data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "There are \\( n \\) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 35200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are \\( n \\) distinct lines in the plane. One of these lines intersects exactly 5 of the \\( n \\) lines, another one intersects exactly 9 of the \\( n \\) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are \\( n \\) pieces of paper, each containing 3 different positive integers no greater than \\( n \\). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 35202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are \\( n \\) players participating in a round-robin chess tournament, where each player competes exactly once against every other player. The winner of a match earns 3 points, a draw gives each player 1 point, and the loser earns 0 points. If the total sum of all players... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 35203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are \\(100\\) countries participating in an olympiad. Suppose \\(n\\) is a positive integers such that each of the \\(100\\) countries is willing to communicate in exactly \\(n\\) languages. If each set of \\(20\\) countries can communicate in exactly one common language,... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are \\(n\\) girls \\(G_{1}, \\ldots, G_{n}\\) and \\(n\\) boys \\(B_{1}, \\ldots, B_{n}\\). A pair \\((G_{i}, B_{j})\\) is called suitable if and only if girl \\(G_{i}\\) is willing to marry boy \\(B_{j}\\). Given that there is exactly one way to pair each girl with a dis... | math | {
"style": "rule",
"ground_truth": "\\frac{n(n+1)}{2}"
} | {
"index": 35205,
"solution": "We represent the problem as a graph with vertices \\(G_{1}, \\ldots, G_{n}, B_{1}, \\ldots, B_{n}\\) such that there is an edge between vertices \\(G_{i}\\) and \\(B_{j}\\) if and only if \\((G_{i}, B_{j})\\) is suitable, so we want to maximize the number of edges while having a uniqu... |
DeepScaleR | [
{
"role": "user",
"content": "There are a batch of wooden strips with lengths of \\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many diff... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are a certain number of identical plastic bags that can be nested within each other. If all the other bags are inside one of the bags, we call this situation a \"bag of bags.\" Calculate the number of ways to make a \"bag of bags\" from 10 bags.\n\nExplanation: Use parent... | math | {
"style": "rule",
"ground_truth": "16796"
} | {
"index": 35207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are a few integers \\( n \\) such that \\( n^{2}+n+1 \\) divides \\( n^{2013}+61 \\). Find the sum of the squares of these integers."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 35208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are a theory part and an experimental part in the assessment of a certain course. Each part of the assessment is graded only as \"pass\" or \"fail\", and the whole course assessment is considered \"pass\" if both parts are passed. The probabilities of passing the theory a... | math | {
"style": "rule",
"ground_truth": "0.254"
} | {
"index": 35209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are a total of $400$ machine parts. If person A works alone for $1$ day, and then person A and person B work together for $2$ days, there will still be $60$ parts unfinished. If both work together for $3$ days, they can produce $20$ parts more than needed. How many parts ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 35210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are a total of 2008 black triangles, marked as \"▲\" and \"△\", arranged in a certain pattern as follows: ▲▲△△▲△▲▲△△▲△▲▲…, then there are black triangles."
}
] | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"index": 35211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 35212,
"solution": "\nGiven 7 arbitrary points in the plane, we need to determine the maximum number of circles that can be drawn through every 4 possible concyclic points.\n\nTo solve this, we consider the combinatorial aspect of selecting 4 points out of 7. The number of ways to choose 4 points from 7 ... |
DeepScaleR | [
{
"role": "user",
"content": "There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"index": 35213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\\alpha+\\beta$?"
