data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A prism has 15 edges. How many faces does the prism have?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A prism is constructed so that its vertical edges are parallel to the $z$-axis. Its cross-section is a square of side length 10.\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C, D, E, F, G, H;\n\nA = (1,1,0);\nB = (1,-1,0);\nC =... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 2801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can n... | math | {
"style": "rule",
"ground_truth": "$5\\text{:}30\\text{ p.m.}$"
} | {
"index": 2802,
"solution": "1. **Initial Setup and Relative Speed Calculation:**\n The privateer starts at position $0$ and the merchantman starts at position $10$ miles away. The privateer chases the merchantman at a speed of $11$ mph, while the merchantman is moving away at $8$ mph. The relative speed of the ... |
DeepScaleR | [
{
"role": "user",
"content": "A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the thir... | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"index": 2803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$."
}
] | math | {
"style": "rule",
"ground_truth": "982,982"
} | {
"index": 2804,
"solution": "$491 \\cdot 1001 \\cdot 2=982,982$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A product originally priced at \\$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount."
}
] | math | {
"style": "rule",
"ground_truth": "8.7\\%"
} | {
"index": 2805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task c... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 2806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students?"
}
] | math | {
"style": "rule",
"ground_truth": "59049"
} | {
"index": 2807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\\circ$ and $90^\\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations\n\\[\nx = vt \\cos \\theta, \\quad y = vt \\sin \\theta - \\frac{1}{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{16}"
} | {
"index": 2808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is fired with an initial velocity of $v$ at an angle of $\\theta$ from the ground. Then its trajectory can modeled by the parametric equations\n\\begin{align*}\nx &= vt \\cos \\theta, \\\\\ny &= vt \\sin \\theta - \\frac{1}{2} gt^2,\n\\end{align*}where $t$ denotes... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 2809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is launched with an initial velocity $u$ at an angle $\\phi$ from the ground. The trajectory can be described by the parametric equations:\n\\[\nx = ut \\cos \\phi, \\quad y = ut \\sin \\phi - \\frac{1}{2} g t^2,\n\\]\nwhere $t$ is the time, $g$ is the acceleration... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 2810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is launched with an initial velocity of $u$ at an angle of $\\alpha$ from the ground. The trajectory can be modeled by the parametric equations:\n\\[\nx = ut \\cos \\alpha, \\quad y = ut \\sin \\alpha - \\frac{1}{2} kt^2,\n\\]\nwhere $t$ denotes time and $k$ denote... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 2811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is launched with an initial velocity of $u$ at an angle of $\\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:\n\\[\nx = ut \\cos \\phi,\n\\]\n\\[\ny = ut \\sin \\phi - \\frac{1}{2} gt^2,\n\\]\nwhere $t$ is time and $... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 2812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projectile is launched, and its height (in meters) over time is represented by the equation $-20t^2 + 50t + 10$, where $t$ is the time in seconds after launch. Determine the maximum height reached by the projectile."
}
] | math | {
"style": "rule",
"ground_truth": "41.25"
} | {
"index": 2813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projection takes $\\begin{pmatrix} 1 \\\\ -2 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{3}{2} \\\\ -\\frac{3}{2} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -4 \\\\ 1 \\end{pmatrix}$ to?"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -5/2 \\\\ 5/2 \\end{pmatrix}"
} | {
"index": 2814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A projection takes $\\begin{pmatrix} 4 \\\\ 4 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{60}{13} \\\\ \\frac{12}{13} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -2 \\\\ 2 \\end{pmatrix}$ to?"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -20/13 \\\\ -4/13 \\end{pmatrix}"
} | {
"index": 2815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A proper divisor of a number is one that is not equal to the number itself. What is the sum of the proper divisors of \\( 450 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "759"
} | {
