data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For how many pairs of consecutive integers in the set $\\{1100, 1101, 1102, \\ldots, 2200\\}$ is no carrying required when the two integers are added?"
}
] | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"index": 11800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many pairs of nonzero integers $(c, d)$ with $-2015 \\leq c, d \\leq 2015$ do the equations $c x=d$ and $d x=c$ both have an integer solution?"
}
] | math | {
"style": "rule",
"ground_truth": "8060"
} | {
"index": 11801,
"solution": "We need both $c / d$ and $d / c$ to be integers, which is equivalent to $|c|=|d|$, or $d= \\pm c$. So there are 4030 ways to pick $c$ and 2 ways to pick $d$, for a total of 8060 pairs."
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integer values of $N$ is the expression $\\dfrac{36}{N+2}$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11802,
"solution": "To solve the problem, we need to determine for how many positive integer values of $N$ the expression $\\frac{36}{N+2}$ results in an integer. This is equivalent to finding the number of positive divisors of $36$ that are greater than $2$ (since $N+2$ must be a divisor of $36$ and $N$... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11803,
"solution": "1. **Identify the conditions for $\\frac{n}{3}$ and $3n$:**\n - $\\frac{n}{3}$ must be a three-digit whole number. This implies $\\frac{n}{3} \\geq 100$ and $\\frac{n}{3} \\leq 999$.\n - $3n$ must also be a three-digit whole number. This implies $3n \\geq 100$ and $3n \\leq 999$.\... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ , $1\\leq n\\leq 2008$ , can the set \\[\\{1,2,3,\\ldots,4n\\}\\] be divided into $n$ disjoint $4$ -element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that sub... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"index": 11804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $a$ does the polynomial $x^{2}-a x+a$ have an integer root?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 11805,
"solution": "Let $r, s$ be the roots of $x^{2}-a x+a=0$. By Vieta's, we have $r+s=r s=a$. Note that if one root is an integer, then both roots must be integers, as they sum to an integer $a$. Then, $$r s-(r+s)+1=1 \\Longrightarrow(r-1)(s-1)=1$$ Because we require $r, s$ to be both integers, we hav... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11806,
"solution": "Suppose that $k$ is a fixed, but unknown, positive integer. Suppose also that the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at the point with positive integer coordinates $(x, y)$. Since $9x+4y=600$ and $kx-4y=24$, adding these equations, we get $9x+kx=624$ and so $(9+... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n \\le 1000$ is$\\left\\lfloor \\dfrac{998}{n} \\right\\rfloor+\\left\\lfloor \\dfrac{999}{n} \\right\\rfloor+\\left\\lfloor \\dfrac{1000}{n}\\right \\rfloor$not divisible by $3$?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 11807,
"solution": "We are tasked with finding how many positive integers \\( n \\leq 1000 \\) make the expression\n\\[\n\\left\\lfloor \\frac{998}{n} \\right\\rfloor + \\left\\lfloor \\frac{999}{n} \\right\\rfloor + \\left\\lfloor \\frac{1000}{n} \\right\\rfloor\n\\]\nnot divisible by 3. We start by ana... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n \\leq 100$ is it true that $10 n$ has exactly three times as many positive divisors as $n$ has?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 11808,
"solution": "Let $n=2^{a} 5^{b} c$, where $2,5 \\nmid c$. Then, the ratio of the number of divisors of $10 n$ to the number of divisors of $n$ is $\\frac{a+2}{a+1} \\frac{b+2}{b+1}=3$. Solving for $b$, we find that $b=\\frac{1-a}{2 a+1}$. This forces $(a, b)=(0,1),(1,0)$. Therefore, the answers ar... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ does $1+2+\\cdots+n$ evenly divide $6n$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ does $\\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 11810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ is $\\frac{n}{30-n}$ also a positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11811,
