data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x, y)$ with $1 \\leq x < y \\leq 200$ such that $i^{x} + i^{y}$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "3651"
} | {
"index": 6600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 100$ such that $i^x+i^y$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "1850"
} | {
"index": 6601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 150$ such that $i^x+i^y$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "3515"
} | {
"index": 6602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 200$ such that $i^x+i^y$ is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "4950"
} | {
"index": 6603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 6604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation $\\operatorname{gcd}(a, b) \\cdot a+b^{2}=10000$"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 6605,
"solution": "Let $\\operatorname{gcd}(a, b)=d, a=d a^{\\prime}, b=d b^{\\prime}$. Then, $d^{2}\\left(a^{\\prime}+b^{\\prime 2}\\right)=100^{2}$. Consider each divisor $d$ of 100. Then, we need to find the number of solutions in coprime integers to $a^{\\prime}+b^{\\prime 2}=\\frac{100^{2}}{d^{2}}$.... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered quintuples of nonnegative integers $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ such that $0 \\leq a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \\leq 7$ and 5 divides $2^{a_{1}}+2^{a_{2}}+2^{a_{3}}+2^{a_{4}}+2^{a_{5}}$."
}
] | math | {
"style": "rule",
"ground_truth": "6528"
} | {
"index": 6606,
"solution": "Let $f(n)$ denote the number of $n$-tuples $(a_{1}, \\ldots, a_{n})$ such that $0 \\leq a_{1}, \\ldots, a_{n} \\leq 7$ and $5 \\mid 2^{a_{1}}+\\ldots+2^{a_{n}}$. To compute $f(n+1)$ from $f(n)$, we note that given any $n$-tuple $(a_{1}, \\ldots, a_{n})$ such that $0 \\leq a_{1}, \\ldot... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$ , inclusive, such that, if $$ q=a+\\frac{1}{b}-\\frac{1}{b+\\frac{1}{c}}, $$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 6607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of permutations $\\pi$ of the set $\\{1,2, \\ldots, 10\\}$ so that for all (not necessarily distinct) $m, n \\in\\{1,2, \\ldots, 10\\}$ where $m+n$ is prime, $\\pi(m)+\\pi(n)$ is prime."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6608,
"solution": "Since $\\pi$ sends pairs $(m, n)$ with $m+n$ prime to pairs $\\left(m^{\\prime}, n^{\\prime}\\right)$ with $m^{\\prime}+n^{\\prime}$ prime, and there are only finitely many such pairs, we conclude that if $m+n$ is composite, then so is $\\pi(m)+\\pi(n)$. Also note that $2 \\pi(1)=\\pi(... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 6609,
"solution": "Let an arbitrary such number be \\(\\overline{a b c d}\\). Then, we desire \\(11 \\mid a+b+c+d\\) and \\(11 \\mid a-b+c-d\\), where the latter comes from the well-known divisibility trick for 11. Sums and differences of multiples of 11 must also be multiples of 11, so this is equivalen... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive integers $n \\leq 1000$ such that \\operatorname{lcm}(n, 9)$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 6610,
"solution": "Suppose $n=3^{a} m$, where $3 \\nmid m$. Then $$\\operatorname{lcm}(n, 9)=3^{\\max (a, 2)} m$$ In order for this to be a square, we require $m$ to be a square, and $a$ to either be even or 1 . This means $n$ is either a square (if $a$ is even) or of the form $3 k^{2}$ where $3 \\nmid k... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive integers less than 10! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"index": 6611,
"solution": "Since $0!=1!=1$, we ignore any possible 0!'s in our sums. Call a sum of factorials reduced if for all positive integers $k$, the term $k$! appears at most $k$ times. It is straightforward to show that every positive integer can be written uniquely as a reduced sum of factorials. Moreov... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ ."
}
] | math | {
"style": "rule",
"ground_truth": "4648"
} | {
"index": 6612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive integers that divide at least two of the integers in the set $\\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6613,
"solution": "For a positive integer $n$, let \\operatorname{rad} n be the product of the distinct prime factors of $n$. Observe that if $n \\mid m^{m}$, all prime factors of $n$ must divide $m$, so \\operatorname{rad} n \\mid m. Therefore, if $n$ is such an integer, \\operatorname{rad} n must divid... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of positive real numbers $x$ that satisfy $\\left(3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x\\right)^{16}=2022 x^{13}$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 6614,
"solution": "Let $f(x)=3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x$. Note that for each integer $i$, if $x \\in\\left[2^{i}, 2^{i+1}\\right)$, then $f(x)=3 \\cdot 2^{i}-x$. This is a line segment from $\\left(2^{i}, 2^{i+1}\\right)$ to $\\left(2^{i+1}, 2^{i}\\right)$, including the fir... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of quadruples $(a, b, c, d)$ of positive integers satisfying $12a+21b+28c+84d=2024$."