"index": 2816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 200 or 600 when divided by 900. Under this proposal, how many leap years will there be that end in double zeroes between 1996 and 4096?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A psychic is faced with a deck of 36 cards placed face down (four suits, with nine cards of each suit). He names the suit of the top card, after which the card is revealed and shown to him. Then the psychic names the suit of the next card, and so on. The goal is for the psychic... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 2818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A public bus departs on schedule at 6:30, 7:00, and 7:30. Student Xiao Ming arrives at the station between 6:50 and 7:30 to catch the bus, and his time of arrival is random. The probability that his waiting time is no more than 10 minutes is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A puppy and two cats together weigh 24 pounds. The puppy and the larger cat together weigh exactly twice as much as the smaller cat, and the puppy and the smaller cat together weigh exactly the same as the larger cat. How many pounds does the puppy weigh?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid \\( S A B C D \\) has a trapezoid \\( A B C D \\) as its base, with bases \\( B C \\) and \\( A D \\). Points \\( P_1, P_2, P_3 \\) lie on side \\( B C \\) such that \\( B P_1 < B P_2 < B P_3 < B C \\). Points \\( Q_1, Q_2, Q_3 \\) lie on side \\( A D \\) such that \\... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"index": 2821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid towar... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 2822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyrami... | math | {
"style": "rule",
"ground_truth": "1280"
} | {
"index": 2823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"index": 2824,
"solution": "1. **Identify the side length of the square base**: Given the area of square $ABCD$ is $196$, we find the side length by taking the square root:\n \\[\n s = \\sqrt{196} = 14\n \\]\n\n2. **Calculate the heights of triangles $ABE$ and $CDE$**: The area of a triangle is given by $\\... |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the vo... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2} - 7"
} | {
"index": 2825,
"solution": "1. **Identify the Geometry of the Pyramid and Cube**: The pyramid has a square base with side length 1 and lateral faces that are equilateral triangles. A cube is placed inside such that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces... |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a square base with sides of length 2 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and the center of its opposite face exactly meets the apex of the pyramid. Calculate the v... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{4}"
} | {
"index": 2826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers... | math | {
"style": "rule",
"ground_truth": "803"
} | {
"index": 2827,
"solution": "Notation is shown on diagram. \\[AM = MB = c = 12, AC = BC = b = 20,\\] \\[DA = DB = DC = a = 25.\\] \\[CM = x + y = \\sqrt{b^2-c^2} = 16,\\] \\[x^2 - y^2 = CD^2 – DM^2 = CD^2 – (BD^2 – BM^2) = c^2 = 144,\\] \\[x – y = \\frac{x^2 – y^2}{x+y} = \\frac {c^2} {16} = 9,\\] \\[x = \\frac {1... |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid is constructed on a $5 \\times 12$ rectangular base. Each of the four edges joining the apex to the corners of the rectangular base has a length of $15$. Determine the volume of this pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 2828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 2829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid is formed on a $6\\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 2830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 2831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid with a square base has a base edge of 20 cm and a height of 40 cm. Two smaller similar pyramids are cut away from the original pyramid: one has an altitude that is one-third the original, and another that is one-fifth the original altitude, stacked atop the first smal... | math | {
"style": "rule",
"ground_truth": "\\frac{3223}{3375}"
} | {
"index": 2832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}"
} | {
"index": 2833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid with a triangular base has edges of unit length, and the angles between its edges are \\(60^{\\circ}, 90^{\\circ},\\) and \\(120^{\\circ}\\). What is the volume of the pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"index": 2834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\\%$, what is the volume of the new pyramid, in cubic inches?"
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 2835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the ... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"index": 2836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 2837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 2838,
"solution": "1. **Form of the Polynomial:**\n The disrespectful function $p(x)$ has leading coefficient $1$, so it can be written in factored form as $p(x) = (x-r)(x-s)$.\n\n2. **Condition for $p(p(x)) = 0$:**\n Plugging our form in, we get:\n \\[\n p(p(x)) = ((x-r)(x-s)-r)((x-r)(x-s)-s) = ... |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .\n."
