data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "We color each number in the set $S = \\{1, 2, ..., 61\\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum ... | math | {
"style": "rule",
"ground_truth": "119"
} | {
"index": 36900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We color some cells in $10000 \\times 10000$ square, such that every $10 \\times 10$ square and every $1 \\times 100$ line have at least one coloring cell. What minimum number of cells we should color ?"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 36901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We colored the $n^2$ unit squares of an $n\\times n$ square lattice such that in each $2\\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used?"
}
] | math | {
"style": "rule",
"ground_truth": "[\\frac{n^2+2n-1}{2}]"
} | {
"index": 36902,
"solution": "\nTo solve the problem, we must determine the largest number of distinct colors that can be used to color an \\( n \\times n \\) square lattice, under the condition that within every \\( 2 \\times 2 \\) sub-square, at least two of the four unit squares share the same color.\n\n### Ana... |
DeepScaleR | [
{
"role": "user",
"content": "We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either\nred or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.\nA triangle formed by vertices of $P$ is called monochromatic if all of its sides ha... | math | {
"style": "rule",
"ground_truth": "859"
} | {
"index": 36903,
"solution": "\nGiven a regular polygon \\( P \\) with 43 vertices, each segment (sides and diagonals) of this polygon is colored either red or blue. We know the following conditions:\n- Every vertex is an endpoint of 20 red segments.\n- Every vertex is an endpoint of 22 blue segments.\n\nSince eve... |
DeepScaleR | [
{
"role": "user",
"content": "We consider a white \\( 5 \\times 5 \\) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 36904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We consider an $n \\times n$ table, with $n\\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish i... | math | {
"style": "rule",
"ground_truth": "\\frac{n(n+1)}{2}"
} | {
"index": 36905,
"solution": "\nGiven an \\( n \\times n \\) table, where \\( n \\geq 1 \\), the task is to determine the maximum number of cells \\( k \\) that can be colored such that there is a unique way to place \\( n \\) tokens on the colored cells. Importantly, no two tokens should be in the same row or col... |
DeepScaleR | [
{
"role": "user",
"content": "We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which ... | math | {
"style": "rule",
"ground_truth": "3\\mid n"
} | {
"index": 36906,
"solution": "\nTo solve the problem, we need to determine which positive integers \\( n \\ge 4 \\) allow a regular \\( n \\)-gon to be dissected into a bicoloured triangulation under the condition that, for each vertex \\( A \\), the number of black triangles having \\( A \\) as a vertex is greate... |
DeepScaleR | [
{
"role": "user",
"content": "We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\\le i\\le k}$ with $$1=d_1<d_2<\\dots <d_k=n\\quad (k\\ge 6).$$\nFind all positive integers $n$ such that $$n=d_5^2+d_6^2.$$"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 36907,
"solution": "\nLet \\( n \\) be a positive integer with at least six positive divisors. The sequence of divisors of \\( n \\) is \\( (d_i)_{1 \\le i \\le k} \\) where:\n\n\\[\n1 = d_1 < d_2 < \\cdots < d_k = n \\quad (k \\ge 6).\n\\]\n\nWe need to find all \\( n \\) such that:\n\n\\[\nn = d_5^2 + ... |
DeepScaleR | [
{
"role": "user",
"content": "We define $N$ as the set of natural numbers $n<10^6$ with the following property:\n\nThere exists an integer exponent $k$ with $1\\le k \\le 43$ , such that $2012|n^k-1$ .\n\nFind $|N|$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"index": 36908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define $|\\begin{array}{l}{a}&{c}\\\\{b}&{d}\\end{array}|=ad-bc$. For example, $|\\begin{array}{l}{1}&{3}\\\\{2}&{4}\\end{array}|=1\\times 4-2\\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \\lt |\\begin{array}{l}{2}&{y}\\\\{x}&{3}\\end{array}| \\lt 3$, then t... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 36909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define \\( a @ b = a \\times (a + 1) \\times \\ldots \\times (a + b - 1) \\). Given \\( x @ y @ 2 = 420 \\), then \\( y @ x = \\) ?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define \\( a \\star b = a \\times a - b \\times b \\). Find the value of \\( 3 \\star 2 + 4 \\star 3 + 5 \\star 4 + \\cdots + 20 \\star 19 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"index": 36911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and\n\n$f(b)=3b+1$ if $b$ is odd\n\n$f(b)=\\frac{b}{2}$ if $b$ is even.\n\nWhat is the smallest possible number of integers in the domain of $f$?"
}... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:\n1. $g(b)=3b+1$ if $b$ is odd\n2. $g(b)=\\frac{b}{2}$ if $b$ is even.\nWhat is the smallest possible number of integers in th... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 36913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 36914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define five-digit numbers like 31024 and 98567 as \"Shenma numbers\", where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different. How many such five-digit numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 36915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define the polynomial $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$ Find the largest prime divisor of $P (2)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 36916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$ Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4/9 \\\\ 4/3 \\end{pmatrix}"
} | {
"index": 36917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We defined an operation denoted by $*$ on the integers, which satisfies the following conditions: \n\n1) $x * 0 = x$ for every integer $x$;\n2) $0 * y = -y$ for every integer $y$;\n3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.\n\nDetermine... | math | {
"style": "rule",
"ground_truth": "1639"
} | {
"index": 36918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$ and gcd $(12, 15)=3$ .) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\\bullet$ gcd $(a, b, c)=1$ , $\\bullet$ gcd $(a, b + c)... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We denote by $\\mathbb{R}^\\plus{}$ the set of all positive real numbers.\n\nFind all functions $f: \\mathbb R^ \\plus{} \\rightarrow\\mathbb R^ \\plus{}$ which have the property:\n\\[f(x)f(y)\\equal{}2f(x\\plus{}yf(x))\\]\nfor all positive real numbers $x$ and $y$.\n\n[i]"
}... | math | {
"style": "rule",
"ground_truth": "f(x) = 2"
} | {
"index": 36920,
"solution": "\nTo solve the problem, we need to find all functions \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) such that for all positive real numbers \\( x \\) and \\( y \\), the functional equation holds:\n\n\\[\nf(x)f(y) = 2f(x + yf(x)).\n\\]\n\n### Step 1: Analyze the Functional Eq... |
DeepScaleR | [
{
"role": "user",
"content": "We divide the height of a cone into three equal parts, and through the division points, we lay planes parallel to the base. How do the volumes of the resulting solids compare to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "1:7:19"
} | {
"index": 36921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 36922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one verte... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 36923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 36925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We draw two lines $(\\ell_1) , (\\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "60^\\circ, 60^\\circ, 60^\\circ"
} | {
"index": 36926,
"solution": "\nWe are given that the lines \\((\\ell_1)\\) and \\((\\ell_2)\\) pass through the orthocenter \\(H\\) of triangle \\(ABC\\) and each line divides the triangle into two figures of equal area and equal perimeters. We need to determine the angles of the triangle.\n\nThe orthocenter \\(H... |
DeepScaleR | [
{
"role": "user",
"content": "We drew the face diagonals of a cube with an edge length of one unit and drew a sphere centered around the cube's center. The sphere intersected the diagonals at the vertices of a convex polyhedron, all of whose faces are regular. What was the radius of the sphere?"
}
] | math | {
"style": "rule",
"ground_truth": "0.579"
} | {
"index": 36927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{128}"
} | {
"index": 36928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{45}{1024}"
} | {
"index": 36929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{1024}"
} | {
"index": 36930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?"
}
] | math | {
"style": "rule",
"ground_truth": "382"
} | {
"index": 36931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have $ 23^2 = 529 $ ordered pairs $ (x, y) $ with $ x $ and $ y $ positive integers from 1 to 23, inclusive. How many of them have the property that $ x^2 + y^2 + x + y $ is a multiple of 6?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 36932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$ . If the chosen sequence is $a_1, a_2, ..., a_k$ ( $k$ is not fixed), then at the $i^{th}$ step, we choo... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"index": 36933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ ."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 36934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have 10 points on a line A_{1}, A_{2} \\cdots A_{10} in that order. Initially there are n chips on point A_{1}. Now we are allowed to perform two types of moves. Take two chips on A_{i}, remove them and place one chip on A_{i+1}, or take two chips on A_{i+1}, remove them, an... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 36935,
"solution": "We claim that n=46 is the minimum possible value of n. As having extra chips cannot hurt, it is always better to perform the second operation than the first operation, except on point A_{1}. Assign the value of a chip on point A_{i} to be i. Then the total value of the chips initially... |
DeepScaleR | [
{
"role": "user",
"content": "We have 10 springs, each originally $0.5 \\mathrm{~m}$ long with a spring constant of $200 \\mathrm{~N}/\\mathrm{m}$. A mass of $2 \\mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Ne... | math | {
"style": "rule",
"ground_truth": "10.39"
} | {
"index": 36936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have 21 pieces of type $\\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \\times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a $100\\times100$ garden and we’ve plant $10000$ trees in the $1\\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 36938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a $6 \\times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?"
