data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $P$ has four roots, $\\frac{1}{4}, \\frac{1}{2}, 2,4$. The product of the roots is 1, and $P(1)=1$. Find $P(0)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 2700,
"solution": "A polynomial $Q$ with $n$ roots, $x_{1}, \\ldots, x_{n}$, and $Q\\left(x_{0}\\right)=1$ is given by $Q(x)=\\frac{\\left(x-x_{1}\\right)\\left(x-x_{2}\\right) \\cdots\\left(x-x_{n}\\right)}{\\left(x_{0}-x_{1}\\right)\\left(x_{0}-x_{2}\\right) \\cdots\\left(x_{0}-x_{4}\\right)}$, so $P(0... |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\\frac{1}{n^{2}}$ for $n=1,2, \\ldots, 2016$. Find \\lfloor 2017 P(2017)\\rfloor."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 2701,
"solution": "Let $Q(x)=x^{2} P(x)-1$. Then $Q(n)=n^{2} P(n)-1=0$ for $n=1,2, \\ldots, 2016$, and $Q$ has degree 2017 . Thus we may write $$Q(x)=x^{2} P(x)-1=(x-1)(x-2) \\ldots(x-2016) L(x)$$ where $L(x)$ is some linear polynomial. Then $Q(0)=-1=(-1)(-2) \\ldots(-2016) L(0)$, so $L(0)=-\\frac{1}{201... |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root. A polynomial is called $k$-tiny if it has degree at most 7 and integer coefficients between $-k$ and $k$, inclusive. A polynomial is called nearly tricky if it is the sum of a tricky polynomial a... | math | {
"style": "rule",
"ground_truth": "64912347"
} | {
"index": 2702,
"solution": "A tricky 7-tiny polynomial takes the form $$\\left(c_{6} x^{6}+\\ldots+c_{1} x+c_{0}\\right)(x-4)$$ For each fixed value of $k, c_{k}-4 c_{k+1}$ should lie in $[-7,7]$, so if we fix $c_{k}$, there are around $15 / 4$ ways of choosing $c_{k+1}$. Therefore if we pick $c_{0}, \\ldots, c_{... |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $P(x)$ with integer coefficients possesses the properties\n\n$$\nP(1)=2019, \\quad P(2019)=1, \\quad P(k)=k,\n$$\n\nwhere $k$ is an integer. Find this integer $k$."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 2703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $f \\in \\mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \\in \\mathbb{Z}[x]$ with $\\operatorname{deg} g_{p}, \\operatorname{deg} h_{p}<\\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are di... | math | {
"style": "rule",
"ground_truth": "\\[ 693 \\]"
} | {
"index": 2704,
"solution": "We claim that $x^{4}+a x^{2}+b$ is splitty if and only if either $b$ or $a^{2}-4 b$ is a perfect square. (The latter means that the polynomial splits into $(x^{2}-r)(x^{2}-s)$ ). Assuming the characterization, one can easily extract the answer. For $a=16$ and $b=n$, one of $n$ and $64-... |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and it satisfies $g(3i)=g(3+i)=0$."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 2705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$ Find $r(6).$"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 2707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial $p(x)$ leaves a remainder of $2$ when divided by $x - 3,$ a remainder of 1 when divided by $x - 4,$ and a remainder of 5 when divided by $x + 4.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 3)(x - 4)(x + 4).$ Find $r(5).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{13}"
} | {
"index": 2708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial \\( P \\) is of the form \\( \\pm x^{6} \\pm x^{5} \\pm x^{4} \\pm x^{3} \\pm x^{2} \\pm x \\pm 1 \\). Given that \\( P(2)=27 \\), what is \\( P(3) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "439"
} | {
"index": 2709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial \\( P(x) \\) of degree 10 with a leading coefficient of 1 is given. The graph of \\( y = P(x) \\) lies entirely above the x-axis. The polynomial \\( -P(x) \\) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two ... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"index": 2710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial of degree $13$ is divided by $d(x)$ to give a quotient of degree $7$ and a remainder of $3x^3+4x^2-x+12$. What is $\\deg d$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?\n\n(A) $\\frac{1 + i \\sqrt{11}}{2}$\n\n(B) $\\frac{1 + i}{2}$\n\n(C) $\\frac{1}{2} + i$\... | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"index": 2712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[12x^3 - 4x^2 + a_1x + 18 = 0.\\]\nDetermine the number of different possible rational roots of this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[8x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\\]Find the number of different possible rational roots for this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\\]Find the number of different possible rational roots of this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\\]Enter all the possible integer roots of this polynomial, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-11, -1, 1, 11"
} | {
"index": 2716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 18.\\]You are told that the integer $r$ is a double root of this polynomial. (In other words, the polynomial is divisible by $(x - r)^2.$) Enter all the possible values of $r,$ separat... | math | {
"style": "rule",
"ground_truth": "-3,-1,1,3"
} | {
"index": 2717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A population consists of $20$ individuals numbered $01$, $02$, $\\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.\n\nWhat is the value of $a+b+2c$?\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=bla... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.\n\nWhat is the value of $8a-4b+2c-d$?\n\n[asy]\nimport graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;\n\npen cqcq... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.\n\nCompute $G(G(G(G(G(1)))))$.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nrea... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.\n\nWhat is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tickspace... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 2722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A portion of the graph of a quadratic function $f(x)$ is shown below.\n\nLet $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, t... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 2723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven?"
