data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.\n\n\\[\\begin{tabular}{| c | c | c | c | c | c | c | c |}\n\\hline\n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \n\\hline\n9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \n\\hline\n17 & 18 & 19... | math | {
"style": "rule",
"ground_truth": "1056"
} | {
"index": 6800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circle centered at $O$ . Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$ . We can write $\\tan \\angle BOD =\\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$ . Compu... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"index": 6801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circle of radius $4$ with center $O_1$ , a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$ , and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$ . The centers of the circle are... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 6802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\\angle AOB = 120^\\circ$, $\\angle BOC = 60^\\circ$, and $\\angle COD = 180^\\circ$. A point $X$ lies on the minor arc $\\overarc{AC}$. If $\\angle AXB... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 6803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circle of radius \\( R \\) centered at the origin on the Cartesian plane. Specify at least one value of \\( R \\) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this c... | math | {
"style": "rule",
"ground_truth": "\\sqrt{1105}"
} | {
"index": 6804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circle with radius $4$, and there are numerous line segments of length $6$ that are tangent to the circle at their midpoints. Compute the area of the region consisting of all such line segments.\nA) $8\\pi$ \nB) $7\\pi$ \nC) $9\\pi$ \nD) $10\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 6805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circular cone with vertex $V$, and let $A B C$ be a triangle inscribed in the base of the cone, such that $A B$ is a diameter and $A C=B C$. Let $L$ be a point on $B V$ such that the volume of the cone is 4 times the volume of the tetrahedron $A B C L$. Find the valu... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4-\\pi}"
} | {
"index": 6806,
"solution": "Let $R$ be the radius of the base, $H$ the height of the cone, $h$ the height of the pyramid and let $B L / L V=x / y$. Let [.] denote volume. Then [cone] $=\\frac{1}{3} \\pi R^{2} H$ and $[A B C L]=\\frac{1}{3} \\pi R^{2} h$ and $h=\\frac{x}{x+y} H$. We are given that $[$ cone $]=4[A ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a circular cone with vertex $V$, and let $ABC$ be a triangle inscribed in the base of the cone, such that $AB$ is a diameter and $AC=BC$. Let $L$ be a point on $BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ABCL$. Find the value of $BL/... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4-\\pi}"
} | {
"index": 6807,
"solution": "Let $R$ be the radius of the base, $H$ the height of the cone, $h$ the height of the pyramid and let $BL/LV=x/y$. Let [.] denote volume. Then [cone] $=\\frac{1}{3} \\pi R^{2} H$ and $[ABCL]=\\frac{1}{3} \\pi R^{2} h$ and $h=\\frac{x}{x+y} H$. We are given that [cone] $=4[ABCL]$, so $x/... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. ... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{77}"
} | {
"index": 6808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a convex pentagon $ABCDE$. Let $P_A, P_B, P_C, P_D,$ and $P_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCD,$ and $ABCE$, respectively. Compute $\\frac{[P_A P_B P_C P_D P_E]}{[ABCDE]}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 6809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a convex pentagon $FGHIJ$ where $\\angle F = \\angle G = 100^\\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4} + 24.62"
} | {
"index": 6810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a coordinate plane where at each lattice point, there is a circle with radius $\\frac{1}{8}$ and a square with sides of length $\\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these... | math | {
"style": "rule",
"ground_truth": "972"
} | {
"index": 6811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a cube $A B C D E F G H$, where $A B C D$ and $E F G H$ are faces, and segments $A E, B F, C G, D H$ are edges of the cube. Let $P$ be the center of face $E F G H$, and let $O$ be the center of the cube. Given that $A G=1$, determine the area of triangle $A O P$."