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\\left\\lfloor\\frac{x}{2}\\right\\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than o... | math | {
"style": "rule",
"ground_truth": "233"
} | {
"index": 36940,
"solution": "We consider the integers from this process written in binary. The first operation truncates the rightmost digit, while the second operation appends 01 to the right. We cannot have a number with a substring 11. For simplicity, call a string valid if it has no consecutive $1^{\\prime} s... |
DeepScaleR | [
{
"role": "user",
"content": "We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36942,
"solution": "This is obtainable by construction. Consider two tetrahedrons glued along a face; this gives us 5 vertices, 9 edges, and 6 faces, for a total of 20 , and one readily checks that the required Eulerian path exists. Now, to see that we cannot do better, first notice that the number $v$ o... |
DeepScaleR | [
{
"role": "user",
"content": "We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 36943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{4165}"
} | {
"index": 36944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a triangle $\\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\\triangle ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 36945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{39}"
} | {
"index": 36946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have an $n$-gon, and each of its vertices is labeled with a number from the set $\\{1, \\ldots, 10\\}$. We know that for any pair of distinct numbers from this set there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible va... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 36947,
"solution": "Each number be paired with each of the 9 other numbers, but each vertex can be used in at most 2 different pairs, so each number must occur on at least $\\lceil 9 / 2\\rceil=5$ different vertices. Thus, we need at least $10 \\cdot 5=50$ vertices, so $n \\geq 50$. To see that $n=50$ is... |
DeepScaleR | [
{
"role": "user",
"content": "We have an equilateral triangle with circumradius $1$ . We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$ , is maximum.\n(The total distance of ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 36948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.<br/>① Grouping method:<br/>For example, $x^{2}-2xy+y^{2}-25... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 36950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 36951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have that $3 \\cdot f(x) + 4 \\cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 36952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have that $\\lceil x \\rceil - \\lfloor x \\rfloor = 0.$ Then, what is $\\lceil x \\rceil - x$?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 36953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We have two concentric circles $C_{1}$ and $C_{2}$ with radii 1 and 2, respectively. A random chord of $C_{2}$ is chosen. What is the probability that it intersects $C_{1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "N/A"
} | {
"index": 36954,
"solution": "The question given at the beginning of the problem statement is a famous problem in probability theory widely known as Bertrand's paradox. Depending on the interpretation of the phrase \"random chord,\" there are at least three different possible answers to this question: - If the ran... |
DeepScaleR | [
{
"role": "user",
"content": "We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 36955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We inscribe spheres with a radius of \\(\\frac{1}{2}\\) around the vertices of a cube with edge length 1. There are two spheres that touch each of these eight spheres. Calculate the difference in volume between these two spheres."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3} \\pi"
} | {
"index": 36956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 36957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 36958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 36959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We know the following to be true:\n\n$\\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$\n\n$\\bullet$ 2. $Z$ = $K \\times K^2.$\n\nWhat is the value of $K$ for which $Z$ is a perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 36960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We label the sides of a dodecagon as \\(C_{1}, C_{2}, \\ldots, C_{12}\\). In how many ways can we color the sides of a dodecagon with four colors such that two adjacent sides are always colored differently? (Two colorings are considered different if any one side \\(C_{i}\\) is ... | math | {
"style": "rule",
"ground_truth": "531444"
} | {
"index": 36961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We make colored cubes according to the following specifications:\n\n1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.\n2. Each face's two triangles must be of different ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We marked the midpoints of all sides and diagonals of a regular 1976-sided polygon. What is the maximum number of these points that can lie on a single circle?"
}
] | math | {
"style": "rule",
"ground_truth": "1976"
} | {
"index": 36963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We measured the angle of elevation of an antenna tower standing on flat terrain from distances of $100 \\text{ m}$, $200 \\text{ m}$, and $300 \\text{ m}$ from its base. The sum of the three angles is $90^\\circ$. How tall is the tower?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 36964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We measured the elevation angles of a tower standing on a horizontal plane from points $50 \\mathrm{~m}$ and $100 \\mathrm{~m}$ away from the base of the tower. The sum of the measured angles is $45^{\\circ}$. How tall is the tower?"