"solution": "\n#### Solution 1: Direct Substitution and Checking\n1. **Condition for Positive Fraction**: We need $\\frac{n}{30-n}$ to be a positive integer. For this to happen, both $n$ and $30-n$ must be positive. This implies $0 < n < 30$.\n2. **Condition for Integer Result**: The fraction $\\f... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "212"
} | {
"index": 11812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to $1000$ is $(\\sin t + i \\cos t)^n = \\sin nt + i \\cos nt$ true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 11813,
"solution": "Let $t=0$. Then, we have $i^n=i$ which means $n\\equiv 1\\pmod{4}$. Thus, the answer is $\\boxed{250}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11814,
"solution": "1. **Sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula:\n \\[\n 1 + 2 + \\cdots + n = \\frac{n(n+1)}{2}\n \\]\n This formula can be derived by pairing terms from the beginning and end of the sequence, each pair summing... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 1000 is $$(\\sin t+i\\cos t)^n=\\sin nt+i\\cos nt$$true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 11815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 11816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\sin (t+\\frac{\\pi}{4})+i\\cos (t+\\frac{\\pi}{4}))^n=\\sin (nt+\\frac{n\\pi}{4})+i\\cos (nt+\\frac{n\\pi}{4})$$ true for all real $t$?"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n$, with $n \\leq 100$, is $n^{3}+5n^{2}$ the square of an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11818,
"solution": "For $n^{3}+5n^{2}$ to be the square of an integer, $\\sqrt{n^{3}+5n^{2}}$ must be an integer. We know that $\\sqrt{n^{3}+5n^{2}}=\\sqrt{n^{2}(n+5)}=\\sqrt{n^{2}} \\sqrt{n+5}=n \\sqrt{n+5}$. For $n \\sqrt{n+5}$ to be an integer, $\\sqrt{n+5}$ must be an integer. In other words, $n+5$ m... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $n\\geq 2$ is $1001_n$ a prime number?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 11819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $x$ is $(x-2)(x-4)(x-6) \\cdots(x-2016)(x-2018) \\leq 0$?"
}
] | math | {
"style": "rule",
"ground_truth": "1514"
} | {
"index": 11820,
"solution": "We count the positive integers $x$ for which the product $(x-2)(x-4)(x-6) \\cdots(x-2016)(x-2018)$ equals 0 and is less than 0 separately. The product equals 0 exactly when one of the factors equals 0. This occurs exactly when $x$ equals one of $2,4,6, \\ldots, 2016,2018$. These are t... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $x$ is $100 \\leq x^2 \\leq 200$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $x$ is $\\log_{10}(x-40) + \\log_{10}(60-x) < 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11822,
"solution": "1. **Identify the domain of the function**: \n The expression $\\log_{10}(x-40) + \\log_{10}(60-x)$ is defined only when both $x-40$ and $60-x$ are positive. This implies:\n \\[\n x-40 > 0 \\quad \\text{and} \\quad 60-x > 0\n \\]\n Simplifying these inequalities, we get:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers \\( n \\) is the sum\n\\[\n(-n)^{3} + (-n+1)^{3} + \\cdots + (n-2)^{3} + (n-1)^{3} + n^{3} + (n+1)^{3}\n\\]\nless than \\( 3129 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers \\( n \\) less than or equal to 500 is \n$$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ \ntrue for all real \\( t \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 11825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integers \\(n\\) less than or equal to \\(50\\) is \\(n!\\) evenly divisible by \\(1 + 2 + \\cdots + n\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 11826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\n\\begin{cases}\n2x>3x-3\\\\\n3x-a>-6\n\\end{cases}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many primes \\( p < 50 \\) is \\( p^{4} + 5p^{3} + 4 \\) divisible by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many real numbers $a$ does the quadratic equation $x^2 + ax + 12a = 0$ have only integer roots for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 11829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11830,