}
... | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{2}}{24}$"
} | {
"index": 6812,
"solution": "From $A G=1$, we get that $A E=\\frac{1}{\\sqrt{3}}$ and $A C=\\frac{\\sqrt{2}}{\\sqrt{3}}$. We note that triangle $A O P$ is located in the plane of rectangle $A C G E$. Since $O P \\| C G$ and $O$ is halfway between $A C$ and $E G$, we get that $[A O P]=\\frac{1}{8}[A C G E]$. Hence,... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 6813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three num... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex w... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 6815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "1059"
} | {
"index": 6816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring?"
}
] | math | {
"style": "rule",
"ground_truth": "802"
} | {
"index": 6817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 uni... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 6818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a decreasing arithmetic sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. If $a\\_3a\\_5=63$ and $a\\_2+{a}\\_{6} =16$,\n\n(1) Find the general term formula of the sequence.\n(2) For what value of $n$ does $S\\_n$ reach its maximum value? A... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 6819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that ver... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is... | math | {
"style": "rule",
"ground_truth": "37/192"
} | {
"index": 6821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a fictional country where license plates are made using an extended Rotokas alphabet with fourteen letters: A, E, G, I, K, M, O, P, R, S, T, U, V, Z. License plates are six letters long and must start with G, end with M, and not contain the letters S, T, or Z. Addition... | math | {
"style": "rule",
"ground_truth": "3024"
} | {
"index": 6822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a fictional language with ten letters in its alphabet: A, B, C, D, F, G, H, J, L, M. Suppose license plates of six letters utilize only letters from this alphabet. How many license plates of six letters are possible that begin with either B or D, end with J, cannot con... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 6823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a five-digit integer in the form $AB,BCA$, where $A$, $B$, and $C$ are distinct digits. What is the largest possible value of $AB,BCA$ that is divisible by 7?"
}
] | math | {
"style": "rule",
"ground_truth": "98,879"
} | {
"index": 6824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a four-digit natural number with the following property: if we swap its first two digits with the second two digits, we get a four-digit number that is 99 less.\n\nHow many such numbers are there in total, and how many of them are divisible by 9?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a function \\( g \\) that maps nonnegative integers to real numbers, with \\( g(1) = 1 \\), and for all nonnegative integers \\( m \\ge n \\),\n\\[ g(m + n) + g(m - n) = \\frac{g(3m) + g(3n)}{3} \\]\nFind the sum of all possible values of \\( g(10) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 6826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a game involving two standard decks of 52 cards each mixed together, making a total of 104 cards. Each deck has 13 ranks and 4 suits, with the suits retaining their colors as in a standard deck. What is the probability that the top card of this combined deck is the Kin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"index": 6827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a geometric sequence where the first term is $\\frac{5}{8}$, and the second term is $25$. What is the smallest $n$ for which the $n$th term of the sequence, multiplied by $n!$, is divisible by one billion (i.e., $10^9$)?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$. \n(1) Determine the range of the real number $a$ and the expression for $S_n$; \n(2) Does there exist a real number $a$ such that $S_1$, $S_3$,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calcul... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 6830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a kite-shaped field with the following measurements and angles: sides AB = 120 m, BC = CD = 80 m, DA = 120 m. The angle between sides AB and BC is 120°. The angle between sides CD and DA is also 120°. The wheat harvested from any location in the field is brought to the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a large square divided into a grid of \\(5 \\times 5\\) smaller squares, each with side length \\(1\\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of th... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{25}"
} | {
"index": 6832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a large square of side length 60 units, subdivided into a grid with non-uniform rows and columns. The rows are divided into segments of 20, 20, and 20 units, and the columns are divided into segments of 15, 15, 15, and 15 units. A shaded region is created by connecting... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 6833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a large square where each side is divided into four equal parts. At each division, a point is placed. An inscribed square is constructed such that its corners are at these division points nearest to the center of each side of the large square. Calculate the ratio of th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a larger grid extending from point $A$ to point $B$, now divided into a 3x2 grid. You can still only move right or down along the drawn segments. How many different routes are there from point $A$ to point $B$?\n\n[asy]\nunitsize(0.09inch);\ndraw((0,0)--(15,0)--(15,10)... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a list of numbers \\[8, 3, x, 3, 7, 3, y\\]. When the mean, median, and mode of this list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible values of \\(y\\) if \\(x=6\\)?\nA) $\\frac{51}{13}$ \nB) $33$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{480}{13}"
} | {
"index": 6836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four num... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 6837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a list where each integer $n$ from 1 to 300 appears exactly $n$ times. What is the median of this number list?"