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 35214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are eight athletes participating in the men's 100m final. It is known that the stadium has eight tracks numbered from inside to outside as 1, 2, 3, 4, 5, 6, 7, 8. If the track numbers of the specified three athletes must be three consecutive numbers, calculate the total n... | math | {
"style": "rule",
"ground_truth": "4320"
} | {
"index": 35215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are eight different symbols designed on $n\\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\\leq k\\leq 7)$ the number of shirts containing at least one of the ... | math | {
"style": "rule",
"ground_truth": "255"
} | {
"index": 35216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are eight envelopes numbered 1 to 8. Find the number of ways in which 4 identical red buttons and 4 identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red butto... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 35217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 35218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are eleven positive integers $n$ such that there exists a convex polygon with $n$ sides whose angles, in degrees, are unequal integers that are in arithmetic progression. Find the sum of these values of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 35219,
"solution": "The sum of the angles of an $n$-gon is $(n-2) 180$, so the average angle measure is $(n-2) 180 / n$. The common difference in this arithmetic progression is at least 1 , so the difference between the largest and smallest angles is at least $n-1$. So the largest angle is at least $(n-1... |
DeepScaleR | [
{
"role": "user",
"content": "There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 35220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\\gcd(a, b, c, d) = 77$ and $\\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "27,720"
} | {
"index": 35221,
"solution": "1. **Define normalized variables**: Let $A = \\frac{a}{77}$, $B = \\frac{b}{77}$, $C = \\frac{c}{77}$, and $D = \\frac{d}{77}$. This transformation ensures that $\\gcd(A, B, C, D) = 1$. The problem then translates to finding the least common multiple of $A, B, C, D$ multiplied by $77$... |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35222,
"solution": "1. **Identify the integer solution condition**: \n The quadratic equation given is:\n \\[\n 5x^2 + kx + 12 = 0\n \\]\n For this equation to have an integer solution, let's denote that integer by $-n$. Thus, the equation can be factored as:\n \\[\n (5x + \\frac{12}{n})(x... |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly 120 ways to color five cells in a $5 \\times 5$ grid such that each row and each column contains exactly one colored cell.\n\nThere are exactly 96 ways to color five cells in a $5 \\times 5$ grid without the corner cell, such that each row and each column cont... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly 120 ways to color five cells in a $5 \\times 5$ grid such that exactly one cell in each row and each column is colored.\n\nThere are exactly 96 ways to color five cells in a $5 \\times 5$ grid without the corner cell, such that exactly one cell in each row and... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly 120 ways to color five cells in a \\( 5 \\times 5 \\) grid such that each row and each column contains exactly one colored cell.\n\nThere are exactly 96 ways to color five cells in a \\( 5 \\times 5 \\) grid without the corner cell such that each row and each ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 35225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are exactly three integers $x$ satisfying the inequality\n\\[x^2 + bx + 2 \\le 0.\\]How many integer values of $b$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 35226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five concentric circles \\(\\Gamma_{0}, \\Gamma_{1}, \\Gamma_{2}, \\Gamma_{3}, \\Gamma_{4}\\) whose radii form a geometric sequence with a common ratio \\(q\\). Find the maximum value of \\(q\\) such that a closed polyline \\(A_{0} A_{1} A_{2} A_{3} A_{4}\\) can be dr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 35227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five distinct nonzero natural numbers; the smallest one is 7. If one of them is decreased by 20, and the other four numbers are each increased by 5, the resulting set of numbers remains the same. What is the sum of these five numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 35228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 35229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in t... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{108}"
} | {
"index": 35230,
"solution": "The desired probability is the number of ways to pick the two isolated people times the probability that the remaining three point at each other. So, $$P=\\binom{5}{2} \\cdot\\left(\\frac{\\binom{2}{2}}{\\binom{4}{2}}\\right)^{3}=10 \\cdot\\left(\\frac{1}{6}\\right)^{3}=\\frac{5}{108}... |
DeepScaleR | [
{
"role": "user",
"content": "There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 35231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five positive integers that are divisors of each number in the list $$30, 90, -15, 135, 45.$$ Find the sum of these five positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 35232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five positive integers that are divisors of each number in the list $$48, 144, 24, 192, 216, 120.$$ Find the sum of these positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 35233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five positive integers that are divisors of each number in the list $$60, 120, -30, 180, 240$$. Find the sum of these five positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 35234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 35235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five students: A, B, C, D, and E.\n\n1. In how many different ways can they line up in a row such that A and B must be adjacent, and C and D cannot be adjacent?\n2. In how many different ways can these five students be distributed into three classes, with each class h... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 35236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five students: A, B, C, D, and E;\n(1) If these five students line up in a row, in how many ways can A not stand in the first position?\n(2) If these five students line up in a row, and A and B must be next to each other while C and D must not be next to each other, i... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 35237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly three people receive the correct letter?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 35238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are five unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly four people get the right letter?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 35239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four balls in a bag, each with the same shape and size, and their numbers are \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), and \\\\(4\\\\).\n\n\\\\((1)\\\\) Draw two balls randomly from the bag. Calculate the probability that the sum of the numbers on the balls drawn is no ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{13}{16}"