"solution": "Let $x^2 + ax + 6a = (x - s)(x - r)$. Vieta's yields $s + r = - a, sr = 6a$. \\begin{eqnarray*}sr + 6s + 6r &=& 0\\\\ sr + 6s + 6r + 36 &=& 36\\\\ (s + 6)(r + 6) &=& 36 \\end{eqnarray*}\nWithout loss of generality let $r \\le s$.\n The possible values of $(r + 6,s + 6)$ are: $( - 36, ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many real values of $c$ do we have $\\left|\\frac12-ci\\right| = \\frac34$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many three-digit positive integers is the sum of the digits equal to $5?$"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 11833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many three-digit whole numbers does the sum of the digits equal $25$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 11834,
"solution": "1. **Identify the range of digits**: Since we are dealing with three-digit numbers, the digits range from $1$ to $9$ for the hundreds place and $0$ to $9$ for the tens and units places.\n\n2. **Set up the equation for the sum of the digits**: Let the digits of the number be $a$, $b$, ... |
DeepScaleR | [
{
"role": "user",
"content": "For how many triples $(x, y, z)$ of integers between -10 and 10 inclusive do there exist reals $a, b, c$ that satisfy $$\\begin{gathered} a b=x \\\\ a c=y \\\\ b c=z ? \\end{gathered}$$"
}
] | math | {
"style": "rule",
"ground_truth": "4061"
} | {
"index": 11835,
"solution": "If none are of $x, y, z$ are zero, then there are $4 \\cdot 10^{3}=4000$ ways, since $x y z$ must be positive. Indeed, $(a b c)^{2}=x y z$. So an even number of them are negative, and the ways to choose an even number of 3 variables to be negative is 4 ways. If one of $x, y, z$ is 0 ,... |
DeepScaleR | [
{
"role": "user",
"content": "For how many two-digit natural numbers \\( n \\) are exactly two of the following three statements true: (A) \\( n \\) is odd; (B) \\( n \\) is not divisible by 3; (C) \\( n \\) is divisible by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 11836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many unordered sets $\\{a, b, c, d\\}$ of positive integers, none of which exceed 168, do there exist integers $w, x, y, z$ such that $(-1)^{w} a+(-1)^{x} b+(-1)^{y} c+(-1)^{z} d=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem ... | math | {
"style": "rule",
"ground_truth": "761474"
} | {
"index": 11837,
"solution": "As an approximation, we assume $a, b, c, d$ are ordered to begin with (so we have to divide by 24 later) and add to 168 with a unique choice of signs; then, it suffices to count $e+f+g+h=168$ with each $e, f, g, h$ in $[-168,168]$ and then divide by 24 (we drop the condition that none... |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $a$ is it true that the line $y=x+a$ passes through the vertex of parabola $y=x^2+a^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $a$ is it true that:\n(1) $a$ is a positive integer such that $a \\le 50$.\n(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 11840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $c$ in the interval $[0, 1000]$ does the equation \\[5 \\lfloor x \\rfloor + 3 \\lceil x \\rceil = c\\]have a solution for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "251"
} | {
"index": 11841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $c$ in the interval $[0, 1000]$ does the equation \\[7 \\lfloor x \\rfloor + 2 \\lceil x \\rceil = c\\]have a solution for $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "223"
} | {
"index": 11842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, and $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 11843,
"solution": "It is evident that $k$ has only 2s and 3s in its prime factorization, or $k = 2^a3^b$.\n$6^6 = 2^6\\cdot3^6$\n$8^8 = 2^{24}$\n$12^{12} = 2^{24}\\cdot3^{12}$\n The LCM of any numbers an be found by writing out their factorizations and taking the greatest power for each factor. $[6^6,8^... |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^9$, $9^9$, and $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 11844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 11845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $n$ in the set $\\{101, 102, 103, ..., 200\\}$ is the tens digit of $n^2$ even?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $n$ with $3 \\leq n \\leq 12$ can a Fano table be created?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11847,