}
] | math | {
"style": "rule",
"ground_truth": "212"
} | {
"index": 6838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a machine that when a positive integer $N$ is entered, the machine's processing rule is:\n\n- If $N$ is odd, output $4N + 2$.\n- If $N$ is even, output $N / 2$.\n\nUsing the above rule, if starting with an input of $N = 9$, after following the machine's process for six... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 6839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a matrix $A = \\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \\in \\{0, 1\\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a memorable $9$-digit telephone number defined as $d_1d_2d_3d_4-d_5d_6d_7d_8d_9$. A number is memorable if the prefix sequence $d_1d_2d_3d_4$ is exactly the same as either of the sequences $d_5d_6d_7d_8$ or $d_6d_7d_8d_9$. Each digit $d_i$ can be any of the ten decimal... | math | {
"style": "rule",
"ground_truth": "199990"
} | {
"index": 6841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a modified function $C' = \\frac{2en}{R + 2nr}$, where $e = 4$, $R = 6$, and $r = 3$. Determine the behavior of $C'$ as $n$ increases from a positive starting point."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 6842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). Fro... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a modified sequence rule:\n1) If a number is 30 or less, triple the number.\n2) If a number is more than 30, subtract 15 from it.\n\nLet $G$ be the first number in a sequence generated by the new rule. $G$ is a \"magic number\" if 18 is not a term in the sequence that ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a modified sequence whose $n$th term is defined by $a_n = (-1)^n \\cdot \\lfloor \\frac{3n+1}{2} \\rfloor$. What is the average of the first $150$ terms of this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{6}"
} | {
"index": 6845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a month with 31 days, where the number of the month is a product of two distinct primes (e.g., July, represented as 7). Determine how many days in July are relatively prime to the month number."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 6846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a new infinite geometric series: $$\\frac{7}{4} + \\frac{28}{9} + \\frac{112}{27} + \\dots$$\nDetermine the common ratio of this series."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"index": 6847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a number line, with a lily pad placed at each integer point. A frog is standing at the lily pad at the point 0 on the number line, and wants to reach the lily pad at the point 2014 on the number line. If the frog stands at the point $n$ on the number line, it can jump ... | math | {
"style": "rule",
"ground_truth": "0.9102805441016536"
} | {
"index": 6848,
"solution": "First, we establish a rough upper bound for the probability $p$. Let $q$ be the probability that the frog can reach the lily pad at the point 2014 on the number line if it is allowed to jump from a point $n$ on the number line to the point $n+1$, in addition to the points $n+2$ and $n+... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"index": 6849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{41}"
} | {
"index": 6850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a pentagonal prism with seven faces, fifteen edges, and ten vertices. One of its faces will be used as the base for a new pyramid. Calculate the maximum value of the sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid)."
... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 6851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a permutation $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ of $\\{1,2,3,4,5\\}$. We say the tuple $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ is flawless if for all $1 \\leq i<j<k \\leq 5$, the sequence $\\left(a_{i}, a_{j}, a_{k}\\right)$ is not an arithm... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6852,
"solution": "We do casework on the position of 3. - If $a_{1}=3$, then the condition is that 4 must appear after 5 and 2 must appear after 1. It is easy to check there are six ways to do this. - If $a_{2}=3$, then there are no solutions; since there must be an index $i \\geq 3$ with $a_{i}=6-a_{1}$... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a polynomial $P(x) \\in \\mathbb{R}[x]$ , with degree $2023$ , such that $P(\\sin^2(x))+P(\\cos^2(x)) =1$ for all $x \\in \\mathbb{R}$ . If the sum of all roots of $P$ is equal to $\\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "4046"
} | {
"index": 6853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a polynomial with integer coefficients given by:\n\\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\\]\nFind the number of different possible rational roots of this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 6854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a quadrilateral ABCD inscribed in a circle with radius 300 meters, where AB = BC = AD = 300 meters, and CD being the side of unknown length. Determine the length of side CD."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 6855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$ with $BC = 2 \\cdot AB$ . Let $\\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\\frac{AK}{KD... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 6856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$ containing three squares. Two smaller squares each occupy a part of rectangle $ABCD$, and each smaller square has an area of 1 square inch. A larger square, also inside rectangle $ABCD$ and not overlapping with the smaller squares, has a side length ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$, and inside it are four squares with non-overlapping interiors. Two squares have the same size and an area of 4 square inches each, located at corners $A$ and $C$ respectively. There is another small square with an area of 1 square inch, and a larger... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle \\( ABCD \\) where the side lengths are \\( \\overline{AB}=4 \\) and \\( \\overline{BC}=8 \\). Points \\( M \\) and \\( N \\) are fixed on sides \\( BC \\) and \\( AD \\), respectively, such that the quadrilateral \\( BMDN \\) is a rhombus. Calculate the ar... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle \\(ABCD\\) which is cut into two parts along a dashed line, resulting in two shapes that resemble the Chinese characters \"凹\" and \"凸\". Given that \\(AD = 10\\) cm, \\(AB = 6\\) cm, and \\(EF = GH = 2\\) cm, find the total perimeter of the two shapes form... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 6860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle with dimensions 6 units by 7 units. A triangle is formed with its vertices on the sides of the rectangle. Vertex $A$ is on the left side, 3 units from the bottom. Vertex $B$ is on the bottom side, 5 units from the left. Vertex $C$ is on the top side, 2 unit... | math | {
"style": "rule",
"ground_truth": "17.5"
} | {
"index": 6861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangle with dimensions 6 units by 8 units. Points $A$, $B$, and $C$ are located on the sides of this rectangle such that the coordinates of $A$, $B$, and $C$ are $(0,2)$, $(6,0)$, and $(3,8)$ respectively. What is the area of triangle $ABC$ in square units?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangular array of single digits \\(d_{i,j}\\) with 10 rows and 7 columns, such that \\(d_{i+1, j} - d_{i, j}\\) is always 1 or -9 for all \\(1 \\leq i \\leq 9\\) and all \\(1 \\leq j \\leq 7\\). For \\(1 \\leq i \\leq 10\\), let \\(m_{i}\\) be the median of \\(d_{... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 6863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a rectangular region of 2x1 unit squares at the center of a large grid of unit squares. Each subsequent ring forms around this rectangle by one unit thickness. Determine the number of unit squares in the $50^{th}$ ring around this central rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "402"
} | {
"index": 6864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a regular hexagon where each of its $6$ sides and the $9$ diagonals are colored randomly and independently either red or blue, each color with the same probability. What is the probability that there exists at least one triangle, formed by three of the hexagon’s vertic... | math | {
"style": "rule",
"ground_truth": "\\frac{1048575}{1048576}"
} | {
"index": 6865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.\nA) $54 + 18\... | math | {
