data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers \\(a_{1}, a_{2}, \\ldots\\) is such that for each \\(m\\) and \\(n\\) the following holds: if \\(m\\) is a divisor of \\(n\\) and \\(m < n\\), then \\(a_{m}\\) is a divisor of \\(a_{n}\\) and \\(a_{m} < a_{n}\\). Find the least possible value of ... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 3400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \\geq 0$. Find $\\operatorname{gcd}(a_{999}, a_{2004})$."
}
] | math | {
"style": "rule",
"ground_truth": "677"
} | {
"index": 3401,
"solution": "If $d$ is the relevant greatest common divisor, then $a_{1000}=a_{999}^{2}+1 \\equiv 1=a_{0}(\\bmod d)$, which implies (by induction) that the sequence is periodic modulo $d$, with period 1000 . In particular, $a_{4} \\equiv a_{2004} \\equiv 0$. So $d$ must divide $a_{4}$. Conversely, ... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers is given by $a_{1}=1$ and $a_{n}=\\operatorname{gcd}\\left(a_{n-1}, n\\right)+1$ for $n>1$. Calculate $a_{2002}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 3402,
"solution": "3. It is readily seen by induction that $a_{n} \\leq n$ for all $n$. On the other hand, $a_{1999}$ is one greater than a divisor of 1999. Since 1999 is prime, we have $a_{1999}=2$ or 2000; the latter is not possible since $2000>1999$, so we have $a_{1999}=2$. Now we straightforwardly c... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\\ge1,$ the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ a... | math | {
"style": "rule",
"ground_truth": "973"
} | {
"index": 3403,
"solution": "Let $x = a_2$. It is apparent that the sequence grows relatively fast, so we start trying positive integers to see what $x$ can be. Finding that $x = 5$ works, after bashing out the rest of the terms we find that $a_{16} = 957$ and $a_{17} = 1089$, hence our answer is $957 + 16 = \\box... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of real numbers $ x_0, x_1, x_2, \\ldots$ is defined as follows: $ x_0 \\equal{} 1989$ and for each $ n \\geq 1$ \r\n\r\n\\[ x_n \\equal{} \\minus{} \\frac{1989}{n} \\sum^{n\\minus{}1}_{k\\equal{}0} x_k.\\]\r\n\r\nCalculate the value of $ \\sum^{1989}_{n\\equal... | math | {
"style": "rule",
"ground_truth": "-1989"
} | {
"index": 3404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty (n=1,2,...)$ has the following property:\n\\begin{align*}\n6a_n+5a_{n-2}=20+11a_{n-1}\\ (\\text{for }n\\geq3).\n\\end{align*}\nThe first two elements are $a_1=0, a_2=1$ . Find the integer closest to $a_{2011}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "40086"
} | {
"index": 3405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.\n(i) The value of $a_0$ is a positive integer.\n(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\\frac{a_i}{a_i + 2} $\n(iii) There exists a... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3406,
"solution": "\nTo solve the given problem, we need to consider how we can construct a sequence of real numbers \\( a_0, a_1, \\ldots \\) such that the three conditions specified hold true, and we need to find the smallest positive integer \\( n \\) for which there exists a good sequence where \\( a... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of real numbers $a_{0}, a_{1}, \\ldots$ is said to be good if the following three conditions hold. (i) The value of $a_{0}$ is a positive integer. (ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\\frac{a_{i}}{a_{i}+2}$. (iii) There exis... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3407,
"solution": "Note that $$ a_{i+1}+1=2\\left(a_{i}+1\\right) \\text { or } a_{i+1}+1=\\frac{a_{i}+a_{i}+2}{a_{i}+2}=\\frac{2\\left(a_{i}+1\\right)}{a_{i}+2} $$ Hence $$ \\frac{1}{a_{i+1}+1}=\\frac{1}{2} \\cdot \\frac{1}{a_{i}+1} \\text { or } \\frac{1}{a_{i+1}+1}=\\frac{a_{i}+2}{2\\left(a_{i}+1\\rig... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of real numbers $a_{0}, a_{1}, \\ldots, a_{9}$ with $a_{0}=0, a_{1}=1$, and $a_{2}>0$ satisfies $$a_{n+2} a_{n} a_{n-1}=a_{n+2}+a_{n}+a_{n-1}$$ for all $1 \\leq n \\leq 7$, but cannot be extended to $a_{10}$. In other words, no values of $a_{10} \\in \\mathbb{R}$ sat... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"index": 3408,