"solution": "First, we calculate the number of pairs that can be formed from the integers from 1 to $n$. One way to form a pair is to choose one number to be the first item of the pair ($n$ choices) and then a different number to be the second item of the pair ($n-1$ choices). There are $n(n-1)$ w... |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of $x$ is the expression $\\frac{x^2-9}{(x^2+2x-3)(x-3)}$ undefined?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of \\( k \\) is \\( 18^{18} \\) the least common multiple of the positive integers \\( 9^9 \\), \\( 12^{12} \\), and \\( k \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 11849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of \\(c\\) in the interval \\([0, 2000]\\) does the equation\n\\[8 \\lfloor x \\rfloor + 3 \\lceil x \\rceil = c\\]\nhave a solution for \\(x\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "363"
} | {
"index": 11850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For how many values of the digit $A$ is it true that $63$ is divisible by $A$ and $273{,}1A2$ is divisible by $4$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a$ and $T,$ $T \\neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$ $B = (2T,0),$ and $C = (2T + 1,28).$ Let $N$ be the sum of the coordinates of the vertex point. Determine the largest value of $N.$"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a$ and $b$ consider the complex number \\[\\frac{\\sqrt{ab+2016}}{ab+100}-\\left({\\frac{\\sqrt{|a+b|}}{ab+100}}\\right)i\\]\n Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 11853,
"solution": "Similar to Solution 1, but concise:\n First, we set the imaginary expression to $0$, so that $|a+b|=0$ or $a=-b$, of which there are $44\\cdot 2+1=89$ possibilities. But $(a,b)\\ne(\\pm 10,\\mp 10)$ because the denominator would be $0$. So this gives $89-2=87$ solutions.\n Then we try... |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a$, $b$, $c$, and $d$, $(x^2+ax+b)(x^2+cx+d)=x^4+x^3-2x^2+17x-5$. What is the value of $a+b+c+d$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \\in \\{1,2,3,4,5\\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 11855,
"solution": "Standard linear algebra over the field $\\mathbb{F}_{5}$ (the integers modulo 5). The dimension of the solution set is at least 0 and at most 2, and any intermediate value can also be attained. So the answer is $1+5+5^{2}=31$. This also can be easily reformulated in more concrete equa... |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a,b,$ and $c$ define $\\fbox{a,b,c}$ to mean $a^b-b^c+c^a$. Then $\\fbox{1,-1,2}$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11856,
"solution": "1. **Substitute the values of $a$, $b$, and $c$ into the expression $\\fbox{a,b,c} = a^b - b^c + c^a$:**\n - Here, $a = 1$, $b = -1$, and $c = 2$.\n - Thus, $\\fbox{1,-1,2} = 1^{-1} - (-1)^2 + 2^1$.\n\n2. **Calculate each term in the expression:**\n - $1^{-1} = \\frac{1}{1} = 1$... |
DeepScaleR | [
{
"role": "user",
"content": "For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$"
}
] | math | {
"style": "rule",
"ground_truth": "510"
} | {
"index": 11857,
"solution": "Either $f(2)=f(4)$ or not. If it is, note that Vieta's forces $a = -6$. Then, $b$ can be anything. However, $c$ can also be anything, as we can set the root of $g$ (not equal to $f(2) = f(4)$) to any integer, producing a possible integer value of $d$. Therefore there are $21^2 = 441$ ... |
DeepScaleR | [
{
"role": "user",
"content": "For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ ."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For letters \\( a \\sim z \\), encode them as $(\\mathbf{a}=1, \\mathbf{b}=2, \\ldots, \\mathbf{z}=26)$. Each English word (assuming all letters in the word are lowercase) can be assigned a product \\(\\mathbf{p}\\) calculated from these letter codes. For example, for the word ... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 11859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 11860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 11861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For n points \\[ P_1;P_2;...;P_n \\] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \\[ P_i;P_{i+1} (i=1;2;...n-1) \\] is the same color or 2 points with at least o... | math | {
"style": "rule",
"ground_truth": "\\frac{3^{n+1} + (-1)^{n+1}}{2}"
} | {
"index": 11862,
