data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of positive integers $(m; n)$ , with $m \\le n$ , such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$ ."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 10300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pairs of union/intersection operations $\\left(\\square_{1}, \\square_{2}\\right) \\in\\{\\cup, \\cap\\}^{2}$ satisfying the condition: for any sets $S, T$, function $f: S \\rightarrow T$, and subsets $X, Y, Z$ of $S$, we have equality of sets $f(X) \\square_... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10301,
"solution": "If and only if $\\square_{1}=\\square_{2}=\\cup$. See http://math.stackexchange.com/questions/359693/overview-of-"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of partitions of the set $\\{1,2,3,\\cdots ,11,12\\}$ into three nonempty subsets such that no subset has two elements which differ by $1$ . \n\n[i]Proposed by Nathan Ramesh"
}
] | math | {
"style": "rule",
"ground_truth": "1023"
} | {
"index": 10302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of pentominoes (5-square polyominoes) that span a 3-by-3 rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 10303,
"solution": "By enumeration, the answer is 6."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1,2,3,4,5,6)$ that satisfy\n\\[\\frac{a_1 + 1}{2} \\cdot \\frac{a_2 + 2}{2} \\cdot \\frac{a_3 + 3}{2} \\cdot \\frac{a_4 + 4}{2} \\cdot \\frac{a_5 + 5}{2} \\cdot \\frac{a_6 + 6}{2} > 6!.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "719"
} | {
"index": 10304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy\n\\[\\frac{a_1 + 1}{2} \\cdot \\frac{a_2 + 2}{2} \\cdot \\frac{a_3 + 3}{2} \\cdot \\frac{a_4 + 4}{2} \\cdot \\frac{a_5 + 5}{2} \\cdot \\frac{a_6 + 6}{2} \\cdot \\frac{a_7 + 7... | math | {
"style": "rule",
"ground_truth": "5039"
} | {
"index": 10305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \\[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\\] is divisible by $3$."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 10306,
"solution": "Since $3$ is one of the numbers, a product with a $3$ in it is automatically divisible by $3,$ so WLOG $x_3=3,$ we will multiply by $5$ afterward since any of $x_1, x_2, \\ldots, x_5$ would be $3,$ after some cancelation we see that now all we need to find is the number of ways that $... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations \\( a_1, a_2, \\ldots, a_{10} \\) of the numbers \\( 1, 2, \\ldots, 10 \\), such that \\( a_{i+1} \\) is not less than \\( a_i - 1 \\) for \\( i = 1, 2, \\ldots, 9 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 10307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations \\((b_1, b_2, b_3, b_4, b_5, b_6)\\) of \\((1,2,3,4,5,6)\\) such that\n\\[\n\\frac{b_1 + 6}{2} \\cdot \\frac{b_2 + 5}{2} \\cdot \\frac{b_3 + 4}{2} \\cdot \\frac{b_4 + 3}{2} \\cdot \\frac{b_5 + 2}{2} \\cdot \\frac{b_6 + 1}{2} > 6!.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "719"
} | {
"index": 10308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations \\(a_1, a_2, \\ldots, a_{10}\\) of the numbers \\(1, 2, \\ldots, 10\\) such that \\(a_{i+1}\\) is not less than \\(a_i - 1\\) for \\(i = 1, 2, \\ldots, 9\\)."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 10309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\\leq$ $k$ $\\leq$ $5$, at least one of the first $k$ terms of the permutation is greater than $k$."
}
] | math | {
"style": "rule",
"ground_truth": "461"
} | {
"index": 10310,
"solution": "If $6$ is the first number, then there are no restrictions. There are $5!$, or $120$ ways to place the other $5$ numbers.\n If $6$ is the second number, then the first number can be $2, 3, 4,$ or $5$, and there are $4!$ ways to place the other $4$ numbers. $4 \\cdot 4! = 96$ ways.\n I... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations of \\( n \\) distinct elements \\( a_1, a_2, \\cdots, a_n \\) (where \\( n \\geqslant 2 \\)) such that \\( a_1 \\) is not in the first position and \\( a_2 \\) is not in the second position."
}
] | math | {
"style": "rule",
"ground_truth": "32,527,596"
} | {
"index": 10311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations of the 6 characters $a, b, c, d, e, f$ such that the subsequences $a c e$ and $d f$ do not appear."