}
] | math | {
"style": "rule",
"ground_truth": "28.08"
} | {
"index": 36965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We need to arrange the performance order for 4 singing programs and 2 skit programs. The requirement is that there must be exactly 3 singing programs between the 2 skit programs. The number of possible arrangements is \\_\\_\\_\\_\\_\\_ . (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 36966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We place planes through each edge and the midpoint of the edge opposite to it in a tetrahedron. Into how many parts do these planes divide the tetrahedron, and what are the volumes of these parts?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touc... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We plotted the graph of the function \\( f(x) = \\frac{1}{x} \\) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \\( g(x) = \\frac{2}{x} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 36969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?"
}
] | math | {
"style": "rule",
"ground_truth": "606"
} | {
"index": 36970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We randomly choose a function $f:[n] \\rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \\geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\\left(f^... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{n}"
} | {
"index": 36971,
"solution": "Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (including 1 itself), if there is one, and zero if there is no such cycle. The problem is equivalent to finding $\\mathbb{E}(N(f)) / n$. Note that $P(N(f)=k)=\\frac{n-1}{n} \\cdot \\frac{n-... |
DeepScaleR | [
{
"role": "user",
"content": "We randomly throw a 10-filér coin onto a $40 \\times 40$ cm chessboard. What is the probability that the coin:\n\na) will be entirely inside one square?\n\nb) will partially cover one and only one square edge?\n\nc) will partially cover two square edges but not a vertex?\n\nd) ... | math | {
"style": "rule",
"ground_truth": "0.1134"
} | {
"index": 36972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We regularly transport goods from city $A$ to city $B$, which is $183 \\mathrm{~km}$ away. City $A$ is $33 \\mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per kilometer is half as much on the river as on land. Where should we ... | math | {
"style": "rule",
"ground_truth": "11\\sqrt{3}"
} | {
"index": 36973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$?"
}
] | math | {
"style": "rule",
"ground_truth": "2/5"
} | {
"index": 36974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{32}"
} | {
"index": 36975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We roll a fair die consecutively until the sum of the numbers obtained, \\( S \\), exceeds 100. What is the most probable value of \\( S \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 36976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We roll five dice, each a different color. In how many ways can the sum of the rolls be 11?"
}
] | math | {
"style": "rule",
"ground_truth": "205"
} | {
"index": 36977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We rolled ten fair six-sided dice at once, and the product of the numbers rolled was 7776. We know that the highest number rolled appeared only once. What is the sum of the numbers rolled?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We say a point is contained in a square if it is in its interior or on its boundary. Three unit squares are given in the plane such that there is a point contained in all three. Furthermore, three points $A, B, C$, are given, each contained in at least one of the squares. Find ... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} / 2"
} | {
"index": 36979,
"solution": "Let $X$ be a point contained in all three squares. The distance from $X$ to any point in any of the three squares is at most $\\sqrt{2}$, the length of the diagonal of the squares. Therefore, triangle $A B C$ is contained in a circle of radius $\\sqrt{2}$, so its circumradius is at mo... |
DeepScaleR | [
{
"role": "user",
"content": "We say a triple $\\left(a_{1}, a_{2}, a_{3}\\right)$ of nonnegative reals is better than another triple $\\left(b_{1}, b_{2}, b_{3}\\right)$ if two out of the three following inequalities $a_{1}>b_{1}, a_{2}>b_{2}, a_{3}>b_{3}$ are satisfied. We call a triple $(x, y, z)$ specia... | math | {
"style": "rule",
"ground_truth": "n \\geq 4"
} | {
"index": 36980,
"solution": "The answer is $n \\geqslant 4$. Consider the following set of special triples $$\\left(0, \\frac{8}{15}, \\frac{7}{15}\\right), \\quad\\left(\\frac{2}{5}, 0, \\frac{3}{5}\\right), \\quad\\left(\\frac{3}{5}, \\frac{2}{5}, 0\\right), \\quad\\left(\\frac{2}{15}, \\frac{11}{15}, \\frac{2}... |
DeepScaleR | [
{
"role": "user",
"content": "We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegat... | math | {
"style": "rule",
"ground_truth": "n\\geq4"
} | {
"index": 36981,
"solution": "\nTo solve this problem, we need to determine for which natural numbers \\( n \\) there exists a set \\( S \\) of special triples, with \\( |S| = n \\), such that any special triple is bettered by at least one element of \\( S \\).\n\n### Understanding the Definitions\n\nA **special t... |
DeepScaleR | [
{
"role": "user",
"content": "We say that a number of 20 digits is *special* if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials."