"solution": "\nTo find the number of acceptable colorings for \\( n \\) points \\( P_1, P_2, \\ldots, P_n \\) on a straight line, we need to adhere to the following rules:\n- Each point is colored with one of five colors: white, red, green, blue, or purple.\n- A coloring is acceptable if, for any ... |
DeepScaleR | [
{
"role": "user",
"content": "For natural numbers $m$ greater than or equal to 2, the decomposition of their cube powers can be represented as follows: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$. Then, \n(1) The smallest number in the decomposition of $8^3$ is; \n(2) Following the above ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 11863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For natural numbers \\\\(m\\\\) greater than or equal to \\\\(2\\\\) and their powers of \\\\(n\\\\), the following decomposition formula is given: \n\\\\(2^{2}=1+3\\\\) \\\\(3^{2}=1+3+5\\\\) \\\\(4^{2}=1+3+5+7\\\\) \\\\(…\\\\) \n\\\\(2^{3}=3+5\\\\) \\\\(3^{3}=7+9+11... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For natural numbers _m_ greater than or equal to 2, the _n_-th power of _m_ can be decomposed as follows: \n2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7… \n2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11… \n2<sup>4</sup> = 7 + 9… \nAccor... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is:"
}
] | math | {
"style": "rule",
"ground_truth": "$R+R'D$"
} | {
"index": 11866,
"solution": "1. **Given Equations**: We start with the given relationships:\n - $P = QD + R$ (where $P$ is divided by $D$)\n - $Q = D'Q' + R'$ (where $Q$ is divided by $D'$)\n\n2. **Substitute $Q$ in the first equation**: Replace $Q$ in the first equation with its expression from the second eq... |
DeepScaleR | [
{
"role": "user",
"content": "For non-negative integers $n$, the function $f(n)$ is defined by $f(0) = 0$, $f(1) = 1$, and $f(n) = f\\left(\\left\\lfloor \\frac{1}{2} n \\right\\rfloor \\right) + n - 2\\left\\lfloor \\frac{1}{2} n \\right\\rfloor$. Find the maximum value of $f(n)$ for $0 \\leq n \\leq 1997$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For non-negative integers \\( x \\), the function \\( f(x) \\) is defined as follows:\n$$\nf(0) = 0, \\quad f(x) = f\\left(\\left\\lfloor \\frac{x}{10} \\right\\rfloor\\right) + \\left\\lfloor \\log_{10} \\left( \\frac{10}{x - 10\\left\\lfloor \\frac{x-1}{10} \\right\\rfloor} \... | math | {
"style": "rule",
"ground_truth": "1111"
} | {
"index": 11868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For non-negative real numbers $x_1, x_2, \\ldots, x_n$ which satisfy $x_1 + x_2 + \\cdots + x_n = 1$, find the largest possible value of $\\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 11869,
"solution": "\nLet \\( x_1, x_2, \\ldots, x_n \\) be non-negative real numbers such that \\( x_1 + x_2 + \\cdots + x_n = 1 \\). We aim to find the largest possible value of \\( \\sum_{j=1}^n (x_j^4 - x_j^5) \\).\n\nTo solve this, we use the method of smoothing. We start by considering small cases ... |
DeepScaleR | [
{
"role": "user",
"content": "For non-zero real numbers \\( x, y, z, w \\), if\n$$\n\\frac{6 x y + 5 y z + 6 z w}{x^{2} + y^{2} + z^{2} + w^{2}} \\leq f,\n$$\n\nfind the minimum value of \\( f \\)."
}
] | math | {
"style": "rule",
"ground_truth": "9/2"
} | {
"index": 11870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$.\... | math | {
"style": "rule",
"ground_truth": "564"
} | {
"index": 11871,
"solution": "Let $c=6-(a+b)$, and note that $\\binom{6}{a + b}=\\binom{6}{c}$. The problem thus asks for the sum $\\binom{6}{a} \\binom{6}{b} \\binom{6}{c}$ over all $a,b,c$ such that $a+b+c=6$. Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression counts the total nu... |
DeepScaleR | [
{
"role": "user",
"content": "For odd primes $p$, let $f(p)$ denote the smallest positive integer $a$ for which there does not exist an integer $n$ satisfying $p \\mid n^{2}-a$. Estimate $N$, the sum of $f(p)^{2}$ over the first $10^{5}$ odd primes $p$. An estimate of $E>0$ will receive $\\left\\lfloor 22 \... | math | {
"style": "rule",
"ground_truth": "2266067"
} | {
"index": 11872,
"solution": "Note that the smallest quadratic nonresidue $a$ is always a prime, because if $a=b c$ with $b, c>1$ then one of $b$ and $c$ is also a quadratic nonresidue. We apply the following heuristic: if $p_{1}$, $p_{2}, \\ldots$ are the primes in increasing order, then given a \"uniform random ... |