}
] | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"index": 6615,
"solution": "Looking at the equation $\\bmod 7$ gives $a \\equiv 3(\\bmod 7)$, so let $a=7a^{\\prime}+3$. Then mod 4 gives $b \\equiv 0(\\bmod 4)$, so let $b=4b^{\\prime}$. Finally, $\\bmod 3$ gives $c \\equiv 2(\\bmod 3)$, so let $c=3c^{\\prime}+2$. Now our equation yields $$84a^{\\prime}+84b^{\\p... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of real solutions $(x, y, z, w)$ to the system of equations: $$\\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\\\ y & =w+x+w x y & w & =y+z+y z w \\end{array}$$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6616,
"solution": "The first equation rewrites as $x=\\frac{w+z}{1-w z}$, which is a fairly strong reason to consider trigonometric substitution. Let $x=\\tan (a), y=\\tan (b), z=\\tan (c)$, and $w=\\tan (d)$, where $-90^{\\circ}<a, b, c, d<90^{\\circ}$. Under modulo $180^{\\circ}$, we find $a \\equiv c+... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of real solutions $(x,y,z,w)$ to the system of equations:\n\\begin{align*}\nx &= z+w+zwx, \\\\ \ny &= w+x+wxy, \\\\\nz &= x+y+xyz, \\\\\nw &= y+z+yzw.\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of sequences of integers $(a_{1}, \\ldots, a_{200})$ such that the following conditions hold. - $0 \\leq a_{1}<a_{2}<\\cdots<a_{200} \\leq 202$. - There exists a positive integer $N$ with the following property: for every index $i \\in\\{1, \\ldots, 200\\}$ t... | math | {
"style": "rule",
"ground_truth": "20503"
} | {
"index": 6618,
"solution": "Let $m:=203$ be an integer not divisible by 3. We'll show the answer for general such $m$ is $m\\left\\lceil\\frac{m-1}{2}\\right\\rceil$. Let $x, y, z$ be the three excluded residues. Then $N$ works if and only if $\\{x, y, z\\} \\equiv\\{N-x, N-y, N-z\\} (\\bmod m)$. Since $x, y, z(\... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x \\in S$ then $(2 x \\bmod 16) \\in S$."
}
] | math | {
"style": "rule",
"ground_truth": "678"
} | {
"index": 6619,
"solution": "For any nonempty $S$ we must have $0 \\in S$. Now if we draw a directed graph of dependencies among the non-zero elements, it creates a balanced binary tree where every leaf has depth 3 . In the diagram, if $a$ is a parent of $b$ it means that if $b \\in S$, then $a$ must also be in $S... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of triples $(f, g, h)$ of permutations on $\\{1,2,3,4,5\\}$ such that $$ \\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\\\ & g(h(f(x)))=f(h(g(x)))=h(x), \\text { and } \\\\ & h(f(g(x)))=g(f(h(x)))=f(x) \\end{aligned} $$ for all $x \\in\\{1,2,3,4,5\\}$."
}... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 6620,
"solution": "Let $f g$ represent the composition of permutations $f$ and $g$, where $(f g)(x)=f(g(x))$ for all $x \\in\\{1,2,3,4,5\\}$. Evaluating fghfh in two ways, we get $$ f=g f h=(f g h) f h=f g h f h=f(g h f) h=f h h, $$ so $h h=1$. Similarly, we get $f, g$, and $h$ are all involutions. Then ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of tuples $\\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ of (not necessarily positive) integers such that $a_{i} \\leq i$ for all $0 \\leq i \\leq 5$ and $$a_{0}+a_{1}+\\cdots+a_{5}=6$$"
}
] | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"index": 6621,
"solution": "Let $b_{i}=i-a_{i}$, so $b_{i} \\geq 0$. Then $$15-\\left(b_{0}+b_{1}+\\cdots+b_{5}\\right)=6 \\Longrightarrow b_{0}+b_{1}+\\cdots+b_{5}=9$$ By stars and bars, the answer is $\\binom{14}{5}=2002$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #2.4*"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 6622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \\leq a, c \\leq 59$ and $1 \\leq b, d \\... | math | {
"style": "rule",
"ground_truth": "366"
} | {
"index": 6623,
"solution": "We instead choose points $(0,0),(1,0),(a, b),(c, d)$ with $0 \\leq a, c \\leq 59$ and $0 \\leq b, d \\leq 5$ with $(c, d)$ in the interior of the triangle formed by the other three points. Any selection of these four points may be connected to form a concave quadrilateral in precisely ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \\times 3 \\times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)"
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 6624,
"solution": "We do casework on the two red unit cubes; they can either be in a corner, an edge, or the center of the face. - If they are both in a corner, they must be adjacent - for each configuration, this corresponds to an edge, of which there are 12. - If one is in the corner and the other is a... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways to color 3 cells in a $3 \\times 3$ grid so that no two colored cells share an edge."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6625,
"solution": "If the middle square is colored, then two of the four corner squares must be colored, and there are $\\binom{4}{2}=6$ ways to do this. If the middle square is not colored, then after coloring one of the 8 other squares, there are always 6 ways to place the other two squares. However, t... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color."
}
] | math | {
"style": "rule",
"ground_truth": "294"
} | {
"index": 6626,
"solution": "Number the vertices 1 through 7 in order. Then, the only way to have three vertices of a regular heptagon that do not form an isosceles triangle is if they are vertices $1,2,4$, rotated or reflected. Thus, it is impossible for have four vertices in the heptagon of one color because it ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways to fill each cell in a $8 \\times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \\times 2$ square in the grid contains the letters $H, M, M, T$ in some order."
}
] | math | {
"style": "rule",
"ground_truth": "1076"
} | {
"index": 6627,
"solution": "We solve the problem for general $n \\times n$ boards where $n$ even. Let the cell in the $i$-th row and $j$-th column be $a_{i, j}$. Claim: In any valid configuration, either the rows (or columns) alternate between ( $\\cdots, H, M, H, M, \\cdots)$ and $(\\cdots, T, M, T, M, \\cdots)$... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways to select 99 cells of a $19 \\times 19$ square grid such that no two selected cells share an edge or vertex."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 6628,
"solution": "We claim the number of ways to select $n^{2}-1$ such cells from a $(2 n-1) \\times(2 n-1)$ grid is exactly $n^{3}$, which implies the answer to this question is 1000 . Partition the board into $n^{2}$ regions, as pictured. Also, shade red every cell in an odd row and column red, so the... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the number of ways to tile a $3 \\times 5$ rectangle with one $1 \\times 1$ tile, one $1 \\times 2$ tile, one $1 \\times 3$ tile, one $1 \\times 4$ tile, and one $1 \\times 5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are conside... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 6629,
"solution": "Our strategy is to first place the $1 \\times 5$ and the $1 \\times 4$ tiles since their size restricts their location. We have three cases: - Case 1: first row. There are 4 ways to place the $1 \\times 4$ tile. There is an empty cell next to the $1 \\times 4$ tile, which can either be... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the ordered pair of positive integers $(x,y)$ such that\n\n\\begin{align*}\nx^y+1&=y^x,\\\\\n2x^y&=y^x+7.\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "(2,3)"
} | {
"index": 6630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the perimeter of the triangle that has area $3-\\sqrt{3}$ and angles $45^\\circ$ , $60^\\circ$ , and $75^\\circ$ ."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2} + 2\\sqrt{3} - \\sqrt{6}"
} | {
"index": 6631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the positive integer less than 1000 which has exactly 29 positive proper divisors."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 6632,
"solution": "Recall that the number $N=p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{k}^{e_{k}}$ (where the $p_{i}$ are distinct primes) has exactly $(e_{1}+1)(e_{2}+1) \\cdots(e_{k}+1)$ positive integer divisors including itself. We seek $N<1000$ such that this expression is 30. Since $30=2 \\cdot 3 \\cd... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the positive real number $x$ satisfying $x^{\\left(2 x^{6}\\right)}=3$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[6]{3}"
} | {
"index": 6633,
"solution": "Let $t=x^{6}$, so $x^{2 t}=3$. Taking this to the third power gives $x^{6 t}=27$, or equivalently $t^{t}=3^{3}$. We can see by inspection that $t=3$, and this is the only solution as for $t>1$, the function $t^{t}$ is monotonically increasing, and if $0<t<1, t^{t}<1$. Solving for $x$ g... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the prime factorization of $1007021035035021007001$ . (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\\ldots p_k^{e_k}$ where $p_1,\\ldots,p_k$ are distinct prime numbers and $e_1,\\ldots,e_k$ are positive integers.)"