}
] | math | {
"style": "rule",
"ground_truth": "582"
} | {
"index": 10312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$ ."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 10313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive divisors $d$ of $15!=15 \\cdot 14 \\cdots 2 \\cdot 1$ such that $\\operatorname{gcd}(d, 60)=5$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 10314,
"solution": "Since $\\operatorname{gcd}(d, 60)=5$, we know that $d=5^{i} d^{\\prime}$ for some integer $i>0$ and some integer $d^{\\prime}$ which is relatively prime to 60. Consequently, $d^{\\prime}$ is a divisor of $(15!) / 5$; eliminating common factors with 60 gives that $d^{\\prime}$ is a fac... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive divisors of 2002."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integer divisors of 12 ! that leave a remainder of 1 when divided by 3."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 10316,
"solution": "First we factor $12!=2^{10} 3^{5} 5^{2} 7^{1} 11^{1}$, and note that $2,5,11 \\equiv-1(\\bmod 3)$ while $7 \\equiv 1$ $(\\bmod 3)$. The desired divisors are precisely $2^{a} 5^{b} 7^{c} 11^{d}$ with $0 \\leq a \\leq 10,0 \\leq b \\leq 2,0 \\leq c \\leq 1,0 \\leq d \\leq 1$, and $a+b+d... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integer solutions \\((x, y, z, w)\\) to the equation \\(x + y + z + w = 25\\) that satisfy \\(x < y\\)."
}
] | math | {
"style": "rule",
"ground_truth": "946"
} | {
"index": 10317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10318,
"solution": "If $n=1$, the relation can not hold, so assume otherwise. If $x>y$, the left hand side factors as $n^{y}\\left(n^{x-y}+1\\right)$ so $n^{x-y}+1$ is a power of $n$. But it leaves a remainder of 1 when divided by $n$ and is greater than 1, a contradiction. We reach a similar contradicti... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ for which \r\n\r\n(i) $n \\leq 1991$ ; \r\n\r\n(ii) 6 is a factor of $(n^2 + 3n +2)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1328"
} | {
"index": 10319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1899"
} | {
"index": 10320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10321,
"solution": "First of all, note that the nonnegative integer condition really does not matter, since even if we have a nonnegative power, there is always a power of $m$ we can multiply to get to non-negative powers. Now we see that our problem is just a matter of m-chopping blocks. What is meant b... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 1000$ such that $15n$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 1000$ that can be expressed in the form\n\\[\\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lfloor 3x \\rfloor = n\\]for some real number $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"index": 10323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 1500$ that can be expressed in the form\n\\[\\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor = n\\] for some real number $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "854"
} | {
"index": 10324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 600$ whose value can be uniquely determined when the values of $\\left\\lfloor \\frac n4\\right\\rfloor$, $\\left\\lfloor\\frac n5\\right\\rfloor$, and $\\left\\lfloor\\frac n6\\right\\rfloor$ are given, where $\\lfloor x \\rfloor$ d... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 10325,
"solution": "1. For $n$ to be uniquely determined, $n$ AND $n + 1$ both need to be a multiple of $4, 5,$ or $6.$ Since either $n$ or $n + 1$ is odd, we know that either $n$ or $n + 1$ has to be a multiple of $5.$ We can state the following cases:\n 1. $n$ is a multiple of $4$ and $n+1$ is a multip... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n=x\\lfloor x \\rfloor$.\n Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$."