DeepScaleR | [
{
"role": "user",
"content": "For point M on the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4}=1$, find the minimum distance from M to the line $x+2y-10=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 11873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For points P and Q on the curve $y = 1 - x^2$, which are situated on opposite sides of the y-axis, find the minimum area of the triangle formed by the tangents at P and Q and the x-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8 \\sqrt{3}}{9}"
} | {
"index": 11874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For polynomial $P(x)=1-\\dfrac{1}{3}x+\\dfrac{1}{6}x^{2}$, define\n\\[Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\\sum_{i=0}^{50} a_ix^{i}.\\]Find $\\sum_{i=0}^{50} |a_i|.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{243}{32}"
} | {
"index": 11875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For polynomial $P(x)=1-\\dfrac{1}{3}x+\\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\\sum_{i=0}^{50} a_ix^{i}$. Then $\\sum_{i=0}^{50} |a_i|=\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "275"
} | {
"index": 11876,
"solution": "Multiply $P(x)P(x^3)$ and notice that the odd degree terms have a negative coefficient. Observing that this is probably true for all polynomials like this (including $P(x)P(x^3)P(x^5)P(x^7)P(x^9)$), we plug in $-1$ to get $\\frac{243}{32} \\implies \\boxed{275}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\\cdot 6,3\\cdot 4, 2\\cdot 2\\cdot 3... | math | {
"style": "rule",
"ground_truth": "\\frac{n}{p}"
} | {
"index": 11877,
"solution": "\nFor a positive integer \\( k > 1 \\), let \\( f(k) \\) represent the number of ways to factor \\( k \\) into a product of positive integers greater than 1. For example, \\( f(12) = 4 \\) because 12 can be factored in these 4 ways: \\( 12 \\), \\( 2 \\cdot 6 \\), \\( 3 \\cdot 4 \\), ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$."
}
] | math | {
"style": "rule",
"ground_truth": "695"
} | {
"index": 11878,
"solution": "First of all, notice that the smallest $n$ with $s(n) = 20$ is $299$. Also, if $s(n + 864) = 20$, $s(n - 136) = 19$ (because subtracting $1000$ from the number removes the $1$ in the thousands place). After checking $s(n - 136)$ for various $n$ with $s(n) = 20$, we see that we need to... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integer $n_{}^{}$, define $S_n^{}$ to be the minimum value of the sum $\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},$ where $a_1,a_2,\\ldots,a_n^{}$ are positive real numbers whose sum is 17. There is a unique positive integer $n^{}_{}$ for which $S_n^{}$ is also an intege... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11879,
"solution": "Consider $n$ right triangles joined at their vertices, with bases $a_1,a_2,\\ldots,a_n$ and heights $1,3,\\ldots, 2n - 1$. The sum of their hypotenuses is the value of $S_n$. The minimum value of $S_n$, then, is the length of the straight line connecting the bottom vertex of the first... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integer \\( n \\), let \\( f(n) \\) denote the unit digit of \\( 1+2+3+\\cdots+n \\). Find the value of \\( f(1)+f(2)+\\cdots+f(2011) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "7046"
} | {
"index": 11880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $i = 2, 3, \\ldots, 2020$ , let \\[\na_i = \\frac{\\sqrt{3i^2+2i-1}}{i^3-i}.\n\\]Let $x_2$ , $\\ldots$ , $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \\cdots + x_{2020}^4 = 1-\\frac{1}{1010\\cdot 2020\\cdot 2021}$ . Let $S$ be the maximu... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\\frac{s(n)}{n}$ ?\n\n*Proposed by Michael Tang*"
}
] | math | {
"style": "rule",
"ground_truth": "1099"
} | {
"index": 11882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $N$ and $k$ define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Determine the quantity of positive integers smaller than $1500$ that are neither $9$-nice nor $10$-nice."
}
] | math | {
"style": "rule",
"ground_truth": "1199"
} | {
"index": 11883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice."