"solution": "Say $a_{2}=a$. Then using the recursion equation, we have $a_{3}=-1, a_{4}=\\frac{a+1}{a-1}, a_{5}=\\frac{-a+1}{a+1}, a_{6}=-\\frac{1}{a}$, $a_{7}=-\\frac{2 a}{a^{2}-1}$, and $a_{8}=1$ Now we have $a_{10} a_{8} a_{7}=a_{10}+a_{8}+a_{7}$. No value of $a_{10}$ can satisfy this equation i... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3409,
"solution": "\n1. **Identify the pattern in the sequence of squares:**\n Each square in the sequence has a side length that is one tile longer than the side of the previous square. Therefore, the side length of the $n$-th square is $n$ tiles.\n\n2. **Calculate the number of tiles in the sixth and... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 3410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the g... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 3411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of twelve \\(0\\)s and/or \\(1\\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \\(1\\)s can be written in the form \\(\\dfrac{m}{n}\\), where \\(m,n\\) are relatively prime positive integers,... | math | {
"style": "rule",
"ground_truth": "2281"
} | {
"index": 3412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arith... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 3413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and \"flips\" every 3rd locker (\"fipping\") a locker means changing its state: if the locker is open he closes it, and if the locker is ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and \"flips\" every 3rd locker (\"flipping\") a locker means changing its state: if the locker is open he closes it, and if the locker is... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A series of numbers were written: \\(100^{100}, 101^{101}, 102^{102}, \\ldots, 234^{234}\\) (i.e., the numbers of the form \\(n^{n}\\) for natural \\(n\\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an inte... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 3416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3417,
"solution": "1. **Identify Symmetry Requirements**: The problem states that the set $S$ is symmetric about the origin, both coordinate axes, and the line $y=x$. This implies:\n - Symmetry about the origin: If $(a, b) \\in S$, then $(-a, -b) \\in S$.\n - Symmetry about the $x$-axis: If $(a, b) \... |
DeepScaleR | [
{
"role": "user",
"content": "A set $\\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\\mathcal{S}$ is an integer. Given that 1 belongs to $\\mathcal{S}$ and that 2002 is t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3418,
"solution": "Let the sum of the integers in $\\mathcal{S}$ be $N$, and let the size of $|\\mathcal{S}|$ be $n+1$. After any element $x$ is removed, we are given that $n|N-x$, so $x\\equiv N\\pmod{n}$. Since $1\\in\\mathcal{S}$, $N\\equiv1\\pmod{n}$, and all elements are congruent to 1 mod $n$. Sinc... |
DeepScaleR | [
{
"role": "user",
"content": "A set $\\mathcal{T}$ of distinct positive integers has the following property: for every integer $y$ in $\\mathcal{T},$ the arithmetic mean of the set of values obtained by deleting $y$ from $\\mathcal{T}$ is an integer. Given that 1 belongs to $\\mathcal{T}$ and that 1764 is t... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 3419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 3420,
"solution": "1. **Define the Set $T$:** \n Let $T$ be the set of all integral triples $(a, b, c)$ such that $a \\ge b \\ge c$, $b+c > a$ (triangle inequality), and $a, b, c < 5$. \n\n2. **Enumerate Possible Triangles:** \n We list all possible triangles that satisfy these conditions:\n - $(... |
DeepScaleR | [
{
"role": "user",
"content": "A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set \\( A \\) consists of 40 elements chosen from \\(\\{1, 2, \\ldots, 50\\}\\), and \\( S \\) is the sum of all elements in the set \\( A \\). How many distinct values can \\( S \\) take?"