}
] | math | {
"style": "rule",
"ground_truth": "769"
} | {
"index": 2839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral \\(A B C D\\) is inscribed in a circle with radius 6 and center at point \\(O\\). Its diagonals \\(A C\\) and \\(B D\\) are mutually perpendicular and intersect at point \\(K\\). Points \\(E\\) and \\(F\\) are the midpoints of \\(A C\\) and \\(B D\\), respective... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral \\(ABCD\\) is inscribed in a circle with a diameter of 1, where \\(\\angle D\\) is a right angle and \\(AB = BC\\).\n\nFind the area of quadrilateral \\(ABCD\\) if its perimeter is \\(\\frac{9\\sqrt{2}}{5}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{25}"
} | {
"index": 2841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral connecting the midpoints of the sides of trapezoid $\\mathrm{ABCD}$ is a rhombus. Find its area if the height of the trapezoid $\\mathrm{BH}=5 \\mathrm{c}$, the smaller base $\\mathrm{BC}=6 \\mathrm{~cm}$, and the angle $\\mathrm{ABC}$ is $120^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral has vertices $P(a,b)$, $Q(2b,a)$, $R(-a, -b)$, and $S(-2b, -a)$, where $a$ and $b$ are integers and $a>b>0$. Find the area of quadrilateral $PQRS$.\n**A)** $16$ square units \n**B)** $20$ square units \n**C)** $24$ square units \n**D)** $28$ square units \n**E)*... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2844,
"solution": "1. **Identify the slopes of the sides to determine the shape of the quadrilateral**:\n - The slope of line segment $PQ$ is calculated as:\n \\[\n \\text{slope of } PQ = \\frac{Q_y - P_y}{Q_x - P_x} = \\frac{a - b}{b - a} = -1\n \\]\n - The slope of line segment $PS$ is ... |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\\sqrt2+b\\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "3008"
} | {
"index": 2846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is divided by its diagonals into four triangles. The areas of three of them are 10, 20, and 30, each being less than the area of the fourth triangle. Find the area of the given quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 2847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"index": 2848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle of radius $200\\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 2849,
"solution": "1. **Setup and Diagram**: Let quadrilateral $ABCD$ be inscribed in a circle $O$ with radius $200\\sqrt{2}$. Assume $AD$ is the side of unknown length, and $AB = BC = CD = 200$. Draw radii $OA$, $OB$, $OC$, and $OD$.\n\n2. **Using the Pythagorean Theorem in $\\triangle BOC$**: Draw alti... |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle of radius $250$. Three sides of this quadrilateral have lengths of $250$, $250$, and $100$ respectively. What is the length of the fourth side?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 2850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \\( 4 \\sqrt{6} \\).\n\nFin... | math | {
"style": "rule",
"ground_truth": "18 \\sqrt{161}"
} | {
"index": 2851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:"
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 2852,
"solution": "Let's consider a quadrilateral $ABCD$ inscribed in a circle. We need to find the sum of the angles inscribed in the four segments outside the quadrilateral.\n\n1. **Identify the Segments and Angles**:\n - The four segments outside the quadrilateral are the regions outside $ABCD$ but ... |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be:"
}
] | math | {
"style": "rule",
"ground_truth": "180^{\\circ}"
} | {
"index": 2853,
"solution": "1. **Understanding the Problem**: We are given a quadrilateral inscribed in a circle. We need to find the sum of the angles inscribed in the four arcs cut off by the sides of the quadrilateral.\n\n2. **Total Arc in a Circle**: The total measure of the arcs in a circle is $360^\\circ$. ... |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral pyramid \\( S A B C D \\) is given, with the base being the parallelogram \\( A B C D \\). A plane is drawn through the midpoint of edge \\( A B \\) that is parallel to the lines \\( A C \\) and \\( S D \\). In what ratio does this plane divide edge \\( S B \\)?... | math | {
"style": "rule",
"ground_truth": "1 : 3"
} | {
"index": 2854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral pyramid \\(SABCD\\) is given, with a base that is a trapezoid \\(ABCD\\). The ratio of the bases \\(AD\\) and \\(BC\\) of this trapezoid is 2. Construct the cross-section of the pyramid with a plane passing through point \\(D\\) and the midpoints of the edges \\... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"index": 2855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quagga is an extinct chess piece whose move is like a knight's, but much longer: it can move 6 squares in any direction (up, down, left, or right) and then 5 squares in a perpendicular direction. Find the number of ways to place 51 quaggas on an $8 \\times 8$ chessboard in su... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 2857,
"solution": "Represent the 64 squares of the board as vertices of a graph, and connect two vertices by an edge if a quagga can move from one to the other. The resulting graph consists of 4 paths of length 5 and 4 paths of length 3 (given by the four rotations of the two paths shown, next page), and... |
DeepScaleR | [
{
"role": "user",