} | {
"index": 35240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four cards, each with a number on both sides. The first card has 0 and 1, the other three cards have 2 and 3, 4 and 5, and 7 and 8 respectively. If any three cards are selected and arranged in a row, how many different three-digit numbers can be formed?"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 35241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four cards, each with one of the numbers $2$, $0$, $1$, $5$ written on them. Four people, A, B, C, and D, each take one card.\n\nA says: None of the numbers you three have differ by 1 from the number I have.\nB says: At least one of the numbers you three have differs ... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"index": 35242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four complex numbers $z$ such that\n\\[z \\overline{z}^3 + \\overline{z} z^3 = 350,\\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four comple... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 35243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used ... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"index": 35244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, ... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"index": 35245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four distinct positive integers $a,b,c,d$ less than $8$ which are invertible modulo $8$. Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $8$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 35246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 35247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four groups of numbers with their respective averages specified as follows:\n1. The average of all multiples of 11 from 1 to 100810.\n2. The average of all multiples of 13 from 1 to 100810.\n3. The average of all multiples of 17 from 1 to 100810.\n4. The average of al... | math | {
"style": "rule",
"ground_truth": "50413.5"
} | {
"index": 35248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four identical balls numbered $1$, $2$, $3$, $4$, and four boxes also numbered $1$, $2$, $3$, $4$.\n\n$(1)$ If each box contains one ball, find the number of ways such that exactly one box has the same number as the ball inside it.\n\n$(2)$ Find the number of ways suc... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 35249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class?"
}
] | math | {
"style": "rule",
"ground_truth": "4 : 3"
} | {
"index": 35250,
"solution": "1. **Define Variables:**\n Let $g$ be the number of girls in Ms. Raub's class. Since there are four more girls than boys, the number of boys, $b$, can be expressed as $b = g - 4$.\n\n2. **Set Up the Equation:**\n The total number of students in the class is given as $28$. Therefor... |
DeepScaleR | [
{
"role": "user",
"content": "There are four pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute the product $\\left(1-\\frac{x_1}{y_1}\\right)\\left(1-\\frac{x_2}{y_2}\\right)\\left(1-\\frac{x_3}{y_3}\\right)\... | math | {
"style": "rule",
"ground_truth": "\\frac{-1}{1008}"
} | {
"index": 35251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 35252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four people in a room. For every two people, there is a $50 \\%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the proba... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{32}"
} | {
"index": 35253,
"solution": "We label the four people in the room $A, B, C$, and $D$. We represent each person by a point. There are six possible pairs of friends: $AB, AC, AD, BC, BD$, and $CD$. We represent a friendship by joining the corresponding pair of points and a non-friendship by not joining the pair of ... |
DeepScaleR | [
{
"role": "user",
"content": "There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 35254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four points that are $7$ units from the line $y = 20$ and $10$ units from the point $(10, 20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 35255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four points that are $7$ units from the line $y=20$ and $15$ units from the point $(10,20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 35256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"index": 35259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are functions $f(x)$ with the following properties: $f(x)=ax^{2}+bx+c$ for some integers $a, b$ and $c$ with $a>0$, and $f(p)=f(q)=17$ and $f(p+q)=47$ for some prime numbers $p$ and $q$ with $p<q$. For each such function, the value of $f(pq)$ is calculated. The sum of all... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 35260,
"solution": "Since $f(p)=17$, then $ap^{2}+bp+c=17$. Since $f(q)=17$, then $aq^{2}+bq+c=17$. Subtracting these two equations, we obtain $a(p^{2}-q^{2})+b(p-q)=0$. Since $p^{2}-q^{2}=(p-q)(p+q)$, this becomes $a(p-q)(p+q)+b(p-q)=0$. Since $p<q$, then $p-q \\neq 0$, so we divide by $p-q$ to get $a(p... |
DeepScaleR | [
{
"role": "user",
"content": "There are infinitely many positive integers $k$ which satisfy the equation\n\\[\\cos^2 (k^2 + 6^2)^\\circ = 1.\\]Enter the two smallest solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 35261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are integers $a, b,$ and $c,$ each greater than $1,$ such that\n\\[\\sqrt[a]{N\\sqrt[b]{N\\sqrt[c]{N}}} = \\sqrt[36]{N^{25}}\\]\nfor all $N \\neq 1$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 35262,
"solution": "1. **Expression Simplification**:\n Start by simplifying the left-hand side of the equation:\n \\[\n \\sqrt[a]{N\\sqrt[b]{N\\sqrt[c]{N}}} = N^{\\frac{1}{a} + \\frac{1}{ab} + \\frac{1}{abc}}\n \\]\n This simplification comes from the properties of exponents, where $\\sqrt[k]{... |
DeepScaleR | [
{
"role": "user",
"content": "There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are integers $x$ that satisfy the inequality $|x-2000|+|x| \\leq 9999$. Find the number of such integers $x$."
