data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "There exists a polynomial $P$ of degree 5 with the following property: if $z$ is a complex number such that $z^{5}+2004 z=1$, then $P(z^{2})=0$. Calculate the quotient $P(1) / P(-1)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2010012 / 2010013"
} | {
"index": 35400,
"solution": "Let $z_{1}, \\ldots, z_{5}$ be the roots of $Q(z)=z^{5}+2004 z-1$. We can check these are distinct (by using the fact that there's one in a small neighborhood of each root of $z^{5}+2004 z$, or by noting that $Q(z)$ is relatively prime to its derivative). And certainly none of the roo... |
DeepScaleR | [
{
"role": "user",
"content": "There exists a positive number $m$ such that the positive roots of the equation $\\sqrt{3} \\sin x - \\cos x = m$ form an arithmetic sequence in ascending order. If the point $A(1, m)$ lies on the line $ax + by - 2 = 0 (a > 0, b > 0)$, find the minimum value of $\\frac{1}{a} + ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 35401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There exists a positive real number $x$ such that $ \\cos (\\arctan (x)) = x $. Find the value of $x^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"index": 35402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There exists a positive real number $x$ such that $\\cos (\\tan^{-1}(x))=x$. Find the value of $x^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "(-1+\\sqrt{5})/2"
} | {
"index": 35403,
"solution": "Draw a right triangle with legs $1, x$; then the angle $\\theta$ opposite $x$ is $\\tan^{-1} x$, and we can compute $\\cos (\\theta)=1 / \\sqrt{x^{2}+1}$. Thus, we only need to solve $x=1 / \\sqrt{x^{2}+1}$. This is equivalent to $x \\sqrt{x^{2}+1}=1$. Square both sides to get $x^{4}+... |
DeepScaleR | [
{
"role": "user",
"content": "There exists a real number $k$ such that the equation\n\\[\\begin{pmatrix} 3 \\\\ 5 \\end{pmatrix} + t \\begin{pmatrix} 4 \\\\ -7 \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ -2 \\end{pmatrix} + s \\begin{pmatrix} -1 \\\\ k \\end{pmatrix}\\]does not have any solutions in $t$ and $s... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"index": 35404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There exists a scalar $c$ so that\n\\[\\mathbf{i} \\times (\\mathbf{v} \\times \\mathbf{i}) + \\mathbf{j} \\times (\\mathbf{v} \\times \\mathbf{j}) + \\mathbf{k} \\times (\\mathbf{v} \\times \\mathbf{k}) = c \\mathbf{v}\\]for all vectors $\\mathbf{v}.$ Find $c.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 35405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There exists a scalar $k$ such that for any vectors $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ such that $\\mathbf{a} + \\mathbf{b} + \\mathbf{c} = \\mathbf{0},$ the equation\n\\[k (\\mathbf{b} \\times \\mathbf{a}) + \\mathbf{b} \\times \\mathbf{c} + \\mathbf{c} \\times \\... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 35406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There exists a unique positive integer $a$ for which the sum \\[U=\\sum_{n=1}^{2023}\\left\\lfloor\\dfrac{n^{2}-na}{5}\\right\\rfloor\\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.\n (Note that $\\lfloor x\\rfloor$ denotes the greatest in... | math | {
"style": "rule",
"ground_truth": "944"
} | {
"index": 35407,
"solution": "Consider the integral \\[\\int_{0}^{2023} \\dfrac{n^2-na}{5} \\, dn.\\] We hope this will give a good enough appoximation of $U$ to find $a.$ However, this integral can be easily evaluated(if you know calculus) to be \\[\\dfrac{1}{15}2023^3-\\dfrac{a}{10}2023^2=2023^2\\left(\\dfrac{20... |
DeepScaleR | [
{
"role": "user",
"content": "There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \\dots < a_k$ such that $\\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \\dots + 2^{a_k}.$ What is $k?$"
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 35408,
"solution": "1. **Understanding the Problem:**\n We need to find the number of terms in the binary expansion of the expression \\(\\frac{2^{289}+1}{2^{17}+1}\\).\n\n2. **Expressing in Binary:**\n The numerator \\(2^{289} + 1\\) in binary is \\(1\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\... |
DeepScaleR | [
{
"role": "user",
"content": "There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \\dots < a_k$ such that\\[\\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \\dots + 2^{a_k}.\\]What is $k?$"
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 35409,
"solution": "1. **Express the problem in binary**: We start by expressing the numbers in binary:\n \\[\n \\frac{2^{289}+1}{2^{17}+1} = \\frac{1\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#0\\#1_2}{1\\#1_2}\n \\]\n where $\\#$ represents $16$ consecutive $0$s.\n\n2. **Expa... |
DeepScaleR | [
{
"role": "user",
"content": "There is a $40\\%$ chance of rain on Saturday and a $30\\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\\frac{a}{b}$, wh... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"index": 35410,
"solution": "Let $x$ be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have $\\dfrac{3}{5}x+\\dfrac{2}{5}2x = \\dfrac{3}{10} \\implies \\dfrac{7}{5}x=\\dfrac{3}{10}$ $\\implies x=\\dfrac{3}{14}$. Therefore, the probability that it doesn't rain on either day... |
DeepScaleR | [
{
"role": "user",
"content": "There is a $6 \\times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the \"on\" position. Compute the number of different configurations of... | math | {
"style": "rule",