"content": "A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\\%$ of the gravel at full price. The following week the quarry drops the price by $10\\%$ , and, again, it sells $60... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"index": 2858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A quartic (4th degree) polynomial \\( p(x) \\) satisfies:\n\\[ p(n) = \\frac{1}{n^2} \\] for \\( n = 1, 2, 3, 4, \\) and \\( 5 \\). Find \\( p(6) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"index": 2859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rabbit and a hedgehog participated in a running race on a 550 m long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of 10 m/s and the hedgehog ran anticlockwise at a speed of 1 m/s. When they met, the rabbit continued as bef... | math | {
"style": "rule",
"ground_truth": "545"
} | {
"index": 2860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 2861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A racer departs from point \\( A \\) along the highway, maintaining a constant speed of \\( a \\) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \\( 1.25a \\) km/h. How many minutes after the start of the first racer was a third racer... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 2862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"index": 2863,
"solution": "To solve this problem, we need to calculate the probability that the contestant guesses at least 2 out of 3 questions correctly. We will consider the following cases:\n\n1. **Case 1: The contestant guesses all three questions correctly.**\n2. **Case 2: The contestant guesses exactly tw... |
DeepScaleR | [
{
"role": "user",
"content": "A radio system consisting of 1000 components (each with a failure rate of $\\lambda_{i} = 10^{-6}$ failures/hour) has been tested and accepted by the customer. Determine the probability of the system operating without failure over the interval $t_{1} < (t = t_{1} + \\Delta t) <... | math | {
"style": "rule",
"ground_truth": "0.367879"
} | {
"index": 2864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A random binary string of length 1000 is chosen. Let \\(L\\) be the expected length of its longest (contiguous) palindromic substring. Estimate \\(L\\)."
}
] | math | {
"style": "rule",
"ground_truth": "23.120"
} | {
"index": 2865,
"solution": "The probability that there exists a palindromic substring of length \\(2n+1\\) is approximately \\(2^{-n} \\cdot 1000\\). Thus, we can expect to often see a length 21 palindrome, and sometimes longer ones. This leads to a guess a bit above 21. \\(L\\) was approximated with \\(10^{7}\\)... |
DeepScaleR | [
{
"role": "user",
"content": "A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ 1 - \\left( \\frac{8}{9} \\right)^n - \\left( \\frac{5}{9} \\right)^n + \\left( \\frac{4}{9} \\right)^n \\]"
} | {
"index": 2866,
"solution": "For the product to be divisible by 10, there must be a factor of 2 and a factor of 5 in there.\nThe probability that there is no 5 is $\\left( \\frac{8}{9}\\right)^n$ .\nThe probability that there is no 2 is $\\left( \\frac{5}{9}\\right)^n$ .\nThe probability that there is neither a 2 ... |
DeepScaleR | [
{
"role": "user",
"content": "A random permutation $a=\\left(a_{1}, a_{2}, \\ldots, a_{40}\\right)$ of $(1,2, \\ldots, 40)$ is chosen, with all permutations being equally likely. William writes down a $20 \\times 20$ grid of numbers $b_{i j}$ such that $b_{i j}=\\max \\left(a_{i}, a_{j+20}\\right)$ for all ... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{13}"
} | {
"index": 2867,
"solution": "We can deduce information about $a$ from the grid $b$ by looking at the largest element of it, say $m$. If $m$ fills an entire row, then the value of $a$ corresponding to this row must be equal to $m$. Otherwise, $m$ must fill an entire column, and the value of $a$ corresponding to thi... |
DeepScaleR | [
{
"role": "user",
"content": "A random point \\(N\\) on a line has coordinates \\((t, -2-t)\\), where \\(t \\in \\mathbb{R}\\). A random point \\(M\\) on a parabola has coordinates \\(\\left( x, x^2 - 4x + 5 \\right)\\), where \\(x \\in \\mathbb{R}\\). The square of the distance between points \\(M\\) and \... | math | {
"style": "rule",
"ground_truth": "\\frac{19\\sqrt{2}}{8}"
} | {
"index": 2868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A random sample of $10$ households was taken from a residential area, and the monthly income data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\\sum_{i=1}^{10} x_i = 80$, $\\sum_{i=1}^{10} y_i = 20$... | math | {
"style": "rule",
"ground_truth": "1.7"
} | {
"index": 2869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Grou... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"index": 2870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A random variable \\(X\\) is given by the probability density function \\(f(x) = \\frac{1}{2} \\sin x\\) within the interval \\((0, \\pi)\\); outside this interval, \\(f(x) = 0\\). Find the variance of the function \\(Y = \\varphi(X) = X^2\\) using the probability density funct... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi^4 - 16\\pi^2 + 80}{4}"
} | {
"index": 2871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the posit... | math | {
"style": "rule",
"ground_truth": "15/8"
} | {
"index": 2872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rare genetic disorder is present in one out of every 1000 people in a certain community, and it typically shows no outward symptoms. There is a specialized blood test to detect this disorder. For individuals who have this disorder, the test consistently yields a positive resu... | math | {
"style": "rule",
"ground_truth": "0.019"
} | {
"index": 2873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rational number written in base eight is $\\underline{ab} . \\underline{cd}$, where all digits are nonzero. The same number in base twelve is $\\underline{bb} . \\underline{ba}$. Find the base-ten number $\\underline{abc}$."