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 3422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set \\( \\mathcal{S} \\) of distinct positive integers has the property that for every integer \\( x \\) in \\( \\mathcal{S}, \\) the arithmetic mean of the set of values obtained by deleting \\( x \\) from \\( \\mathcal{S} \\) is an integer. Given that 1 belongs to \\( \\mat... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set \\( \\mathcal{T} \\) of distinct positive integers has the following property: for every integer \\( y \\) in \\( \\mathcal{T}, \\) the arithmetic mean of the set of values obtained by deleting \\( y \\) from \\( \\mathcal{T} \\) is an integer. Given that 2 belongs to \\(... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set \\( \\mathcal{T} \\) of distinct positive integers has the property that for every integer \\( y \\) in \\( \\mathcal{T}, \\) the arithmetic mean of the set of values obtained by deleting \\( y \\) from \\( \\mathcal{T} \\) is an integer. Given that 2 belongs to \\( \\mat... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood sma... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 3426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \\neq 0$ and $A \\neq B$. (The hundreds digit of the pr... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3427,
"solution": "Solution 1: Let $N=AB0AB$ and let $t$ be the two-digit integer $AB$. We note that $N=1001t$, and that $1001=11 \\cdot 91=11 \\cdot 7 \\cdot 13$. Therefore, $N=t \\cdot 7 \\cdot 11 \\cdot 13$. We want to write $N$ as the product of 5 distinct odd integers, each greater than 2, and to co... |
DeepScaleR | [
{
"role": "user",
"content": "A set containing three real numbers can be represented as $\\{a, \\frac{b}{a}, 1\\}$, and also as $\\{a^2, a+b, 0\\}$. Find the value of $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 3428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set containing three real numbers can be represented as $\\{a, \\frac{b}{a}, 1\\}$, and also as $\\{a^2, a+b, 0\\}$. Find the value of $a^{2003} + b^{2004}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 3429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set containing three real numbers can be represented as $\\{a,\\frac{b}{a},1\\}$, or as $\\{a^{2}, a+b, 0\\}$. Find the value of $a^{2023}+b^{2024}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 3430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "791"
} | {
"index": 3431,
"solution": "Let the four numbers be $a$, $b$, $c$, and $d$, in no particular order. Adding the pairwise sums, we have $3a+3b+3c+3d=1030+x+y$, so $x+y=3(a+b+c+d)-1030$. Since we want to maximize $x+y$, we must maximize $a+b+c+d$.\n Of the four sums whose values we know, there must be two sums that ... |
DeepScaleR | [
{
"role": "user",
"content": "A set of $25$ square blocks is arranged into a $5 \\times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?"
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3432,
"solution": "1. **Choosing Rows and Columns:**\n The problem requires selecting 3 blocks such that no two blocks are in the same row or column. We start by choosing 3 rows and 3 columns from the $5 \\times 5$ grid. The number of ways to choose 3 rows from 5 is given by the combination formula ${5... |
DeepScaleR | [
{
"role": "user",
"content": "A set of $36$ square blocks is arranged into a $6 \\times 6$ square. How many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column?"
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"index": 3433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is:"
}
] | math | {
"style": "rule",
"ground_truth": "$5s + 80n$"
} | {
"index": 3434,
"solution": "Let the original set of numbers be $\\{x_1, x_2, \\ldots, x_n\\}$, and the sum of these numbers is given by $s = x_1 + x_2 + \\ldots + x_n$.\n\nEach number in the set undergoes the following transformations:\n1. Increased by $20$: $x_i + 20$\n2. Multiplied by $5$: $5(x_i + 20) = 5x_i +... |
DeepScaleR | [
{
"role": "user",
"content": "A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of... | math | {
"style": "rule",
"ground_truth": "557"
} | {
"index": 3435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?\n<details><summary>Clarifications</summary>\n\n- $S$ is the ``set of 10 distinct integers'' from th... | math | {
"style": "rule",
"ground_truth": "511"
} | {
"index": 3436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of 36 square blocks is arranged into a 6 × 6 square. How many different combinations of 4 blocks can be selected from that set so that no two blocks are in the same row or column?"