"style": "rule",
"ground_truth": "54 + 36\\sqrt{2} - 9\\pi"
} | {
"index": 6866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a regular octagon. How many triangles can be formed whose vertices are the vertices of the octagon?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 6867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a regular polygon with $2^n$ sides, for $n \\ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\\prod_{i=2}^{\\infty}\\frac{A_i}{A_{i+1}}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\pi}"
} | {
"index": 6868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a regular tetrahedron $ABCD$. Find $\\sin \\angle BAC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"index": 6869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:\n\n\\begin{tabular}{|c|c|}\\hline\n\\textbf{Tens} & \\textbf{Units} \\\\ \\hline\n2 & $0 \\hspace{2mm} 0 \\hspace{2mm} 1 \\hspace{2mm} 1 \\hspace{2mm} 2$ \\\\ \\hline\n3 & $3 \\hspa... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a right rectangular prism \\(B\\) with edge lengths \\(2,\\ 5,\\) and \\(6\\), including its interior. For any real \\(r \\geq 0\\), let \\(T(r)\\) be the set of points in 3D space within a distance \\(r\\) from some point in \\(B\\). The volume of \\(T(r)\\) is expres... | math | {
"style": "rule",
"ground_truth": "\\frac{8112}{240}"
} | {
"index": 6871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon rec... | math | {
"style": "rule",
"ground_truth": "1/2013"
} | {
"index": 6872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\\cdot2$ . If each term of the sequence can be written in the form $a\\cdot r_1^n+b\\cdot r_2^n$ , what is the positive difference between $r_1$ and $r_2$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{2}"
} | {
"index": 6873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $x_1,x_2,\\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\\dots,10$ let \\[ x_{n+2}=\\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \\] Suppose $x_n>0$ for $n=1,2,\\dots,11$ and $x_{12}=0$ . Then the value of $x_2$ can be written as $\\fr... | math | {
"style": "rule",
"ground_truth": "622"
} | {
"index": 6874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $\\{a_n\\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\\ldots+a_n$. Find $a_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 6875,
"solution": "\nWe begin with the sequence \\(\\{a_n\\}\\) of integers defined such that \\(a_1 = 1\\), \\(a_2 = 2\\), and for \\(n \\geq 3\\), \\(a_{n+1}\\) is the largest prime divisor of the sum \\(S_n = a_1 + a_2 + \\ldots + a_n\\). We are tasked with finding the value of \\(a_{100}\\).\n\n### S... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $\\{a_n\\}$ where the sum of the first $n$ terms, $S_n$, satisfies $S_n = 2a_n - a_1$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence.\n\n(1) Find the general formula for the sequence $\\{a_n\\}$.\n(2) Let $b_n = \\log_2 a_n$ and $c_n = \\frac{3}{b_... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $\\{a_n\\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \\cdots + |a_{10}|$."
}
] | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 6877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $\\{a_n\\}$ with property P: if $a_p = a_q$ for $p, q \\in \\mathbb{N}^{*}$, then it must hold that $a_{p+1} = a_{q+1}$. Suppose the sequence $\\{a_n\\}$ has property P, and it is given that $a_1=1$, $a_2=2$, $a_3=3$, $a_5=2$, and $a_6+a_7+a_8=21$. Determine... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\\dots$ defined by\n\\begin{align*}\nx_1 &= \\sqrt[3]{3}, \\\\\nx_2 &= (\\sqrt[3]{3})^{\\sqrt[3]{3}},\n\\end{align*}and in general,\n\\[x_n = (x_{n - 1})^{\\sqrt[3]{3}}\\]for $n > 1.$ What is the smallest value of $n$ for which $x_n$ ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\\frac{2}{3}$. Suppose that $x_{n}=\\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \\geq 4$. Find the least $n$ such that $x_{n} \\leq \\frac{1}{10^{6}}$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6880,
"solution": "The recursion simplifies to $\\frac{x_{n-1}}{x_{n}}+\\frac{x_{n-3}}{x_{n-2}}=2 \\frac{x_{n-2}}{x_{n-1}}$. So if we set $y_{n}=\\frac{x_{n-1}}{x_{n}}$ for $n \\geq 2$ then we have $y_{n}-y_{n-1}=y_{n-1}-y_{n-2}$ for $n \\geq 3$, which means that $\\left\\{y_{n}\\right\\}$ is an arithmet... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence defined as $500, x, 500 - x, \\ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ th... | math | {
"style": "rule",
"ground_truth": "309"
} | {
"index": 6881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence of complex numbers $\\left\\{z_{n}\\right\\}$ defined as \"interesting\" if $\\left|z_{1}\\right|=1$ and for every positive integer $n$, the following holds:\n$$\n4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.\n$$\n\nFind the largest constant $C$ such that for a... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 6882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}."