}
] | math | {
"style": "rule",
"ground_truth": "321"
} | {
"index": 2874,
"solution": "The parts before and after the decimal points must be equal. Therefore $8a + b = 12b + b$ and $c/8 + d/64 = b/12 + a/144$. Simplifying the first equation gives $a = (3/2)b$. Plugging this into the second equation gives $3b/32 = c/8 + d/64$. Multiplying both sides by 64 gives $6b = 8c +... |
DeepScaleR | [
{
"role": "user",
"content": "A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2875,
"solution": "1. **Understanding the Reflections**: Each time the light is reflected, it forms an angle with the line equal to the angle of incidence. Given that $\\measuredangle CDA = 8^\\circ$, the light will reflect off line $CD$ and $AD$ such that the angle of incidence equals the angle of refle... |
DeepScaleR | [
{
"role": "user",
"content": "A ray of light passing through the point $A = (-3,9,11),$ reflects off the plane $x + y + z = 12$ at $B,$ and then passes through the point $C = (3,5,9).$ Find the point $B.$\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C;\n\nA... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{5}{3}, \\frac{16}{3}, \\frac{25}{3} \\right)"
} | {
"index": 2876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ray passing through the focus $F$ of the parabola $y^2 = 4x$ intersects the parabola at point $A$. Determine the equation of the line to which a circle with diameter $FA$ must be tangent."
}
] | math | {
"style": "rule",
"ground_truth": "y=0"
} | {
"index": 2877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ray starting from point $A(-4,1)$ reflects off the line $l_{1}: x-y+3=0$ and the reflected ray passes through point $B(-3,2)$. Find the slope of the line containing the reflected ray."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 2878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A reader mentioned that his friend's house in location $A$, where he was invited for lunch at 1 PM, is located 1 km from his own house in location $B$. At 12 PM, he left $B$ in his wheelchair heading towards location $C$ for a stroll. His friend, intending to join him and help ... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 2879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?"
}
] | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"index": 2880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-10, 15]$. Find the probability that the roots of the polynomial\n\\[ x^4 + 3ax^3 + (3a - 3)x^2 + (-5a + 4)x - 3 \\]\nare all real."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{25}"
} | {
"index": 2881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial\n\\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\\]are all real."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{19}"
} | {
"index": 2882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial\n$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$\n are all real can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime pos... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 2883,
"solution": "The polynomial we are given is rather complicated, so we could use Rational Root Theorem to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, $x = 1, -1, 2, -2$ are all possible rational roots. Upon plugging these roots into the polynomial, $x = -2$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "A real number $x$ is chosen uniformly at random from the interval $(0,10)$. Compute the probability that $\\sqrt{x}, \\sqrt{x+7}$, and $\\sqrt{10-x}$ are the side lengths of a non-degenerate triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{22}{25}"
} | {
"index": 2884,
"solution": "For any positive reals $a, b, c$, numbers $a, b, c$ is a side length of a triangle if and only if $$(a+b+c)(-a+b+c)(a-b+c)(a+b-c)>0 \\Longleftrightarrow \\sum_{\\text {cyc }}\\left(2 a^{2} b^{2}-a^{4}\\right)>0$$ (to see why, just note that if $a \\geq b+c$, then only the factor $-a+b+... |
DeepScaleR | [
{
"role": "user",
"content": "A real number $x$ satisfies $9^{x}+3^{x}=6$. Compute the value of $16^{1 / x}+4^{1 / x}$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 2885,
"solution": "Setting $y=3^{x}$ in the given equation yields $$y^{2}+y=6 \\Longrightarrow y^{2}+y-6=0 \\Longrightarrow y=-3,2$$ Since $y>0$ we must have $$3^{x}=2 \\Longrightarrow x=\\log _{3}(2) \\Longrightarrow 1 / x=\\log _{2}(3)$$ This means that $$16^{1 / x}+4^{1 / x}=\\left(2^{1 / x}\\right)^{... |
DeepScaleR | [
{
"role": "user",
"content": "A real number \\(x\\) is chosen uniformly at random from the interval \\([0,1000]\\). Find the probability that \\(\\left\\lfloor\\frac{\\left\\lfloor\\frac{x}{2.5}\\right\\rfloor}{2.5}\\right\\rfloor=\\left\\lfloor\\frac{x}{6.25}\\right\\rfloor\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 2886,
"solution": "Let \\(y=\\frac{x}{2.5}\\), so \\(y\\) is chosen uniformly at random from [0,400]. Then we need \\(\\left\\lfloor\\frac{\\lfloor y\\rfloor}{2.5}\\right\\rfloor=\\left\\lfloor\\frac{y}{2.5}\\right\\rfloor\\). Let \\(y=5a+b\\), where \\(0 \\leq b<5\\) and \\(a\\) is an integer. Then \\(\... |
DeepScaleR | [
{
"role": "user",
"content": "A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high?"