"ground_truth": "3970"
} | {
"index": 35411,
"solution": "Take any configuration of switches such that there exists at least one row and one column which are switched on. There are $\\left(2^{6}-1\\right)^{2}=3969$ such configurations. We prove that any two such configurations $A$ and $B$ lead to a different set of lights. Without loss of ge... |
DeepScaleR | [
{
"role": "user",
"content": "There is a $6 \\times 6$ square in which all cells are white. In one move, you are allowed to change the color of both cells in any domino (rectangle consisting of two cells) to the opposite color. What is the minimum number of moves required to obtain a square with a checkerbo... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 35412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a \\(4 \\times 4\\) square. Its cells are called neighboring if they share a common side. All cells are painted in two colors: red and blue. It turns out that each red cell has more red neighbors than blue ones, and each blue cell has an equal number of red and blue ne... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 35414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a basket of apples. If Class A shares the apples such that each person gets 3 apples, 10 apples remain. If Class B shares the apples such that each person gets 4 apples, 11 apples remain. If Class C shares the apples such that each person gets 5 apples, 12 apples remai... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 35415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a beach soccer tournament with 17 teams, where each team plays against every other team exactly once. A team earns 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win in a penalty shootout. The losing team earns no points. What i... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 35416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 35417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number o... | math | {
"style": "rule",
"ground_truth": "19/60"
} | {
"index": 35418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a box containing red, blue, green, and yellow balls. It is known that the number of red balls is twice the number of blue balls, the number of blue balls is twice the number of green balls, and the number of yellow balls is more than seven. How many yellow balls are in... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \\( N \\) be the number of two-humped camels. How ma... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 35420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a card game called \"Twelve Months\" that is played only during the Chinese New Year. The rules are as follows:\n\nStep 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.\n\nStep 2: Lay ou... | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"index": 35421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a certain regularity in the operation between rational numbers and irrational numbers. For example, if $a$ and $b$ are rational numbers, and $a(\\pi +3)+b=0$, then $a=0$, $b=0$. Given that $m$ and $n$ are rational numbers:<br/>$(1)$ If $(m-3)×\\sqrt{6}+n-3=0$, then the... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{9}"
} | {
"index": 35422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a championship where 16 football teams participate, each playing with every other team exactly once. What is the minimum number of games that must be played so that among any three teams, there are at least two that have already played against each other?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 35423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a city with $n$ citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following:\n\n[list]\n[*][i]Virus positive[/i]: there is at least one currentl... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"index": 35424,
"solution": "\nTo determine the smallest number of tests required to ascertain if the sceptervirus is currently present or has been present in the city, let's analyze the given conditions for the test results:\n\n1. **Virus positive**: Indicates there is at least one currently infected individual ... |
DeepScaleR | [
{
"role": "user",
"content": "There is a committee composed of 10 members who meet around a table: 7 women, 2 men, and 1 child. The women sit in indistinguishable rocking chairs, the men on indistinguishable stools, and the child on a bench, also indistinguishable from any other benches. How many distinct w... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 35425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a committee composed of eight women and two men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways are there for me to arrange the eight chairs and two stools for a meeting... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 35426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that\n\\[\\frac {z}{z + n} = 4i.\\]\n Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "697"
} | {
"index": 35427,
"solution": "\\[\\frac {z}{z+n}=4i\\]\n\\[1-\\frac {n}{z+n}=4i\\]\n\\[1-4i=\\frac {n}{z+n}\\]\n\\[\\frac {1}{1-4i}=\\frac {z+n}{n}\\]\n\\[\\frac {1+4i}{17}=\\frac {z}{n}+1\\]\n Since their imaginary part has to be equal,\n\\[\\frac {4i}{17}=\\frac {164i}{n}\\]\n\\[n=\\frac {(164)(17)}{4}=697\\]\n\... |
DeepScaleR | [
{
"role": "user",
"content": "There is a cube of size \\(10 \\times 10 \\times 10\\) made up of small unit cubes. A grasshopper is sitting at the center \\(O\\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, prov... | math | {