}
] | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"index": 35264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights on the island than liars?\". They answered in turn such that ... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 35265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\\frac{1}{2}$ chance to hop to the next pad, and an equal... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{256}"
} | {
"index": 35266,
"solution": "1. **Define the problem**: Fiona the frog starts at lily pad $0$ and can hop to the next pad or jump two pads forward with equal probability of $\\frac{1}{2}$. She must reach pad $10$ without landing on pads $3$ or $6$.\n\n2. **Calculate probability to reach pad $2$**:\n - Fiona can... |
DeepScaleR | [
{
"role": "user",
"content": "There are many ways in which the list \\(0,1,2,3,4,5,6,7,8,9\\) can be separated into groups. For example, this list could be separated into the four groups \\(\\{0,3,4,8\\}\\), \\(\\{1,2,7\\}\\), \\{6\\}, and \\{5,9\\}. The sum of the numbers in each of these four groups is \\... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 35267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are more than 20 and fewer than 30 children in Miss Tree's class. They are all standing in a circle. Anna notices that there are six times as many children between her and Zara going round the circle clockwise, as there are going round anti-clockwise. How many children ar... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 35268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of dra... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 35269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ${p}_{a,b}$ be the sum of the zeros of $P(x)$. Find the sum of the ${p}_{a,b}$'s fo... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 35270,
"solution": "Since $r+si$ is a root, by the Complex Conjugate Root Theorem, $r-si$ must be the other imaginary root. Using $q$ to represent the real root, we have\n$(x-q)(x-r-si)(x-r+si) = x^3 -ax^2 + bx -65$\n Applying difference of squares, and regrouping, we have\n$(x-q)(x^2 - 2rx + (r^2 + s^2)... |
DeepScaleR | [
{
"role": "user",
"content": "There are numbers $1, 2, \\cdots, 36$ to be filled into a $6 \\times 6$ grid, with each cell containing one number. Each row must be in increasing order from left to right. What is the minimum sum of the six numbers in the third column?"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 35271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are positive integers $x$ and $y$ that satisfy the system of equations \\begin{align*} \\log_{10} x + 2 \\log_{10} (\\text{gcd}(x,y)) &= 60\\\\ \\log_{10} y + 2 \\log_{10} (\\text{lcm}(x,y)) &= 570. \\end{align*} Let $m$ be the number of (not necessarily distinct) prime f... | math | {
"style": "rule",
"ground_truth": "880"
} | {
"index": 35273,
"solution": "Let $x=10^a$ and $y=10^b$ and $a<b$. Then the given equations become $3a=60$ and $3b=570$. Therefore, $x=10^{20}=2^{20}\\cdot5^{20}$ and $y=10^{190}=2^{190}\\cdot5^{190}$. Our answer is $3(20+20)+2(190+190)=\\boxed{880}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "There are positive integers $x$ and $y$ that satisfy the system of equations\\begin{align*} \\log_{10} x + 2 \\log_{10} (\\text{gcd}(x,y)) &= 60\\\\ \\log_{10} y + 2 \\log_{10} (\\text{lcm}(x,y)) &= 570. \\end{align*}Let $m$ be the number of (not necessarily distinct) prime fac... | math | {
"style": "rule",
"ground_truth": "880"
} | {
"index": 35274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are positive integers that have these properties:\n\n$\\bullet$ I. The sum of the squares of their digits is $50,$ and\n\n$\\bullet$ II. Each digit is larger than the one on its left.\n\nWhat is the product of the digits of the largest integer with both properties?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$ , if $x_1 = ax + b$ and $y_1 = cy + d$ , then $y_1^2 = x_1^3 - 36x_1$ . What is $a + b + c + d$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 35276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are real numbers $a$ and $b$ for which the function $f$ has the properties that $f(x) = ax + b$ for all real numbers $x$, and $f(bx + a) = x$ for all real numbers $x$. What is the value of $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 35277,
"solution": "Since $f(x) = ax + b$ for all real numbers $x$, then $f(t) = at + b$ for some real number $t$. When $t = bx + a$, we obtain $f(bx + a) = a(bx + a) + b = abx + (a^{2} + b)$. We also know that $f(bx + a) = x$ for all real numbers $x$. This means that $abx + (a^{2} + b) = x$ for all real... |
DeepScaleR | [
{
"role": "user",
"content": "There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \\sqrt{n} \\cdot i,$ where $m$ and $n$ are positive integers and $i = \\sqrt{-1}.$ Find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"index": 35278,
"solution": "We plug -20 into the equation obtaining $(-20)^3-20a+b$, likewise, plugging -21 into the second equation gets $(-21)^3+441c+d$.\n Both equations must have 3 solutions exactly, so the other two solutions must be $m + \\sqrt{n} \\cdot i$ and $m - \\sqrt{n} \\cdot i$.\n By Vieta's, the s... |
DeepScaleR | [
{
"role": "user",
"content": "There are red and blue socks in a drawer, with a total number not exceeding 2017. If two socks are randomly drawn, the probability that they are of the same color is $\\frac{1}{2}$. What is the maximum number of red socks in the drawer?"