}
] | math | {
"style": "rule",
"ground_truth": "749"
} | {
"index": 11884,
"solution": "All integers $a$ will have factorization $2^a3^b5^c7^d...$. Therefore, the number of factors in $a^7$ is $(7a+1)(7b+1)...$, and for $a^8$ is $(8a+1)(8b+1)...$. The most salient step afterwards is to realize that all numbers $N$ not $1 \\pmod{7}$ and also not $1 \\pmod{8}$ satisfy the ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice."
}
] | math | {
"style": "rule",
"ground_truth": "266"
} | {
"index": 11885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $a$ and $N$, let $r(a, N) \\in\\{0,1, \\ldots, N-1\\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \\leq 1000000$ for which $r(n, 1000)>r(n, 1001)$."
}
] | math | {
"style": "rule",
"ground_truth": "499500"
} | {
"index": 11886,
"solution": "Note that $0 \\leq r(n, 1000) \\leq 999$ and $0 \\leq r(n, 1001) \\leq 1000$. Consider the $\\binom{1000}{2}=499500$ ways to choose pairs $(i, j)$ such that $i>j$. By the Chinese Remainder Theorem, there is exactly one $n$ such that $1 \\leq n \\leq 1000 \\cdot 1001$ such that $n \\eq... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $a$ and $b$ such that $a$ is coprime to $b$, define $\\operatorname{ord}_{b}(a)$ as the least positive integer $k$ such that $b \\mid a^{k}-1$, and define $\\varphi(a)$ to be the number of positive integers less than or equal to $a$ which are coprime to $a... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 11887,
"solution": "The maximum order of an element modulo $n$ is the Carmichael function, denoted $\\lambda(n)$. The following properties of the Carmichael function are established: - For primes $p>2$ and positive integers $k, \\lambda\\left(p^{k}\\right)=(p-1) p^{k-1}$. - For a positive integer $k$, $$... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $a$ and $b$, let $M(a, b)=\\frac{\\operatorname{lcm}(a, b)}{\\operatorname{gcd}(a, b)}$, and for each positive integer $n \\geq 2$, define $$x_{n}=M(1, M(2, M(3, \\ldots, M(n-2, M(n-1, n)) \\ldots)))$$ Compute the number of positive integers $n$ such that ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11888,
"solution": "The desired condition is that $x_{n}=5 x_{n+1}$ or $x_{n+1}=5 x_{n}$. Note that for any prime $p$, we have $\\nu_{p}(M(a, b))=\\left|\\nu_{p}(a)-\\nu_{p}(b)\\right|$. Furthermore, $\\nu_{p}(M(a, b)) \\equiv \\nu_{p}(a)+\\nu_{p}(b) \\bmod 2$. So, we have that $$\\nu_{p}\\left(x_{n}\\ri... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole numb... | math | {
"style": "rule",
"ground_truth": "188"
} | {
"index": 11889,
"solution": "Notice that we must have $a = 1$, otherwise $1$ cent stamp cannot be represented. At least $b-1$ numbers of $1$ cent stamps are needed to represent the values less than $b$. Using at most $c-1$ stamps of value $1$ and $b$, it can have all the values from $1$ to $c-1$ cents. Plus $\\lf... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $a, b, a \\uparrow \\uparrow b$ is defined as follows: $a \\uparrow \\uparrow 1=a$, and $a \\uparrow \\uparrow b=a^{a \\uparrow \\uparrow(b-1)}$ if $b>1$. Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \\u... | math | {
"style": "rule",
"ground_truth": "283"
} | {
"index": 11890,
"solution": "We see that the smallest such $n$ must be a prime power, because if two numbers are distinct mod $n$, they must be distinct mod at least one of the prime powers that divide $n$. For $k \\geq 2$, if $a \\uparrow \\uparrow k$ and $a \\uparrow \\uparrow(k+1)$ are distinct $\\bmod p^{r}$,... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\\{m, m+4, m+10, n+1, n+2, 2n\\}$ are equal to $n$. What is $m+n$?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 11891,