"style": "rule",
"ground_truth": "\\frac{27!}{(9!)^3}"
} | {
"index": 35428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "83159"
} | {
"index": 35429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a four-digit number \\( A \\). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \\( A \\), and the smallest possible number is 594 less than \\( A \\). What is \\( A \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1963"
} | {
"index": 35430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a four-digit positive integer whose thousand's place is 2. If the digit 2 is moved to the unit's place, the new number formed is 66 greater than twice the original number. Let x be the original number in the units and tens places. Express the original number as 2000 + ... | math | {
"style": "rule",
"ground_truth": "2508"
} | {
"index": 35431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a game called \"Twenty-Four Points\" with the following rules: Choose four natural numbers between $1$ and $13$, and perform addition, subtraction, multiplication, and division operations on these four numbers (each number can be used only once and parentheses can be u... | math | {
"style": "rule",
"ground_truth": "3 \\times \\left(10 - 6 + 4\\right)"
} | {
"index": 35432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area \\( S \\) of the path in square feet?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.\nWhat is the smallest number of the edges in the initial graph with 30 vertices?"
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 35434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\\frac{1}{2}$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compu... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{7}"
} | {
"index": 35435,
"solution": "For clarity, we will number our grid, with $(0,0)$ being the corner that Philip starts in, and the grid stretching in the positive $x$ direction, i.e. all elements of the grid are of the form $(x, y)$, with $y \\in\\{0,1\\}$ and $x \\in \\mathbb{N}$. We will use recursion and casework... |
DeepScaleR | [
{
"role": "user",
"content": "There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arri... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 35436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure: he looks at the coin (or lack thereof) underneath him, and then, - If the coi... | math | {
"style": "rule",
"ground_truth": "6098"
} | {
"index": 35437,
"solution": "We keep track of the following quantities: Let $N$ be the sum of $2^{k}$, where $k$ ranges over all nonnegative integers such that position $-1-k$ on the number line contains a tails-up coin. Let $M$ be the sum of $2^{k}$, where $k$ ranges over all nonnegative integers such that posit... |
DeepScaleR | [
{
"role": "user",
"content": "There is a hemispherical raw material. If this material is processed into a cube through cutting, the maximum value of the ratio of the volume of the obtained workpiece to the volume of the raw material is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {6}}{3\\pi }"
} | {
"index": 35438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that t... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 35439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\\frac{4}{7}$, then the number common to both sets of four numbers is"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 35440,
"solution": "1. **Calculate the total of the first four numbers**: \n Given that the average of the first four numbers is $5$, the sum of these numbers is:\n \\[\n 4 \\times 5 = 20\n \\]\n\n2. **Calculate the total of the last four numbers**: \n Given that the average of the last four ... |
DeepScaleR | [
{
"role": "user",
"content": "There is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensive than the next most expensive one. Each time Sam presses the big red button on the machine, the machine randomly selects one of the remaining toys and ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{6}{7}"
} | {
"index": 35441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a magical tree with 58 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if on any given day the number of fruits on the tree is less than the ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a magical tree with 63 fruits. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number that ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a mathematical operator \"$\\odot$\" that satisfies the following equations: $2 \\odot 4=8, 4 \\odot 6=14, 5 \\odot 3=13, 8 \\odot 7=23$. According to this rule, what is $9 \\odot 3=?$"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 35445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a moving point \\( M \\) on the base \\( A_{1}B_{1}C_{1}D_{1} \\) of the cube \\( ABCD - A_{1}B_{1}C_{1}D_{1} \\), and \\( BM \\parallel \\) plane \\( ADC \\). Find the maximum value of \\( \\tan \\angle D_{1}MD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 35446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a moving point \\( P \\) on the \\( x \\)-axis. Given the fixed points \\( A(0,2) \\) and \\( B(0,4) \\), what is the maximum value of \\( \\sin \\angle APB \\) when \\( P \\) moves along the entire \\( x \\)-axis?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 35447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a parking lot with $10$ empty spaces. Three different cars, A, B, and C, are going to park in such a way that each car has empty spaces on both sides, and car A must be parked between cars B and C. How many different parking arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 35448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without look... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 35449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divi... | math | {