}
] | math | {
"style": "rule",
"ground_truth": "496"
} | {
"index": 10326,
"solution": "Bounding gives $x^2\\le n<x^2+x$. Thus there are a total of $x$ possible values for $n$, for each value of $x^2$. Checking, we see $31^2+31=992<1000$, so there are \\[1+2+3+...+32= \\boxed{496}\\] such values for $n$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"index": 10327,
"solution": "Note that $1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}$ will have a denominator that divides $5!$. Therefore, for the expression to be an integer, $\\frac{n^6}{6!}$ must have a denominator that divides $5!$. Thus, $6\\mid n^6$, and $6\\mid n$. Let $n=6m$. Subst... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ such that \n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) > 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 3)(n - 5) \\dotsm (n - 97) < 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 5)(n - 9) \\dotsm (n - 101) < 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 10330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n,$ $1 \\le n \\le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 10332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n,$ $1 \\le n \\le 2000,$ for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 10333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers $n,$ $1 \\le n \\le 2600,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers \\( n \\) that satisfy\n\\[\n(n - 1)(n - 3)(n - 5) \\dotsm (n - 99) < 0.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 10335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers \\( x \\), where \\( x \\neq 9 \\), such that\n\\[\n\\log _{\\frac{x}{9}}\\left(\\frac{x^{2}}{3}\\right)<6+\\log _{3}\\left(\\frac{9}{x}\\right) .\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "223"
} | {
"index": 10336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers \\(n \\le 500\\) that can be expressed in the form\n\\[\n\\lfloor x \\rfloor + \\lfloor 3x \\rfloor + \\lfloor 4x \\rfloor = n\n\\]\nfor some real number \\(x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "248"
} | {
"index": 10337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10338,
"solution": "We look for all positive integers of the form $2^a-2^b<1000,$ where $0\\leq b<a.$ Performing casework on $a,$ we can enumerate all possibilities in the table below: \\[\\begin{array}{c|c} & \\\\ [-2.25ex] \\boldsymbol{a} & \\boldsymbol{b} \\\\ \\hline & \\\\ [-2ex] 1 & 0 \\\\ 2 & 0,1 ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than $2000$ that are neither $5$-nice nor $6$-nice."
}
] | math | {
"style": "rule",
"ground_truth": "1333"
} | {
"index": 10339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than 10000 that contain the digit 1."
}
] | math | {
"style": "rule",
"ground_truth": "3439"
} | {
"index": 10340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than 1000000 that are divisible by some perfect cube greater than 1."
}
] | math | {
"style": "rule",
"ground_truth": "168089"
} | {
"index": 10341,
"solution": "Using the following code, we get the answer (denoted by the variable ans): ans $=0$ for $n$ in xrange $(1,1000000)$ : ``` divisible_by_cube = True for i in xrange(2,101): if n%(i*i*i)==0: divisible_by_cube = False break if divisible_by_cube: ans = ans + 1 print ans ``` This gives the ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than 1000000 which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\\max \\left(0,20-80\\left|1-\\frac{X}{Y}\\right|\\right)$ rounded to the nearest inte... | math | {
"style": "rule",
"ground_truth": "247548"
} | {
"index": 10342,
"solution": "$\\mathrm{N}=1000000$ $\\mathrm{s}=[0] * \\mathrm{~N}$ ans $=0$ for i in range(1, N): if i <= s[i]: ans $+=1$ for $j$ in range(i + i, N, i): $s[j]+=$ i print(ans)"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 10343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$."
}
] | math | {
"style": "rule",
"ground_truth": "222"
} | {
"index": 10344,
"solution": "Note that $2017=2202201_{3}$, and $2187=3^7=10000000_{3}$. There can be a $1,2,...,7$ digit number less than $2187$, and each digit can either be $1$ or $2$. So $2^1$ one digit numbers and so on up to $2^7$ $7$ digit.\n Now we have to subtract out numbers from $2018$ to $2187$\n Then ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$"
}
] | math | {
"style": "rule",
"ground_truth": "435"
} | {
"index": 10345,
"solution": "$10^{10} = 2^{10}\\cdot 5^{10}$ so $10^{10}$ has $11\\cdot11 = 121$ divisors.\n$15^7 = 3^7\\cdot5^7$ so $15^7$ has $8\\cdot8 = 64$ divisors.\n$18^{11} = 2^{11}\\cdot3^{22}$ so $18^{11}$ has $12\\cdot23 = 276$ divisors.\n Now, we use the Principle of Inclusion-Exclusion. We have $121 +... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 10346,