}
] | math | {
"style": "rule",
"ground_truth": "7^7 \\times 11^7 \\times 13^7"
} | {
"index": 6634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the prime factorization of 1007021035035021007001."
}
] | math | {
"style": "rule",
"ground_truth": "7^{7} \\cdot 11^{7} \\cdot 13^{7}"
} | {
"index": 6635,
"solution": "The number in question is $$\\sum_{i=0}^{7}\\binom{7}{i} 1000^{i}=(1000+1)^{7}=1001^{7}=7^{7} \\cdot 11^{7} \\cdot 13^{7}$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the prime factorization of \\(1007021035035021007001\\). (You should write your answer in the form \\(p_{1}^{e_{1}} p_{2}^{e_{2}} \\ldots p_{k}^{e_{k}}\\), where \\(p_{1}, \\ldots, p_{k}\\) are distinct prime numbers and \\(e_{1}, \\ldots, e_{k}\\) are positive integers... | math | {
"style": "rule",
"ground_truth": "7^7 * 11^7 * 13^7"
} | {
"index": 6636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product\n\\[\n\\prod_{n = 1}^{15} \\frac{n^2 + 5n + 6}{n+2}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18!}{6}"
} | {
"index": 6637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product of $0.\\overline{123}$ and $9$, and write your result as a fraction in simplified form."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{37}"
} | {
"index": 6638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product of all positive integers $b \\geq 2$ for which the base $b$ number $111111_{b}$ has exactly $b$ distinct prime divisors."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 6639,
"solution": "Notice that this value, in base $b$, is $$\\frac{b^{6}-1}{b-1}=(b+1)\\left(b^{2}-b+1\\right)\\left(b^{2}+b+1\\right)$$ This means that, if $b$ satisfies the problem condition, $(b+1)\\left(b^{2}-b+1\\right)\\left(b^{2}+b+1\\right)>p_{1} \\ldots p_{b}$, where $p_{i}$ is the $i$ th small... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product of the roots of the equation \\[3x^3 - x^2 - 20x + 27 = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 6640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product of the roots of the equation \\[x^3 - 12x^2 + 48x + 28 = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-28"
} | {
"index": 6641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the product of the sums of the squares and the cubes of the roots of the equation \\[x\\sqrt{x} - 8x + 9\\sqrt{x} - 1 = 0,\\] given that all roots are real and nonnegative."
}
] | math | {
"style": "rule",
"ground_truth": "13754"
} | {
"index": 6642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the radius of the inscribed circle of a triangle with sides 15,16 , and 17 ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"index": 6643,
"solution": "Hero's formula gives that the area is $\\sqrt{24 \\cdot 9 \\cdot 8 \\cdot 7}=24 \\sqrt{21}$. Then, using the result that the area of a triangle equals the inradius times half the perimeter, we see that the radius is $\\sqrt{21}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the range of $y=|x+7|-|x-2|$."
}
] | math | {
"style": "rule",
"ground_truth": "[-9, 9]"
} | {
"index": 6644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the remainder when\n${2007 \\choose 0} + {2007 \\choose 3} + \\cdots + {2007 \\choose 2007}$\nis divided by 1000."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 6645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the remainder when $$\\sum_{k=1}^{30303} k^{k}$$ is divided by 101."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 6646,
"solution": "The main idea is the following lemma: Lemma. For any non-negative integer $n$ and prime $p, \\sum_{k=n+1}^{n+p^{2}-p} k^{k} \\equiv 1(\\bmod p)$. Proof. Note that $a^{b}$ depends only on the value of $a(\\bmod p)$ and the value of $b(\\bmod p-1)$. Since $p$ and $p-1$ are relatively pri... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the remainder when 10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032."
