+2+3+\\dotsb$ and let $D(...TRUNCATED) | "Let $C_n$ be the product of the first $n$ terms of the arithmetic series $1+2+3+\\dotsb$ and let $D(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999191 | "Alright, let's try to solve this problem. So, we need to find the value of n (where n is at least 1(...TRUNCATED) | "To solve the problem, we need to find the value of \\( n \\geq 1 \\) for which \\( C_n = D_n \\), w(...TRUNCATED) | 16,853 | "<think>\nAlright, let's try to solve this problem. So, we need to find the value of n (where n is a(...TRUNCATED) | [{"from":"human","value":"Let $C_n$ be the product of the first $n$ terms of the arithmetic series $(...TRUNCATED) |
[{"content":"In a certain triangle, the length of the hypotenuse is numerically equal to the sum of (...TRUNCATED) | "In a certain triangle, the length of the hypotenuse is numerically equal to the sum of the squares (...TRUNCATED) | "In a certain triangle, the length of the hypotenuse is numerically equal to the sum of the squares (...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999505 | "Okay, so I need to find the radius of the inscribed circle in a right triangle where the hypotenuse(...TRUNCATED) | "Given a right triangle where the length of the hypotenuse \\( c \\) is numerically equal to the sum(...TRUNCATED) | 24,551 | "<think>\nOkay, so I need to find the radius of the inscribed circle in a right triangle where the h(...TRUNCATED) | [{"from":"human","value":"In a certain triangle, the length of the hypotenuse is numerically equal t(...TRUNCATED) |
[{"content":"Given a right triangle, if a circle is drawn so that its diameter lies on the hypotenus(...TRUNCATED) | "Given a right triangle, if a circle is drawn so that its diameter lies on the hypotenuse of the tri(...TRUNCATED) | "Given a right triangle, if a circle is drawn so that its diameter lies on the hypotenuse of the tri(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998814 | "Okay, so I need to find the radius of a circle that's drawn such that its diameter is on the hypote(...TRUNCATED) | "Given a right triangle with the hypotenuse \\( AB \\) of length \\( c \\), we need to find the radi(...TRUNCATED) | 16,924 | "<think>\nOkay, so I need to find the radius of a circle that's drawn such that its diameter is on t(...TRUNCATED) | [{"from":"human","value":"Given a right triangle, if a circle is drawn so that its diameter lies on (...TRUNCATED) |
[{"content":"A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of t(...TRUNCATED) | "A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ ver(...TRUNCATED) | "A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ ver(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999288 | "Okay, so I need to solve this problem about a 100-gon, P1, where the sum of the x-coordinates of it(...TRUNCATED) | "Given a 100-gon \\( P_1 \\) with the sum of the \\( x \\)-coordinates of its vertices equal to 2009(...TRUNCATED) | 10,535 | "<think>\nOkay, so I need to solve this problem about a 100-gon, P1, where the sum of the x-coordina(...TRUNCATED) | [{"from":"human","value":"A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coo(...TRUNCATED) |
[{"content":"Find the number of four-digit positive integers where the sum of the digits equals 7.",(...TRUNCATED) | Find the number of four-digit positive integers where the sum of the digits equals 7. | Find the number of four-digit positive integers where the sum of the digits equals 7. | ai2-adapt-dev/openmath-2-math | 0.998468 | "Okay, so I need to figure out how many four-digit numbers there are where the sum of the digits equ(...TRUNCATED) | "To find the number of four-digit positive integers where the sum of the digits equals 7, we start b(...TRUNCATED) | 27,393 | "<think>\nOkay, so I need to figure out how many four-digit numbers there are where the sum of the d(...TRUNCATED) | [{"from":"human","value":"Find the number of four-digit positive integers where the sum of the digit(...TRUNCATED) |
[{"content":"In a right triangle, the length of the hypotenuse is 10 cm and the length of one of the(...TRUNCATED) | "In a right triangle, the length of the hypotenuse is 10 cm and the length of one of the legs is 6 c(...TRUNCATED) | "In a right triangle, the length of the hypotenuse is 10 cm and the length of one of the legs is 6 c(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.998721 | "Okay, so I need to find the tangent of the angle between the hypotenuse and the longer leg in a rig(...TRUNCATED) | "Given a right triangle with a hypotenuse of 10 cm and one leg of 6 cm, we need to find the tangent (...TRUNCATED) | 24,835 | "<think>\nOkay, so I need to find the tangent of the angle between the hypotenuse and the longer leg(...TRUNCATED) | [{"from":"human","value":"In a right triangle, the length of the hypotenuse is 10 cm and the length (...TRUNCATED) |
[{"content":"Homewood Middle School has 1200 students, and 730 of these students attend a summer pic(...TRUNCATED) | "Homewood Middle School has 1200 students, and 730 of these students attend a summer picnic. If two-(...TRUNCATED) | "Homewood Middle School has 1200 students, and 730 of these students attend a summer picnic. If two-(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.997558 | "Okay, so I need to figure out how many girls attend the summer picnic at Homewood Middle School. Le(...TRUNCATED) | "Let \\( G \\) be the number of girls in the school and \\( B \\) be the number of boys. We know the(...TRUNCATED) | 274 | "<think>\nOkay, so I need to figure out how many girls attend the summer picnic at Homewood Middle S(...TRUNCATED) | [{"from":"human","value":"Homewood Middle School has 1200 students, and 730 of these students attend(...TRUNCATED) |
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