"style": "rule",
"ground_truth": "2601"
} | {
"index": 35450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a pentagon ABCDE. If the vertices A, B, C, D, E are colored with one of three colors: red, yellow, green, such that adjacent vertices are of different colors, then there are a total of different coloring methods."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 35451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$ 's place. Tom counted in the eggs, but his count was off by $1$ in the $10$ 's place. Raoul counted the eggs, but his count was off by $1$ in the $100$ 's place. Sasha, Jose, Pe... | math | {
"style": "rule",
"ground_truth": "439"
} | {
"index": 35452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a point inside an equilateral triangle with side length \\( d \\) whose distances from the vertices are 3, 4, and 5 units. Find the side length \\( d \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{25 + 12 \\sqrt{3}}"
} | {
"index": 35453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a polynomial $P(x)$ with integer coefficients such that\\[P(x)=\\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 35454,
"solution": "Note that $2022 = 210\\cdot 9 +132$. Since the only way to express $132$ in terms of $105$, $70$, $42$, or $30$ is $135 = 30+30+30+42$, we are essentially just counting the number of ways to express $210*9$ in terms of these numbers. Since $210 = 2*105=3*70=5*42=7*30$, it can only be ... |
DeepScaleR | [
{
"role": "user",
"content": "There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \\cdot 440$. What is the sum of the digits of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35455,
"solution": "\nWe start with the given equation:\n\\[\n(n+1)! + (n+2)! = n! \\cdot 440\n\\]\n\nFirst, we express the factorials in terms of \\(n!\\):\n\\[\n(n+1)! = (n+1)n!\n\\]\n\\[\n(n+2)! = (n+2)(n+1)n!\n\\]\n\nSubstituting these into the equation, we get:\n\\[\n(n+1)n! + (n+2)(n+1)n! = 440n!\n... |
DeepScaleR | [
{
"role": "user",
"content": "There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\\frac{\\pi}{2})$ all of the form $\\frac{m_k}{n_k}\\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3,... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 35456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a positive real number $x$ not equal to either $\\tfrac{1}{20}$ or $\\tfrac{1}{2}$ such that\\[\\log_{20x} (22x)=\\log_{2x} (202x).\\]The value $\\log_{20x} (22x)$ can be written as $\\log_{10} (\\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers.... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 35457,
"solution": "By the change of base rule, we have $\\frac{\\log 22x}{\\log 20x}=\\frac{\\log 202x}{\\log 2x}$, or $\\frac{\\log 22 +\\log x}{\\log 20 +\\log x}=\\frac{\\log 202 +\\log x}{\\log 2 +\\log x}=k$. We also know that if $a/b=c/d$, then this also equals $\\frac{a-b}{c-d}$. We use this iden... |
DeepScaleR | [
{
"role": "user",
"content": "There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.\n ~ pi_is_3.14"
}
] | math | {
"style": "rule",
"ground_truth": "307"
} | {
"index": 35458,
"solution": "If $16p+1 =k$, we have $k \\equiv 1 \\mod 16$, so $k^3 \\equiv 1 \\mod 16$. If $k=1$ we have $p=0$, which is not prime. If $k=17$ we have $16p+1=4913$, or $p=\\boxed{307}$"
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a prime number which is a factor of every sum of three consecutive integers. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 35459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a question: \"If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\\left(a+b\\right)+4\\left(2a+b\\right)$?\" Tang, who loves to use his brain, solved the problem as follows: \nOriginal expression $=2a+2b+8a+4... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 35460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$ . $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\\frac{AE}{EB} = 1$ and $\\frac{CF}{FD} = \\frac{1}{2}$ . Call $X$ the intersection of $AF$ and $DE$ . What is the area of pentagon $B... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 35461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a rectangular field that measures $20\\text{m}$ by $15\\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\\text{m}$ sides such that he is $6\\text{m}$ from a corner. He flies in a straight line to another point on the per... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat ... | math | {
"style": "rule",
"ground_truth": "100/3"
} | {
"index": 35463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a room with four doors. Determine the number of different ways for someone to enter and exit this room."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 35464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a round table with 9 chairs, and 4 people are seated randomly. What is the probability that no two people are sitting next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "1/14"
} | {