... | math | {
"style": "rule",
"ground_truth": "13515"
} | {
"index": 6883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence of hexagons where each new hexagon adds two layers of dots instead of one. The first hexagon has 1 dot. The second hexagon has 1 central dot, 6 dots in the first layer, and 12 dots in the second layer, making a total of 19 dots. Determine the total number of... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 6884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence of positive real numbers where \\( a_1, a_2, \\dots \\) satisfy\n\\[ a_n = 9a_{n-1} - n \\]\nfor all \\( n > 1 \\). Find the smallest possible value of \\( a_1 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{64}"
} | {
"index": 6885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sequence of real numbers \\(\\{a_n\\}\\) defined by \\(a_1 = 1\\) and \\(a_{n+1} = \\frac{a_n}{1 + n a_n}\\) for \\(n \\geq 1\\). Find the value of \\(\\frac{1}{a_{2005}} - 2000000\\)."
}
] | math | {
"style": "rule",
"ground_truth": "9011"
} | {
"index": 6886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a series of squares where each square has a border of dots around it, each border containing dots on all four sides. The smallest square (first in the series) has a single dot at the center. Each successive square surrounds the previous one with an additional border of... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 6887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a set of 150 cards numbered from 1 to 150. Each card is randomly placed in a box. A card is selected at random from this box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"index": 6888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a set of circles in the upper half-plane, all tangent to the $x$-axis. Begin with two circles of radii $50^2$ and $53^2$, externally tangent to each other, defined as Layer $L_0$. For each pair of consecutive circles in $\\bigcup_{j=0}^{k-1}L_j$, a new circle in Layer ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1517}"
} | {
"index": 6889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\\sqrt{d} - b$ cm. Determine the value of $b+d$."
}
] | math | {
"style": "rule",
"ground_truth": "19.5"
} | {
"index": 6890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shade... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 6891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inwar... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 6892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of ... | math | {
"style": "rule",
"ground_truth": "\\frac{26}{37}"
} | {
"index": 6893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square flag with a red cross of uniform width and a blue triangular central region on a white background. The cross is symmetric with respect to each of the diagonals of the square. Let's say the entire cross, including the blue triangle, occupies 45% of the area of ... | math | {
"style": "rule",
"ground_truth": "1.08\\%"
} | {
"index": 6894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square in the coordinate plane with vertices at $(2, 1)$, $(5, 1)$, $(5, 4)$, and $(2, 4)$. A line joining $(2, 3)$ and $(5, 1)$ divides the square shown into two parts. Determine the fraction of the area of the square that is above this line."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 6895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square pyramid $S-ABCD$ with a height of $h$. The base $ABCD$ is a square with side length 1. Points $S$, $A$, $B$, $C$, and $D$ all lie on the surface of a sphere with radius 1. The task is to find the distance between the center of the base $ABCD$ and the vertex $S... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 6896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 6897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1. Find the difference between the sum of the areas of the squares and the sum of the areas of ... | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\pi}{2}"
} | {
"index": 6898,
"solution": "The ratio of the area of each square and the circle immediately inside it is $\\frac{4}{\\pi}$. The total sum of the areas of the squares is $1+\\frac{1}{2}+\\frac{1}{4}+\\ldots=2$. Difference in area is then $2-2 \\cdot \\frac{4}{\\pi} = 2 - \\frac{\\pi}{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a square-based pyramid (with base vertices $A, B, C, D$) with equal side edges, and let the apex be $E$. Let $P$ be the point that divides the side edge $A E$ in a ratio of 3:1, such that $E P : P A = 3$, and let $Q$ be the midpoint of the side edge $C E$. In what rati... | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 6899,
"solution": ""
} |
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