}
] | math | {
"style": "rule",
"ground_truth": "10^9 - 1"
} | {
"index": 36982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We say that a positive real number $d$ is good if there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \\ldots \\in(0, d)$ such that for each $n$, the points $a_{1}, \\ldots, a_{n}$ partition the interval $[0, d]$ into segments of length at most $1 / n$ each. Find $\\sup \\{... | math | {
"style": "rule",
"ground_truth": "\\ln 2"
} | {
"index": 36983,
"solution": "Let $d^{\\star}=\\sup \\{d \\mid d$ is good $\\}$. We will show that $d^{\\star}=\\ln (2) \\doteq 0.693$. 1. $d^{\\star} \\leq \\ln 2:$ Assume that some $d$ is good and let $a_{1}, a_{2}, \\ldots$ be the witness sequence. Fix an integer $n$. By assumption, the prefix $a_{1}, \\ldots, ... |
DeepScaleR | [
{
"role": "user",
"content": "We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \\cdots, a_n \\in S$, all integer roots of the polynomial $a_0+a_1x+\\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form ... | math | {
"style": "rule",
"ground_truth": "\\mathbb{Z}"
} | {
"index": 36984,
"solution": "\nTo find all rootiful sets of integers \\( S \\) that contain all numbers of the form \\( 2^a - 2^b \\) for positive integers \\( a \\) and \\( b \\), we need to analyze the properties of such sets.\n\n### Step 1: Understand the Definition\n\nA set \\( S \\) is rootiful if, for any p... |
DeepScaleR | [
{
"role": "user",
"content": "We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We say that two natural numbers form a perfect pair when the sum and the product of these two numbers are perfect squares. For example, 5 and 20 form a perfect pair because $5+20=25=5^{2}$ and $5 \\times 20=100=10^{2}$. Does 122 form a perfect pair with any other natural number... | math | {
"style": "rule",
"ground_truth": "122 \\times 121"
} | {
"index": 36986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We seek the true statement. There are one hundred statements written in a notebook:\n\n1) There is exactly one false statement in this notebook.\n2) There are exactly two false statements in this notebook.\n...\n100) There are exactly one hundred false statements in this notebo... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 36987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\\frac{1}{3}$. W... | math | {
"style": "rule",
"ground_truth": "17/27"
} | {
"index": 36989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$. \n\nWhat is the length of the broken line?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 36990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We toss a die 30 times. Determine, with an accuracy of 8 decimal places, the probability that there will be at least one six among the numbers rolled, and that no five or four appears before the first six."
}
] | math | {
"style": "rule",
"ground_truth": "0.33333333"
} | {
"index": 36991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 ho... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 36992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We use \\( S_{k} \\) to represent an arithmetic sequence with the first term \\( k \\) and common difference \\( k^{2} \\). For example, \\( S_{3} \\) is \\( 3, 12, 21, \\cdots \\). If 306 is a term in \\( S_{k} \\), the sum of all possible \\( k \\) that satisfy this condition... | math | {
"style": "rule",
"ground_truth": "326"
} | {
"index": 36993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_{1}$ attacks another American $A_{2}$, then $A_{2}$ also attacks $A_{1}$. Let $m$ be the number of squares that an American attacks whe... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 36994,
"solution": "Since one of the Americans must be in the top left corner, that eliminates $m$ squares from consideration for placing additional Americans. So $m+n$ is at most 64, which implies $m n$ can be at most 1024. To achieve 1024, we can color a chessboard the normal way, and say that an Ameri... |
DeepScaleR | [
{
"role": "user",
"content": "We want to set up an electric bell. The location of the bell is at a distance of $30 \\mathrm{~m}$ from the ringing spot. The internal resistance of the bell is 2 ohms. We plan to use 2 Leclanché cells connected in series, each with an electromotive force of 1.5 Volts and an in... | math | {
"style": "rule",
"ground_truth": "0.63"
} | {
"index": 36995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We will call a ticket with a number from 000000 to 999999 excellent if the difference between some two adjacent digits of its number is 5.\n\nFind the number of excellent tickets."
}
] | math | {
"style": "rule",
"ground_truth": "409510"
} | {
"index": 36996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \\(4 = 2^{2}\\). Which of the following is a common divisor of the smallest and the largest ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \\(4 = 2^{2}\\). Which of the following is a common divisor of the smallest and the largest ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We wish to color the integers $1,2,3, \\ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "186"
} | {
"index": 36999,
"solution": "The condition is equivalent to never having an odd number and an even number in the same color. We can choose one of the three colors for the odd numbers and distribute the other two colors freely among the 5 even numbers; this can be done in $3 \\cdot 2^{5}=96$ ways. We can also choo... |
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