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"index": 3437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of 6 distinct lattice points is chosen uniformly at random from the set $\\{1,2,3,4,5,6\\}^{2}$. Let $A$ be the expected area of the convex hull of these 6 points. Estimate $N=\\left\\lfloor 10^{4} A\\right\\rfloor$. An estimate of $E$ will receive $\\max \\left(0,\\left\... | math | {
"style": "rule",
"ground_truth": "104552"
} | {
"index": 3438,
"solution": "The main tools we will use are linearity of expectation and Pick's theorem. Note that the resulting polygon is a lattice polygon, and this the expected area $A$ satisfies $$A=I+\\frac{B}{2}-1$$ where $I$ is the expected number of interior points and $B$ is the expected number of bounda... |
DeepScaleR | [
{
"role": "user",
"content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 3439,
"solution": "1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:\n \\[\n S = \\frac{n(n+1)}{2}\n \\]\n\n2. **Expression for the average after erasing one number**: If one number $x$... |
DeepScaleR | [
{
"role": "user",
"content": "A set of marbles can be divided in equal shares among $2$, $3$, $4$, $5$, or $6$ children with no marbles left over. What is the least number of marbles that the set could have?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of numbers $\\{-3, 1, 5, 8, 10, 14\\}$ needs to be rearranged with new rules:\n\n1. The largest isn't in the last position, but it is in one of the last four places.\n2. The smallest isn’t in the first position, but it is in one of the first four places.\n3. The median is... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"index": 3441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that ther... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3442,
"solution": "\nTo solve this problem, we need to find the smallest positive integer \\( b \\) such that there exists a non-negative integer \\( a \\) for which the set \n\\[\n\\{P(a+1), P(a+2), \\ldots, P(a+b)\\}\n\\]\nis fragrant. The polynomial \\( P(n) = n^2 + n + 1 \\).\n\nA set is considered f... |
DeepScaleR | [
{
"role": "user",
"content": "A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets \\(F\\) and \\(T\\), each with at least 2 elements, such that the elements of \\(F\\) are consecutive Fibonacci numbers, and the elements of \\(T\\) are consecutive triangular number... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the tr... | math | {
"style": "rule",
"ground_truth": "253"
} | {
"index": 3444,
"solution": "Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset $\\mathcal{S}$. Without loss of generality, consider any $a, b, c \\,\\in \\mathcal{S}$ with $a < b < c$. $\\,\\m... |
DeepScaleR | [
{
"role": "user",
"content": "A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 3445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3446,
"solution": "Call the octahedron $A B C D E F$, where $A, B$, and $C$ are opposite $D, E$, and $F$, respectively. Note that each Hamiltonian cycle can be described in terms of the order it visits vertices in exactly 12 different ways. Conversely, listing the six vertices in some order determines a ... |
DeepScaleR | [
{
"role": "user",
"content": "A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A... | math | {
"style": "rule",
"ground_truth": "385"
} | {
"index": 3447,
"solution": "1. **Determine the total number of teams**: Given that each team won $10$ games and lost $10$ games, and there were no ties, each team played $20$ games. Since each game involves two teams, the total number of games played is $\\frac{21 \\times 20}{2} = 210$. This confirms there are $2... |
DeepScaleR | [
{
"role": "user",
"content": "A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 3449,
"solution": "1. **Initial Set and Operation Definition**: We start with a set of tiles numbered from 1 to 100. The operation, denoted as $P(x)$, involves removing all tiles numbered with a perfect square and renumbering the remaining tiles consecutively starting with 1.\n\n2. **Understanding Perfec... |
DeepScaleR | [
{
"role": "user",
"content": "A seven-digit natural number \\( N \\) is called interesting if:\n\n- It consists of non-zero digits;\n- It is divisible by 4;\n- Any number obtained from \\( N \\) by permuting its digits is also divisible by 4.\n\nHow many interesting numbers exist?"