"solution": "1. **Identify the median**: The median of the set $\\{m, m+4, m+10, n+1, n+2, 2n\\}$ is the average of the third and fourth smallest numbers. Given that the median is $n$, we have:\n \\[\n \\frac{(m+10) + (n+1)}{2} = n\n \\]\n Simplifying this equation:\n \\[\n m + 10 + n ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $m, n$, let \\operatorname{gcd}(m, n) denote the largest positive integer that is a factor of both $m$ and $n$. Compute $$\\sum_{n=1}^{91} \\operatorname{gcd}(n, 91)$$"
}
] | math | {
"style": "rule",
"ground_truth": "325"
} | {
"index": 11892,
"solution": "Since $91=7 \\times 13$, we see that the possible values of \\operatorname{gcd}(n, 91) are 1, 7, 13, 91. For $1 \\leq n \\leq 91$, there is only one value of $n$ such that \\operatorname{gcd}(n, 91)=91. Then, we see that there are 12 values of $n$ for which \\operatorname{gcd}(n, 91)=... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$ and $k$, let $\\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\\sum_{n=1}^{\\infty} \\sum_{k=1}^{\\infty} \\frac{\\mho(n, k)}{3^{n+k-7}}$$"
}
] | math | {
"style": "rule",
"ground_truth": "167"
} | {
"index": 11893,
"solution": "A prime $p$ is counted in $\\mho(n, k)$ if $p \\mid n$ and $k \\leq p$. Thus, for a given prime $p$, the total contribution from $p$ in the sum is $$3^{7} \\sum_{m=1}^{\\infty} \\sum_{k=1}^{p} \\frac{1}{3^{p m+k}}=3^{7} \\sum_{i \\geq p+1} \\frac{1}{3^{i}}=\\frac{3^{7-p}}{2}$$ Therefo... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 11894,
"solution": "$Lemma:$ Highest remainder when $n$ is divided by $1\\leq k\\leq n/2$ is obtained for $k_0 = (n + (3 - n \\pmod{3}))/3$ and the remainder thus obtained is $(n - k_0*2) = [(n - 6)/3 + (2/3)*n \\pmod{3}]$.\n$Note:$ This is the second highest remainder when $n$ is divided by $1\\leq k\\l... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, define $S_n$ to be the minimum value of the sum\n\\[\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},\\]where $a_1,a_2,\\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less th... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11896,
"solution": "To solve for the number of positive integers less than or equal to $97$ for which $D(n) = 2$, we analyze the binary representations of numbers and count those with exactly two transitions between adjacent digits (from 0 to 1 or from 1 to 0).\n\n#### Case Analysis:\nFor $D(n) = 2$, the... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11897,
"solution": "Let $x$ be an integer, and let $p_{x}$ be the smallest prime factor of $x$. Then, if $L(a)=x$, we note that we must have $a=p x$ for some prime $p \\leq p_{x}$. (Otherwise, if $p>p_{x}$, then $\\frac{p x}{p_{x}}>x$. If $p$ is composite, then $k x>x$ for some factor $k$ of $x$.)\n\nSo ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\\cdots+c_{210}$?"
}
] | math | {
"style": "rule",
"ground_truth": "329"
} | {
"index": 11898,
"solution": "In order for $c_{n} \\neq 0$, we must have $\\operatorname{gcd}(n, 210)=1$, so we need only consider such $n$. The number $n^{c_{n}}-1$ is divisible by 210 iff it is divisible by each of 2, 3, 5, and 7, and we can consider the order of $n$ modulo each modulus separately; $c_{n}$ will ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let $f(n)$ be the product of the digits of $n$. Find the largest positive integer $m$ such that $$\\sum_{n=1}^{\\infty} \\frac{f(n)}{m\\left\\lfloor\\log _{10} n\\right\\rfloor}$$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "2070"
} | {
"index": 11899,
"solution": "We know that if $S_{\\ell}$ is the set of all positive integers with $\\ell$ digits, then $$\\begin{aligned} & \\sum_{n \\in S_{\\ell}} \\frac{f(n)}{k^{\\left\\lfloor\\log _{10}(n)\\right\\rfloor}}=\\sum_{n \\in S_{\\ell}} \\frac{f(n)}{k^{\\ell-1}}=\\frac{(0+1+2+\\ldots+9)^{\\ell}}{k^... |
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