}
] | math | {
"style": "rule",
"ground_truth": "40968192"
} | {
"index": 6647,
"solution": "Let $X_{k}$ denote $2^{k}$ except with leading zeroes added to make it four digits long. Let $\\overline{a b c \\cdots}$ denote the number obtained upon concatenating $a, b, c, \\ldots$ We have $$2^{6} \\cdot \\overline{X_{0} X_{1} \\ldots X_{5}}=\\overline{X_{6} X_{7} \\ldots X_{11}}$... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the side length of the largest cube contained in the region $\\{(x, y, z): x^{2}+y^{2}+z^{2} \\leq 25 \\text{ and } x \\geq 0\\}$ of three-dimensional space."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{6}}{3}"
} | {
"index": 6648,
"solution": "The given region is a hemisphere, so the largest cube that can fit inside it has one face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be $s$. Then, the radius of the circle is the hypotenuse of a triangle ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest base-10 positive integer greater than 15 that is a palindrome both in base 2 and base 4."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 6649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest base-10 positive integer greater than 5 that is a palindrome when written in both base 2 and 4."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 6651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest multiple of 63 with an odd number of ones in its base two representation."
}
] | math | {
"style": "rule",
"ground_truth": "4221"
} | {
"index": 6652,
"solution": "Notice that $63=2^{6}-1$, so for any $a$ we know $$63 a=64 a-a=2^{6}(a-1)+(64-a)$$ As long as $a \\leq 64$, we know $a-1$ and $64-a$ are both integers between 0 and 63 , so the binary representation of $63 a$ is just $a-1$ followed by $64-a$ in binary (where we append leading 0 s to ma... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive angle $x,$ in degrees, such that\n\\[\\tan 4x = \\frac{\\cos x - \\sin x}{\\cos x + \\sin x}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "9^\\circ"
} | {
"index": 6653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $k$ such that 49 divides $\\binom{2 k}{k}$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 6654,
"solution": "The largest $a$ such that $7^{a} \\left\\lvert\\,\\binom{ 2 k}{k}\\right.$ is equal to the number of carries when you add $k+k$ in base 7 , by Kummer's Theorem. Thus, we need two carries, so $2 k$ must have at least 3 digits in base 7 . Hence, $k \\geq 25$. We know $k=25$ works because... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ for which $$0<\\sqrt[4]{n}-\\lfloor\\sqrt[4]{n}\\rfloor<\\frac{1}{2015}$$"
}
] | math | {
"style": "rule",
"ground_truth": "4097"
} | {
"index": 6655,
"solution": "Let $n=a^{4}+b$ where $a, b$ are integers and $0<b<4 a^{3}+6 a^{2}+4 a+1$. Then $$\\begin{aligned} \\sqrt[4]{n}-\\lfloor\\sqrt[4]{n}\\rfloor & <\\frac{1}{2015} \\\\ \\sqrt[4]{a^{4}+b}-a & <\\frac{1}{2015} \\\\ \\sqrt[4]{a^{4}+b} & <a+\\frac{1}{2015} \\\\ a^{4}+b & <\\left(a+\\frac{1}{2... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ for which $\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "6156"
} | {
"index": 6656,
"solution": "The number $\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}}$ is a positive integer if and only if its square is a perfect square. We have $$(\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}})^{2} =(100+\\sqrt{n})+(100-\\sqrt{n})+2 \\sqrt{(100+\\sqrt{n})(100-\\sqrt{n})} =200+2 \\sqrt{10000-n}$$ T... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that\n\\[\\sum_{k = 0}^n \\log_2 \\left( 1 + \\frac{1}{2^{2^k}} \\right) \\ge 1 + \\log_2 \\frac{2014}{2015}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 6657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that $(n + i), (n + i)^3,$ and $(n + i)^4$ are the vertices of a triangle in the complex plane whose area is greater than 3000."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 6659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63."
}
] | math | {
"style": "rule",
"ground_truth": "111888"
} | {
"index": 6660,
"solution": "First, the number must be a multiple of 9 and 7. The first is easy to check and holds for all permutations. Note that when two adjacent digits $a$ and $b$ are swapped, the number changes by $9(a-b) \\cdot 10^{k}$ (we disregard sign), so $9(a-b)$ must also be a multiple of 63 for all di... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive integer that does not appear in any problem statement on any round at HMMT November 2023."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6661,
"solution": "The number 22 does not appear on any round. On the other hand, the numbers 1 through 21 appear as follows. \\begin{tabular}{c|c|c} Number & Round & Problem \\\\ \\hline 1 & Guts & 21 \\\\ 2 & Guts & 13 \\\\ 3 & Guts & 17 \\\\ 4 & Guts & 13 \\\\ 5 & Guts & 14 \\\\ 6 & Guts & 2 \\\\ 7 & ... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the smallest positive value of $x,$ in degrees, for which the function\n\\[f(x) = \\sin \\frac{x}{3} + \\sin \\frac{x}{11}\\]achieves its maximum value."