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 35279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can g... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 35280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are seven students taking a graduation photo in a row. Student A must stand in the middle, and students B and C must stand together. How many different arrangements are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 35281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 35282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several consecutive natural numbers. If we select 4 different numbers from them and add them together, we can obtain 385 different sums. How many such natural numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 35283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 35284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several positive numbers written on the board. The sum of the five largest numbers is 0.29 of the sum of all the numbers, and the sum of the five smallest numbers is 0.26 of the sum of all the numbers. How many numbers in total are written on the board?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 35285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several sets of three different numbers whose sum is $15$ which can be chosen from $\\{ 1,2,3,4,5,6,7,8,9 \\}$. How many of these sets contain a $5$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 35286,
"solution": "1. **Define the set and condition**: Let the three-element set be $\\{a, b, c\\}$, where $a, b, c$ are distinct elements from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. We are given that the sum of the elements in the set is $15$.\n\n2. **Condition on including $5$**: Suppose $a = 5$.... |
DeepScaleR | [
{
"role": "user",
"content": "There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation?"
}
] | math | {
"style": "rule",
"ground_truth": "136"
} | {
"index": 35287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles."
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"index": 35288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$ . What is the maximum possible number of cups in the kitchen?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 35289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are several white rabbits and gray rabbits. When 6 white rabbits and 4 gray rabbits are placed in a cage, there are still 9 more white rabbits remaining, and all the gray rabbits are placed. When 9 white rabbits and 4 gray rabbits are placed in a cage, all the white rabbi... | math | {
"style": "rule",
"ground_truth": "159"
} | {
"index": 35290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are six clearly distinguishable frogs sitting in a row. Two are green, three are red, and one is blue. Green frogs refuse to sit next to the red frogs, for they are highly poisonous. In how many ways can the frogs be arranged?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 35291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are six empty slots corresponding to the digits of a six-digit number. Claire and William take turns rolling a standard six-sided die, with Claire going first. They alternate with each roll until they have each rolled three times. After a player rolls, they place the numb... | math | {
"style": "rule",
"ground_truth": "\\frac{43}{192}"
} | {
"index": 35292,
"solution": "A number being divisible by 6 is equivalent to the following two conditions: - the sum of the digits is divisible by 3 - the last digit is even Regardless of Claire and William's strategies, the first condition is satisfied with probability $\\frac{1}{3}$. So Claire simply plays to ma... |
DeepScaleR | [
{
"role": "user",
"content": "There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are six students with unique integer scores in a mathematics exam. The average score is 92.5, the highest score is 99, and the lowest score is 76. What is the minimum score of the student who ranks 3rd from the highest?"
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 35295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed among these six people, with each person getting one letter, what is the probability that exactly three people get the right letter?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"index": 35296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are some bullfinches in a pet store. One of the children exclaimed, \"There are more than fifty bullfinches!\" Another replied, \"Don't worry, there are fewer than fifty bullfinches.\" The mother added, \"At least there is one!\" The father concluded, \"Only one of your s... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 35297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"index": 35298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. \n$(1)$ How many unique three-digit numbers can be formed without repetition? \n$(2)$ How many unique four-digit even numbers can be formed without repetition?"
}
] | math | {
"style": "rule",
"ground_truth": "2296"
} | {
"index": 35299,
"solution": ""
} |
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