"solution": "A number is divisible by four if its last two digits are divisible by 4. Thus, we require that $10b + a$ and $10b + c$ are both divisible by $4$. If $b$ is odd, then $a$ and $c$ must both be $2 \\pmod 4$ meaning that $a$ and $c$ are $2$ or $6$. If $b$ is even, then $a$ and $c$ must be... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of positive solutions to\n\\[\\sin (\\arccos( \\tan (\\arcsin x) ) ) ) = x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\\equiv 2013\\pmod p$ has a solution in integers $x$ and $y$ ."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 10348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most 5 such that $\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)^{2}=(a+b+c+d)(a-b+c-d)\\left((a-c)^{2}+(b-d)^{2}\\right)$"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 10349,
"solution": "Let $x=a+c, y=a-c, w=b+d$, and $z=b-d$. Then $$\\left(w^{2}+x^{2}+y^{2}+z^{2}\\right)^{2}=4\\left(x^{2}-w^{2}\\right)\\left(y^{2}+z^{2}\\right)$$ and since $\\left|x^{2}+w^{2}\\right| \\geq\\left|x^{2}-w^{2}\\right|$ it follows that $w=0$ or $y=z=0$. Now $y=z=0$ implies $a=b=c=d=0$. N... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 10350,
"solution": "Our fraction can be written in the form $\\frac{1000 - a}{a} = \\frac{1000}{a} - 1.$ Thus the fraction is reducible when $a$ divides $1000.$ We also want $500 < a < 1000.$ By PIE, the total values of $a$ that make the fraction reducible is, \\[249 + 99 - 49 = 299.\\] By complementary ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of rationals $\\frac{m}{n}$ such that \n\n(i) $0 < \\frac{m}{n} < 1$ ;\n\n(ii) $m$ and $n$ are relatively prime;\n\n(iii) $mn = 25!$ ."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 10351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions of the equation\n\\[\\frac{4x}{x^2 + x + 3} + \\frac{5x}{x^2 - 5x + 3} = -\\frac{3}{2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions of the equation\n\\[\\frac{x}{100} = \\sin x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 10353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions of the equation\n\\[\\frac{x}{50} = \\cos x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 10354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to\n\\[(x^{2006} + 1)(x^{2004} + x^{2002} + x^{2000} + \\dots + x^2 + 1) = 2006x^{2005}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\n\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{10}{x - 10} = 2x.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 10356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\n\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{50}{x - 50} = 2x.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 10357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\n\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{50}{x - 50} = x + 5.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 10358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{100}{x - 100} = x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 10359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{120}{x - 120} = x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 10360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real solutions to the equation:\n\\[\n\\frac{1^2}{x - 1} + \\frac{2^2}{x - 2} + \\frac{3^2}{x - 3} + \\dots + \\frac{120^2}{x - 120} = x.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 10361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of real zeros of $x^{3}-x^{2}-x+2$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10362,
"solution": "Let $f(x)=x^{3}-x^{2}-x+2$, so $f^{\\prime}(x)=3 x^{2}-2 x-1$. The slope is zero when $3 x^{2}-2 x-1=0$, where $x=-\\frac{1}{3}$ and $x=1$. Now $f\\left(\\frac{1}{3}\\right)>0$ and $f(1)>0$, so there are no zeros between $x=-\\frac{1}{3}$ and $x=1$. Since \\lim _{x \\rightarrow+\\inft... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET."
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 10363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$."
}
] | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 10364,
"solution": "We use Burnside's Lemma. The set being acted upon is the set of integer triples $(a,r,s)$ such that $ars=2010$. Because $r$ and $s$ are indistinguishable, the permutation group consists of the identity and the permutation that switches $r$ and $s$. In cycle notation, the group consist... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of sequences $a_{1}, a_{2}, \\ldots, a_{10}$ of positive integers with the property that $a_{n+2}=a_{n+1}+a_{n}$ for $n=1,2, \\ldots, 8$, and $a_{10}=2002$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 10365,
"solution": "3 Let $a_{1}=a, a_{2}=b$; we successively compute $a_{3}=a+b ; \\quad a_{4}=a+$ $2 b ; \\quad \\ldots ; \\quad a_{10}=21 a+34 b$. The equation $2002=21 a+34 b$ has three positive integer solutions, namely $(84,7),(50,28),(16,49)$, and each of these gives a unique sequence."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \\( k \\) letters, the number of S's is strictly more than 20 times the number of R's for all \\( 1 \\leq k \\leq 2111 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2111} \\binom{2111}{100}"
} | {
"index": 10366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of sets $A$ that satisfy the three conditions: $\\star$ $A$ is a set of two positive integers $\\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\\star$ $A$ contains the number $30.$"
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 10367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$."