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 3450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit numbe... | math | {
"style": "rule",
"ground_truth": "10002"
} | {
"index": 3451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A seven-digit phone number \\(d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7}\\) is called \"memorable\" if the initial three digits \\(d_{1} d_{2} d_{3}\\) match either the middle three digits \\(d_{4} d_{5} d_{6}\\) or the last three digits \\(d_{5} d_{6} d_{7}\\) (it is possible f... | math | {
"style": "rule",
"ground_truth": "19990"
} | {
"index": 3452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "75,000,000"
} | {
"index": 3453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shape is created by aligning five unit cubes in a straight line. Then, one additional unit cube is attached to the top of the second cube in the line and another is attached beneath the fourth cube in the line. Calculate the ratio of the volume to the surface area."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 3454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?\n[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), n... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{30}"
} | {
"index": 3455,
"solution": "1. **Calculate the Volume:**\n Each unit cube has a volume of $1$ cubic unit. Since there are seven unit cubes joined together, the total volume \\( V \\) is:\n \\[\n V = 7 \\times 1 = 7 \\text{ cubic units}\n \\]\n\n2. **Calculate the Surface Area:**\n To find the surface ar... |
DeepScaleR | [
{
"role": "user",
"content": "A shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.\n\nWhat is the ratio of the areas of the original hexagon to the cut-out shape?"
}
] | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"index": 3456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters?"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 3457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ship sails eastward at a speed of 15 km/h. At point A, the ship observes a lighthouse B at an angle of 60° northeast. After sailing for 4 hours, the ship reaches point C, where it observes the lighthouse at an angle of 30° northeast. At this time, the distance between the shi... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ship sails on a river. After 6 hours, it returns to its starting point, having covered a distance of 36 km according to the map (naturally, the ship had to move in one direction and then in the opposite direction).\n\nWhat is the speed of the ship if we assume that it did not... | math | {
"style": "rule",
"ground_truth": "3 + 3\\sqrt{2}"
} | {
"index": 3459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ship travels from Port A to Port B against the current at a speed of 24 km/h. After arriving at Port B, it returns to Port A with the current. It is known that the journey with the current takes 5 hours less than the journey against the current. The speed of the current is 3 ... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 3460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shipbuilding company has an annual shipbuilding capacity of 20 ships. The output function of building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (unit: ten thousand yuan), and the cost function is $C(x) = 460x + 5000$ (unit: ten thousand yuan). In economics, the marginal fun... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shirt regularly priced at $\\$30$ is on sale for $20\\%$ off. Mary has a coupon that will take another $25\\%$ off of the sale price. What single percent discount would give the same final price as the two successive discounts?"
}
] | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"index": 3462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________."
}
] | math | {
"style": "rule",
"ground_truth": "0.032"
} | {
"index": 3463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter fires at a target until the first hit, with a hit rate of 0.6 for each shot. If there are 4 bullets in total, what is the expected number of remaining bullets $\\xi$?"
}
] | math | {
"style": "rule",
"ground_truth": "2.376"
} | {
"index": 3464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \\_\\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "2.376"
} | {
"index": 3465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"index": 3466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________."