}
] | math | {
"style": "rule",
"ground_truth": "8910^\\circ"
} | {
"index": 6662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the square of 1017 without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "1034289"
} | {
"index": 6663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the square of 1023 without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "1046529"
} | {
"index": 6664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the square of 1033 without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "1067089"
} | {
"index": 6665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the square of 1085 without using a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "1177225"
} | {
"index": 6666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum $\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor$."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 6667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum $i^{-103} + i^{-102} + \\cdots + i^{-1} + i^0 + i^1 + \\cdots + i^{102} + i^{103} + \\sum_{n=1}^{103} n$."
}
] | math | {
"style": "rule",
"ground_truth": "5355"
} | {
"index": 6668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum \\( S = \\sum_{i=0}^{101} \\frac{x_{i}^{3}}{1 - 3x_{i} + 3x_{i}^{2}} \\) for \\( x_{i} = \\frac{i}{101} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 6669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 6670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #3.4*"
}
] | math | {
"style": "rule",
"ground_truth": "164"
} | {
"index": 6671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of $302^2 - 298^2$ and $152^2 - 148^2$."
}
] | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"index": 6672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of \\(0.\\overline{25} + 0.\\overline{87}\\) as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{112}{99}"
} | {
"index": 6673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "179"
} | {
"index": 6674,
"solution": "All squares must end with $0,1,4,5,6$, or 9, meaning that $p$ must end with 1 and 9. Moreover, since all odd squares are $1 \\bmod 4$, we know that $p$ must be $1 \\bmod 4$. This rules all primes except for $41,61,29,89$. Since $17^{2}=289,19^{2}=361,23^{2}=529,89,61$, and 29 all work.... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all integers $1 \\leq a \\leq 10$ with the following property: there exist integers $p$ and $q$ such that $p, q, p^{2}+a$ and $q^{2}+a$ are all distinct prime numbers."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6675,
"solution": "Odd $a$ fail for parity reasons and $a \\equiv 2(\\bmod 3)$ fail for $\\bmod 3$ reasons. This leaves $a \\in\\{4,6,10\\}$. It is easy to construct $p$ and $q$ for each of these, take $(p, q)=(3,5),(5,11),(3,7)$, respectively."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive integers $a \\leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 6676,
"solution": "Assume $b$ and $c$ exist. Considering the two values modulo 13, we find $$\\begin{cases}a+10 b+2 c \\equiv 2 & (\\bmod 13) \\\\ 2 a+5 b+c \\equiv 8 & (\\bmod 13)\\end{cases}$$ Subtracting twice the second equation from the first, we get $-3 a \\equiv-14(\\bmod 13)$. So, we have $a \\eq... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive integers $n$ for which $9 \\sqrt{n}+4 \\sqrt{n+2}-3 \\sqrt{n+16}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6677,
"solution": "For the expression to be an integer at least one of $n$ and $n+2$ must be a perfect square. We also note that at most one of $n$ and $n+2$ can be a square, so exactly one of them is a square. Case 1: $n$ is a perfect square. By our previous observation, it must be that $4 \\sqrt{n+2}=3... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive integers $n$ such that $50 \\leq n \\leq 100$ and $2 n+3$ does not divide $2^{n!}-1$."
}
] | math | {
"style": "rule",
"ground_truth": "222"
} | {
"index": 6678,
"solution": "We claim that if $n \\geq 10$, then $2 n+3 \\nmid 2^{n!}-1$ if and only if both $n+1$ and $2 n+3$ are prime. If both $n+1$ and $2 n+3$ are prime, then assume $2 n+3 \\mid 2^{n!}-1$. By Fermat Little Theorem, $2 n+3 \\mid 2^{2 n+2}+1$. However, since $n+1$ is prime, $\\operatorname{gcd}... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive integers $n$ such that $n^{2}-3000$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "1872"
} | {
"index": 6679,
"solution": "Suppose $n^{2}-3000=x^{2}$, so $n^{2}-x^{2}=3000$. This factors as $(n-x)(n+x)=3000$. Thus, we have $n-x=2a$ and $n+x=2b$ for some positive integers $a, b$ such that $ab=750$ and $a<b$. Therefore, we have $n=a+b$, so the sum will be just the sum of divisors of $750=2 \\cdot 3 \\cdot 5^... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive integers $n<2048$ such that $n$ has an even number of 1's in its binary representation."