}
] | math | {
"style": "rule",
"ground_truth": "728"
} | {
"index": 10368,
"solution": "Note that the prime factorization of the product is $3^{2}\\cdot 7 \\cdot 11 \\cdot 17 \\cdot 31 \\cdot 41 \\cdot 61$. Ignoring overcounting, by stars and bars there are $6$ ways to choose how to distribute the factors of $3$, and $3$ ways to distribute the factors of the other primes... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of sets of composite numbers less than 23 that sum to 23."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 10369,
"solution": "Because 23 is odd, we must have an odd number of odd numbers in our set. Since the smallest odd composite number is 9, we cannot have more than 2 odd numbers, as otherwise the sum would be at least 27. Therefore, the set has exactly one odd number. The only odd composite numbers less ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of six-digit palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"index": 10370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions in natural numbers for the equation \\(\\left\\lfloor \\frac{x}{10} \\right\\rfloor = \\left\\lfloor \\frac{x}{11} \\right\\rfloor + 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 10371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions in natural numbers to the equation \\(\\left\\lfloor \\frac{x}{10} \\right\\rfloor = \\left\\lfloor \\frac{x}{11} \\right\\rfloor + 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 10372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \\ldots, a_{k} ; b_{1}, b_{2}, \\ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\\cdots+a_{k}(b_{1}+b_{2}+\\cdots+b_{k})=7$$"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 10373,
"solution": "Let $k, a_{1}, \\ldots, a_{k}, b_{1}, \\ldots, b_{k}$ be a solution. Then $b_{1}, b_{1}+b_{2}, \\ldots, b_{1}+\\cdots+b_{k}$ is just some increasing sequence of positive integers. Considering the $a_{i}$ as multiplicities, the $a_{i}$ 's and $b_{i}$ 's uniquely determine a partition o... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\cos 4x + \\cos^2 3x + \\cos^3 2x + \\cos^4 x = 0\\]for $-\\pi \\le x \\le \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{2} \\right)^x\\]on the interval $(0,100 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 10375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,100\\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 10376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,150 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 10377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,200 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 10378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{3}{4} \\right)^x\\]\non the interval \\( (0, 50\\pi) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\]\non the interval \\( (0, 150 \\pi) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 10380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,150 \\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 10381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\tan (3 \\pi \\cos \\theta) = \\cot (3 \\pi \\sin \\theta)\\]\nwhere $\\theta \\in (0, 3\\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\tan (5 \\pi \\cos \\theta) = \\cot (5 \\pi \\sin \\theta)\\]where $\\theta \\in (0, 2 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\tan (7 \\pi \\cos \\theta) = \\cot (7 \\pi \\sin \\theta)\\]\nwhere $\\theta \\in (0, 4 \\pi).$"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to the equation $x+y+z=525$ where $x$ is a multiple of 7, $y$ is a multiple of 5, and $z$ is a multiple of 3."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 10385,
"solution": "First, note that $525=3 \\times 7 \\times 5 \\times 5$. Then, taking the equation modulo 7 gives that $7 \\mid x$; let $x=7 x^{\\prime}$ for some nonnegative integer $x^{\\prime}$. Similarly, we can write $y=5 y^{\\prime}$ and $z=3 z^{\\prime}$ for some nonnegative integers $y^{\\prim... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to:\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\]\non the interval $(0,50 \\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of solutions to:\n\\[\\sin x = \\left(\\frac{1}{3}\\right)^x\\]\non the interval $(0, 50\\pi)$."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 10387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of square units in the area of the shaded region. Consider a rectangular plot with vertices at points (0,0), (40,0), (40,20), and (0,20). Within this rectangle, a shaded polygon is formed with vertices at (0,0), (20,0), (40,10), (40,20), and (10,20). Determine t... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 10388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of square units in the area of the shaded region. [asy]\nsize(200);\ndraw((0,0)--(30,0)--(30,30)--(0, 30)--cycle);\ndot((30,0));dot((0,0));dot((10,0));dot((30,20));dot((30,30)); dot((20,30));dot((0,30));dot((0,10));\nfilldraw((0,0)--(10,0)--(30,20)--(30,30)--(20... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 10389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of square units in the area of the triangle.\n\n[asy]size(125);\ndraw( (-10,-2) -- (2,10), Arrows);\ndraw( (0,-2)-- (0,10) ,Arrows);\ndraw( (5,0) -- (-10,0),Arrows);\n\nlabel(\"$l$\",(2,10), NE);\nlabel(\"$x$\", (5,0) , E);\nlabel(\"$y$\", (0,-2) , S);\nfilldraw... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 10390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 10391,
"solution": "For a natural number $n$, let $A_{n}$ be a set containing all sequences which satisfy the problem conditions but which 12 is replaced by $n$. Also, let $a_{n}$ be the size of $A_{n}$. We first consider $a_{1}$ and $a_{2}$. We get $a_{1}=1$, as the only sequence satisfying the problem ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets $S$ of $\\{1,2, \\ldots 63\\}$ the sum of whose elements is 2008."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 10392,
"solution": "Note that $1+2+\\cdots+63=2016$. So the problem is equivalent to finding the number of subsets of $\\{1,2, \\cdots 63\\}$ whose sum of elements is 8. We can count this by hand: $\\{8\\},\\{1,7\\},\\{2,6\\}$, $\\{3,5\\},\\{1,2,5\\},\\{1,3,4\\}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets $S$ of $\\{1,2, \\ldots 6\\}$ satisfying the following conditions: - $S$ is non-empty. - No subset of $S$ has the property that the sum of its elements is 10."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 10393,
"solution": "We do casework based on the largest element of $S$. Call a set $n$-free if none of its subsets have elements summing to $n$. Case 1: The largest element of $S$ is 6. Then $4 \\notin S$. If $5 \\notin S$, then we wish to find all 4-free subsets of $\\{1,2,3\\}$ (note that $1+2+3=6<10$)... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets $S$ of $\\{1,2, \\ldots, 48\\}$ satisfying both of the following properties: - For each integer $1 \\leq k \\leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$. - There are exactly nine integers $1 \\leq m \\leq 47$ so that both $m$ and $m+1$ are in ... | math | {
"style": "rule",
"ground_truth": "177100"
} | {
"index": 10394,
"solution": "This problem can be thought of as laying down a series of $1 \\times 2$ dominoes, with each one having either the left or right square marked. The second condition states that exactly 9 pairs of consecutive dominoes will have the leftmost one with the right square marked and the right... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets $\\{a, b, c\\}$ of $\\{1,2,3,4, \\ldots, 20\\}$ such that $a<b-1<c-3$."
}
] | math | {
"style": "rule",
"ground_truth": "680"
} | {
"index": 10395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ that are subsets of neither $\\{1, 2, 3, 4, 5\\}$ nor $\\{4, 5, 6, 7, 8\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"index": 10396,
"solution": "Note that by Principle of Inclusion and Exclusion, the total number of subsets must be $2^8-2^5-2^5+2^2$ as denoted by above. Thus our answer is $64(3)+4 = \\boxed{196}$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of subsets of $\\{1,2,3,\\ldots,10\\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\\{\\mathbf{1},\\mathbf{2},5\\}$ and $\\{1,3,\\mathbf{6},\\mathbf{7},10\\}.$"
}
] | math | {
"style": "rule",
"ground_truth": "235"
} | {
"index": 10397,
"solution": "The problem is the same as laying out a line of polynomoes to cover spots $0,1,...10$: 1 triomino ($RGG$), $n$ dominoes ($RG$), and $8-2n$ monominoes ($R$). The $G$ spots cover the members of the subset. The total number spots is 11, because one $R$ spot always covers the 0, and the o... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of terms $n \\leq 2012$ such that $a_{n}=\\frac{3^{n+1}-1}{2}$ is divisible by 7."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 10398,
"solution": "To demonstrate that $7 \\mid a_{n}=\\frac{3^{n+1}-1}{2}$ is equivalent to showing that $3^{n+1}-1 \\equiv 0 \\bmod 7$. However $3^{n+1}-1=0 \\bmod 7 \\Longrightarrow n+1 \\equiv 0 \\bmod 6$, hence the values of $n \\leq 2012$ are $5,11,17, \\ldots, 2009$, which is in total $$\\frac{20... |
DeepScaleR | [
{
"role": "user",
"content": "Find the number of the form $7x36y5$ that is divisible by 1375."
}
] | math | {
"style": "rule",
"ground_truth": "713625"
} | {
"index": 10399,
"solution": ""
} |
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