}
] | math | {
"style": "rule",
"ground_truth": "0.032"
} | {
"index": 3467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter's probabilities of hitting the 10, 9, 8, 7 rings, and below 7 rings in a shooting are 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that the shooter in a single shot:\n(1) Hits the 10 or 9 rings,\n(2) Hits at least the 7 ring,\n(3) Hits les... | math | {
"style": "rule",
"ground_truth": "0.29"
} | {
"index": 3468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 3469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:\n\n0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:\n\n(1) Hits either the 10 or 9 ring; (2) Scores less than 7 ri... | math | {
"style": "rule",
"ground_truth": "0.03"
} | {
"index": 3470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shop advertises everything is \"half price in today's sale.\" In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3471,
"solution": "1. **Calculate the initial discount:** The shop advertises everything at \"half price in today's sale.\" This means that the price is reduced to half of the original price. Mathematically, this can be expressed as:\n \\[\n \\text{Sale Price} = 50\\% \\times \\text{Original Price} =... |
DeepScaleR | [
{
"role": "user",
"content": "A shop sells two kinds of products $A$ and $B$ at the price $\\$ 2100$. Product $A$ makes a profit of $20\\%$, while product $B$ makes a loss of $20\\%$. Calculate the net profit or loss resulting from this deal."
}
] | math | {
"style": "rule",
"ground_truth": "-175"
} | {
"index": 3472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a \"storef... | math | {
"style": "rule",
"ground_truth": "1824"
} | {
"index": 3473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"index": 3474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shopkeeper set up incorrect scales in his shop, where one side of the balance beam is longer than the other. During one weighing, 3 cans balanced with 8 packets, and during another, 1 packet balanced with 6 cans. Given that the true weight of one can is 1 kg, how much do 8 pa... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shopper buys a $100$ dollar coat on sale for $20\\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\\%$ is paid on the final selling price. The total amount the shopper pays for the coat is"
}
] | math | {
"style": "rule",
"ground_truth": "81.00"
} | {
"index": 3476,
"solution": "1. **Calculate the price after the initial discount**: \n The original price of the coat is $100$ dollars. The discount offered is $20\\%$. \n The price after the discount is calculated as follows:\n \\[\n 100 \\times (1 - 0.20) = 100 \\times 0.80 = 80 \\text{ dollars}\n \\... |
DeepScaleR | [
{
"role": "user",
"content": "A shopper plans to purchase an item that has a listed price greater than $\\$100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\$30$ off the listed price, and Coupon C gives $25\\%$ off the amount by which the listed pr... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3477,
"solution": "Let the listed price be $P$, where $P > 100$. We can express $P$ as $P = 100 + p$ where $p > 0$.\n\n1. **Calculate the savings from each coupon:**\n - **Coupon A:** This coupon gives $15\\%$ off the listed price. Therefore, the savings from Coupon A are:\n \\[\n 0.15P = 0.15(... |
DeepScaleR | [
{
"role": "user",
"content": "A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost... | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 3479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A showroom has 150 lights, all of which are initially turned on. Each light has an individual switch, numbered from 1 to 150. A student first toggles all switches that are multiples of 3, and then toggles all switches that are multiples of 5. How many lights remain on in the sh... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 3480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sign at the fish market says, \"50% off, today only: half-pound packages for just $3 per package.\" What is the regular price for a full pound of fish, in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3481,
"solution": "1. **Understanding the problem**: The problem states that half-pound packages of fish are being sold for $3 per package after a 50% discount. We need to find the regular price for a full pound of fish.\n\n2. **Setting up the equation**: Let $x$ be the regular price for a full pound of ... |
DeepScaleR | [
{
"role": "user",
"content": "A sign has 31 spaces on a single line. The word RHOMBUS is written from left to right in 7 consecutive spaces. There is an equal number of empty spaces on each side of the word. Counting from the left, in what space number should the letter $R$ be put?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3482,
"solution": "Since the letters of RHOMBUS take up 7 of the 31 spaces on the line, there are $31-7=24$ spaces that are empty. Since the numbers of empty spaces on each side of RHOMBUS are the same, there are $24 \\div 2=12$ empty spaces on each side. Therefore, the letter R is placed in space number... |
DeepScaleR | [
{
"role": "user",