"index": 35465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a school that offers 10 courses for students to elect. Among them, Courses A, B, and C have conflicting schedules and thus at most one of these can be chosen. The school mandates that each student must elect three courses. How many different elective schemes are availa... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 35466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are la... | math | {
"style": "rule",
"ground_truth": "650"
} | {
"index": 35467,
"solution": "For each $i$th switch (designated by $x_{i},y_{i},z_{i}$), it advances itself only one time at the $i$th step; thereafter, only a switch with larger $x_{j},y_{j},z_{j}$ values will advance the $i$th switch by one step provided $d_{i}= 2^{x_{i}}3^{y_{i}}5^{z_{i}}$ divides $d_{j}= 2^{x_... |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of data: $a_{1}=\\frac{3}{1×2×3}$, $a_{2}=\\frac{5}{2×3×4}$, $a_{3}=\\frac{7}{3×4×5}$, $\\ldots $, $a_{n}=\\frac{2n+1}{n(n+1)(n+2)}$. Let $S_{n}=a_{1}+a_{2}+a_{3}+\\ldots +a_{n}$. Find the value of $S_{12}$. To solve this problem, Xiao Ming first simplified $a_{n... | math | {
"style": "rule",
"ground_truth": "\\frac{201}{182}"
} | {
"index": 35468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of natural numbers (it is known that there are at least seven numbers) such that the sum of every seven of them is less than 15, and the sum of all numbers in the set is 100. What is the smallest number of numbers that can be in the set?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 35469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\cdots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\), circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\), and so on, with circle \\( C_{1}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\cdots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\), circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\), and so on, with circle \\( C_{1}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\dots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\); circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\); ..., circle \\( C_{1} \\) passes ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a settlement $C$ (a point) located at the intersection of roads $A$ and $B$ (straight lines). Sasha walks along road $A$ towards $C$, taking 45 steps per minute with a step length of 60 cm. At the start, Sasha is 290 m away from $C$. Dania walks along road $B$ towards ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 35473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a simple pendulum with a period of $T=1$ second in summer. In winter, the length of the pendulum shortens by 0.01 centimeters. How many seconds faster is this pendulum in winter over a 24-hour period compared to summer? (Round to the nearest second).\n\nNote: The formu... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 35475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{9}"
} | {
"index": 35476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a solution of table salt in a flask. From the flask, $\\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 35477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a sphere with a radius of $\\frac{\\sqrt{3}}{2}$, on which 4 points $A, B, C, D$ form a regular tetrahedron. What is the maximum distance from the center of the sphere to the faces of the regular tetrahedron $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"index": 35478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a stack of 200 cards, numbered from 1 to 200 from top to bottom. Starting with the top card, the following operations are performed in sequence: remove the top card, then place the next card at the bottom of the stack; remove the new top card, then place the next card ... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"index": 35479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a string of lights with a recurrent pattern of three blue lights followed by four yellow lights, spaced 7 inches apart. Determine the distance in feet between the 4th blue light and the 25th blue light, given that 1 foot equals 12 inches."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 35480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the ch... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 35482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points awarded for hitting each zone is inversely proportional to the probab... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 35483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 35484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a ten-digit number. From left to right:\n- Its first digit indicates how many zeros are in the number.\n- Its second digit indicates how many ones are in the number.\n- Its third digit indicates how many twos are in the number.\n- $\\cdots \\cdots$\n- Its tenth digit i... | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"index": 35485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a ten-digit number. From left to right:\n- Its first digit indicates the number of zeros in the ten-digit number.\n- Its second digit indicates the number of ones in the ten-digit number.\n- Its third digit indicates the number of twos in the ten-digit number.\n- ...\n... | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"index": 35486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a three-digit number \\( A \\). By placing a decimal point in front of one of its digits, we get a number \\( B \\). If \\( A - B = 478.8 \\), find \\( A \\)."