}
] | math | {
"style": "rule",
"ground_truth": "1048064"
} | {
"index": 6680,
"solution": "Note that the positive integers less than 2047 are those with at most 11 binary digits. Consider the contribution from any one of those digits. If we set that digit to 1, then the remaining 10 digits can be set in $2^{9}=512$ ways so that the number of 1's is even. Therefore the answer... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all positive real numbers \\(x \\leq 5\\) satisfying \\(x=\\frac{\\left\\lceil x^{2}\\right\\rceil+\\lceil x\\rceil \\cdot\\lfloor x\\rfloor}{\\lceil x\\rceil+\\lfloor x\\rfloor}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 6681,
"solution": "Note that all integer \\(x\\) work. If \\(x\\) is not an integer then suppose \\(n<x<n+1\\). Then \\(x=n+\\frac{k}{2n+1}\\), where \\(n\\) is an integer and \\(1 \\leq k \\leq 2n\\) is also an integer, since the denominator of the fraction on the right hand side is \\(2n+1\\). We now s... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all possible distinct values of \\( m+n \\) if \\( m \\) and \\( n \\) are positive integers such that\n$$\n\\operatorname{lcm}(m, n) + \\operatorname{gcd}(m, n) = 2(m+n) + 11\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 6682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all the roots of $(2x+3)(x-4)+(2x+3)(x-6)=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 6683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers \\underline{a} \\underline{b} \\underline{c}, if \\underline{a} \\underline{b} \\underline{c} is a multiple of $x$, then the three-digit (base 10) number \\underline... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 6684,
"solution": "Note that $\\overline{a b c 0}-\\overline{b c a}=a\\left(10^{4}-1\\right)$ must also be a multiple of $x$. Choosing $a=1$ means that $x$ divides $10^{3}-1$, and this is clearly a necessary and sufficient condition. The only two-digit factors of $10^{3}-1$ are 27 and 37, so our answer i... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the fourth powers of the roots of the equation \\[ x\\sqrt{x} - 8x + 9\\sqrt{x} - 2 = 0, \\] assuming all roots are real and nonnegative."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"index": 6685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \\cdots + 512$."
}
] | math | {
"style": "rule",
"ground_truth": "341"
} | {
"index": 6686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-1 -3-9-27 -81-243-729$."
}
] | math | {
"style": "rule",
"ground_truth": "-1093"
} | {
"index": 6687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-3 + 6 - 12 + 24 - \\cdots - 768$."
}
] | math | {
"style": "rule",
"ground_truth": "514"
} | {
"index": 6688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the number $10 - \\sqrt{2018}$ and its radical conjugate."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the roots of the equation \\[x\\sqrt{x} - 6x + 7\\sqrt{x} - 1 = 0,\\]given that all of the roots are real and nonnegative."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the series:\n\n\\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "305175780"
} | {
"index": 6691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the series:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "510"
} | {
"index": 6692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the series:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 6693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the squares of cosine for the angles progressing by 10 degrees starting from 0 degrees up to 180 degrees:\n\\[\\cos^2 0^\\circ + \\cos^2 10^\\circ + \\cos^2 20^\\circ + \\dots + \\cos^2 180^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{2}"
} | {
"index": 6694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \\cdots + 10_6^2$."
}
] | math | {
"style": "rule",
"ground_truth": "231_6"
} | {
"index": 6695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the squares of the roots of the equation \\[x\\sqrt{x} - 8x + 9\\sqrt{x} - 2 = 0,\\] given that all of the roots are real and nonnegative."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 6696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum of the squares of the roots of the equation \\[x^{2018} +44x^{2015} + 3x^3 + 404 = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum:\n\\[\n\\sin^2 3^\\circ + \\sin^2 9^\\circ + \\sin^2 15^\\circ + \\dots + \\sin^2 177^\\circ.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"index": 6699,
"solution": ""
} |
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