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 2887,
"solution": "1. **Identify the thickness per sheet**: Given that $500$ sheets of paper have a total thickness of $5$ cm, we can calculate the thickness of one sheet of paper. This is done by dividing the total thickness by the number of sheets:\n \\[\n \\text{Thickness per sheet} = \\frac{5 \\t... |
DeepScaleR | [
{
"role": "user",
"content": "A recent report about the amount of plastic created in the last 65 years stated that the 8.3 billion tonnes produced is as heavy as 25000 Empire State Buildings in New York or a billion elephants. On that basis, how many elephants have the same total weight as the Empire State ... | math | {
"style": "rule",
"ground_truth": "40000"
} | {
"index": 2888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recent study found that 60$\\%$ of men and 80$\\%$ of women surveyed support increased funding for particular medical research. The study surveyed 100 men and 900 women. What was the overall percent of the people surveyed who supported increased funding?"
}
] | math | {
"style": "rule",
"ground_truth": "78\\%"
} | {
"index": 2889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recipe calls for $4 \\frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need? Express your answer as a mixed number."
}
] | math | {
"style": "rule",
"ground_truth": "2\\frac{1}{4}"
} | {
"index": 2890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recipe calls for $5 \\frac{3}{4}$ cups of flour and $2 \\frac{1}{2}$ cups of sugar. If you make one-third of the recipe, how many cups of flour and sugar do you need? Express your answers as mixed numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 2891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recipe calls for $\\frac{1}{3}$ cup of sugar. If you already have $\\frac{1}{6}$ cup, how much more sugar is required to reach the initial quantity? Once you find this quantity, if you need a double amount for another recipe, how much sugar would that be in total?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recipe for 30 cookies requires two cups of flour among its ingredients. Eduardo wants to bake five dozen cookies. How many cups of flour will he need to use in his adjusted recipe?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredi... | math | {
"style": "rule",
"ground_truth": "8 \\frac{3}{4}"
} | {
"index": 2894,
"solution": "To determine the maximum number of servings Jordan can make, we need to calculate the number of servings each ingredient can produce and then find the minimum of these values, as this will be the limiting factor.\n\n1. **Chocolate**: \n - The recipe requires $2$ squares of chocolate ... |
DeepScaleR | [
{
"role": "user",
"content": "A reconnaissance team has 12 soldiers, including 3 radio operators. The 12 soldiers are randomly divided into three groups, with group sizes of 3, 4, and 5 soldiers, respectively. What is the probability that each group has exactly 1 radio operator?"
}
] | math | {
"style": "rule",
"ground_truth": "3/11"
} | {
"index": 2895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypoten... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 2896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{74}"
} | {
"index": 2897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle and a regular pentagon have the same perimeter. Let $A$ be the area of the circle circumscribed about the rectangle, with the rectangle having side lengths in the ratio 3:1, and $B$ the area of the circle circumscribed around the pentagon. Find $A/B$.\nA) $\\frac{12... | math | {
"style": "rule",
"ground_truth": "\\frac{5(5-\\sqrt{5})}{64}"
} | {
"index": 2898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 2899,
"solution": ""
} |
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