"content": "A sign painter paints individual numerals for a row of 100 houses. The houses are numbered with consecutive integers from 1 to 100. How many 9s are painted, and what is the total count of all digits used in painting?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 3483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sign painter paints individual numerals for a row of 50 houses. The houses are numbered with consecutive integers from 1 to 50. How many 9s are painted?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sign painter paints numbers for each of 150 houses, numbered consecutively from 1 to 150. How many times does the digit 9 appear in total on all the house numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 3485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 3486,
"solution": "1. **Understanding the problem**: Each bench section can hold either $7$ adults or $11$ children. When $N$ bench sections are connected, the same number of adults and children must fill all the seats exactly.\n\n2. **Setting up the equation**: Let $x$ be the number of adults and $y$ be... |
DeepScaleR | [
{
"role": "user",
"content": "A single-elimination ping-pong tournament has $2^{2013}$ players, seeded in order of ability. If the player with seed $x$ plays the player with seed $y$, then it is possible for $x$ to win if and only if $x \\leq y+3$. For how many players $P$ it is possible for $P$ to win? (In... | math | {
"style": "rule",
"ground_truth": "6038"
} | {
"index": 3487,
"solution": "We calculate the highest seed $n$ that can win. Below, we say that a player $x$ vicariously defeats a player $y$ if $x$ defeats $y$ directly or indirectly through some chain (i.e. $x$ defeats $x_{1}$, who defeated $x_{2}, \\ldots$, who defeated $x_{n}$, who defeated $y$ for some player... |
DeepScaleR | [
{
"role": "user",
"content": "A six digit number (base 10) is squarish if it satisfies the following conditions:\n(i) none of its digits are zero;\n(ii) it is a perfect square; and\n(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when consid... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 3488,
"solution": "To find the number of squarish numbers, we need to satisfy the following conditions:\n1. The number is a six-digit number.\n2. None of its digits are zero.\n3. It is a perfect square.\n4. The first two digits, the middle two digits, and the last two digits of the number are all perfect... |
DeepScaleR | [
{
"role": "user",
"content": "A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by:"
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"index": 3489,
"solution": "1. **Express the number in a general form**: \n Let the three-digit number be represented as $\\overline{abc}$, where $a$, $b$, and $c$ are its digits. The six-digit number formed by repeating $\\overline{abc}$ is $\\overline{abcabc}$.\n\n2. **Convert the six-digit number into a mat... |
DeepScaleR | [
{
"role": "user",
"content": "A six-digit number 111aaa is the product of two consecutive positive integers \\( b \\) and \\( b+1 \\). Find the value of \\( b \\)."
}
] | math | {
"style": "rule",
"ground_truth": "333"
} | {
"index": 3490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"index": 3491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A six-digit number can be tripled by reducing the first digit by three and appending a three at the end. What is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "428571"
} | {
"index": 3492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 3493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A six-digit palindrome is a positive integer with respective digits $abcdcba$, where $a$ is non-zero. Let $T$ be the sum of all six-digit palindromes. Calculate the sum of the digits of $T$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A slant asymptote of the rational expression $y = \\frac{2x^2 + 3x - 7}{x-3}$ is the line approached by the equation as $x$ approaches $\\infty$ or $-\\infty$. If this line is of the form $y = mx + b$, find $m+b$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sled loaded with children starts from rest and slides down a snowy $25^\\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope?"
}
] | math | {
"style": "rule",
"ground_truth": "0.40"
} | {
"index": 3496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A small ball is released from a height \\( h = 45 \\) m without an initial velocity. The collision with the horizontal surface of the Earth is perfectly elastic. Determine the moment in time after the ball starts falling when its average speed equals its instantaneous speed. Th... | math | {
"style": "rule",
"ground_truth": "4.24"
} | {
"index": 3497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3498,
"solution": "To find the minimum number of small bottles necessary to completely fill a large bottle, we need to determine how many times $35$ milliliters (the capacity of one small bottle) goes into $500$ milliliters (the capacity of one large bottle).\n\n1. **Calculate the number of small bottles... |
DeepScaleR | [
{
"role": "user",
"content": "A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged sea... | math | {
"style": "rule",
"ground_truth": "1/32"
} | {
"index": 3499,
"solution": ""
} |
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