}
] | math | {
"style": "rule",
"ground_truth": "532"
} | {
"index": 35487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a tram with a starting station A and an ending station B. A tram departs from station A every 5 minutes towards station B, completing the journey in 15 minutes. A person starts cycling along the tram route from station B towards station A just as a tram arrives at stat... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 35488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a type of four-digit number where the sum of any two adjacent digits is no greater than 2. When these numbers are arranged in ascending order, what is the second to last number?"
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 35489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. H... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 35490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q... | math | {
"style": "rule",
"ground_truth": "547"
} | {
"index": 35491,
"solution": "Note that if $\\tan \\theta$ is positive, then $\\theta$ is in the first or third quadrant, so $0^{\\circ} < \\theta < 90^{\\circ} \\pmod{180^{\\circ}}$. Also notice that the only way $\\tan{\\left(2^{n}\\theta\\right)}$ can be positive for all $n$ that are multiples of $3$ is when $2... |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique polynomial $P(x)$ of degree $4$ with rational coefficients and leading coefficient $1$ which has $\\sqrt{2}+\\sqrt{5}$ as a root. What is $P(1)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 35492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique positive integer $n$ such that $\\log_2{(\\log_{16}{n})} = \\log_4{(\\log_4{n})}.$ What is the sum of the digits of $n?$"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 35493,
"solution": "1. **Rewrite the equation using logarithmic identities:**\n Given the equation \\(\\log_2{(\\log_{16}{n})} = \\log_4{(\\log_4{n})}\\), we use the identity \\(\\log_{a^b} c = \\frac{1}{b} \\log_a c\\) to simplify the bases:\n \\[\n \\log_2{(\\log_{2^4}{n})} = \\log_{2^2}{(\\log_{... |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35494,
"solution": "If we set $x=2^y$, we can obtain three terms of a geometric sequence through logarithm properties. The three terms are \\[\\frac{y+1}{3}, \\frac{y}{2}, y.\\] In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms, so we obtain the fo... |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique quadruple of positive integers $(a, b, c, k)$ such that $c$ is not a perfect square and $a+\\sqrt{b+\\sqrt{c}}$ is a root of the polynomial $x^{4}-20 x^{3}+108 x^{2}-k x+9$. Compute $c$."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 35495,
"solution": "The four roots are $a \\pm \\sqrt{b \\pm \\sqrt{c}}$, so the sum of roots is 20 , so $a=5$. Next, we compute the sum of squares of roots: $$(a+\\sqrt{b \\pm \\sqrt{c}})^{2}+(a-\\sqrt{b \\pm \\sqrt{c}})^{2}=2 a^{2}+2 b \\pm 2 \\sqrt{c}$$ so the sum of squares of roots is $4 a^{2}+4 b$.... |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique triplet of positive integers \\((a, b, c)\\) such that \\(a \\leq b \\leq c\\) and\n$$\n\\frac{25}{84}=\\frac{1}{a}+\\frac{1}{a b}+\\frac{1}{a b c}.\n$$\nDetermine \\(a + b + c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique two-digit positive integer $t$ for which the last two digits of $11\\cdot t$ are $36$.\n\nWhat is $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 35497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique two-digit positive integer $t$ for which the last two digits of $13 \\cdot t$ are $26$."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 35498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There is a unique two-digit positive integer $u$ for which the last two digits of $15\\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 35499,
"solution": ""
} |
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