}
] | math | {
"style": "rule",
"ground_truth": "96,\\!569"
} | {
"index": 2724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . What is the smallest special integer in decimal representation?\n\n*Proposed by Michael Ren*"
... | math | {
"style": "rule",
"ground_truth": "793"
} | {
"index": 2725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \\ldots$ The smallest part is equal to $\\frac{1}{400} N$ . Find the value of $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1109"
} | {
"index": 2727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is *funny* if for all positive divisors $d$ of $n$ , $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 2728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is called*bad*if it cannot be expressed as the product of two distinct positive integers greater than $1$ . Find the number of bad positive integers less than $100. $ *Proposed by Michael Ren*"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n\\geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \\[1+z+z^2+z^{n-1}+z^n=0.\\] Find how many interesting numbers are smaller than $2022.$"
}
] | math | {
"style": "rule",
"ground_truth": "404"
} | {
"index": 2730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $N$ has base-eleven representation $\\underline{a}\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}$ and base-eight representation $\\underline1\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}\\kern 0.1em\\underline{a},$ where $a,b,$ and $c$ represent ... | math | {
"style": "rule",
"ground_truth": "621"
} | {
"index": 2731,
"solution": "From the given information, $121a+11b+c=512+64b+8c+a \\implies 120a=512+53b+7c$. Since $a$, $b$, and $c$ have to be positive, $a \\geq 5$. Since we need to minimize the value of $n$, we want to minimize $a$, so we have $a = 5$. Then we know $88=53b+7c$, and we can see the only solution... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:\n(a) It is a palindrome\n(b) It factors as a product of a 2-digit pr... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2732,
"solution": "1. **Identify the properties of a four-digit palindrome**: A four-digit palindrome has the form $abba$, where $a$ and $b$ are digits and $a \\neq 0$. For example, $1991$ is a palindrome because reversing the digits gives the same number.\n\n2. **Check divisibility by 11**: A number $ab... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that $\\log_2 N$ is an integer is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{300}"
} | {
"index": 2733,
"solution": "1. **Identify the Condition for $\\log_2 N$ to be an Integer:**\n To have $\\log_2 N$ as an integer, $N$ must be a power of $2$, i.e., $N = 2^k$ for some integer $k$.\n\n2. **Determine the Range of $k$:**\n Since $N$ is a three-digit number, $100 \\leq N \\leq 999$. We need to find... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $a$ is input into a machine. If $a$ is odd, the output is $a+3$. If $a$ is even, the output is $a+5$. This process can be repeated using each successive output as the next input. If the input is $a=15$ and the machine is used 51 times, what is the final outpu... | math | {
"style": "rule",
"ground_truth": "218"
} | {
"index": 2734,
"solution": "If $a$ is odd, the output is $a+3$, which is even because it is the sum of two odd integers. If $a$ is even, the output is $a+5$, which is odd, because it is the sum of an even integer and an odd integer. Starting with $a=15$ and using the machine 2 times, we obtain $15 \\rightarrow 15... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2735,
"solution": "1. **Express $n$ in terms of its prime factors**: Let $n = 7^k \\cdot m$, where $m$ is an integer not divisible by 7. This ensures that all factors of 7 in $n$ are accounted for by $7^k$.\n\n2. **Use the multiplicative property of the divisor function**: The number of divisors function... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$?"
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 2736,
"solution": "\nWe are tasked with determining how many positive integers \\( n \\leq 2022 \\) are $inversosimil$, which means \\( n \\) can be expressed with \\( n \\) integers such that both the sum and the product of these integers equal \\( n \\).\n\nTo solve this problem, let's first consider t... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is a multiple of 7. The square root of $n$ is between 17 and 18. How many possible values of $n$ are there?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2737,
"solution": "Since the square root of $n$ is between 17 and 18, then $n$ is between $17^2 = 289$ and $18^2 = 324$. Since $n$ is a multiple of 7, we need to count the number of multiples of 7 between 289 and 324. Since $41 \\times 7 = 287$ and $42 \\times 7 = 294$, then 294 is the smallest multiple ... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is infallible if it is possible to select $n$ vertices of a regular 100-gon so that they form a convex, non-self-intersecting $n$-gon having all equal angles. Find the sum of all infallible integers $n$ between 3 and 100, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "262"
} | {
"index": 2738,
"solution": "Suppose $A_{1} A_{2} \\ldots A_{n}$ is an equiangular $n$-gon formed from the vertices of a regular 100-gon. Note that the angle $\\angle A_{1} A_{2} A_{3}$ is determined only by the number of vertices of the 100-gon between $A_{1}$ and $A_{3}$. Thus in order for $A_{1} A_{2} \\ldots A... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies\n\\[{\\ \\{\\frac{n}{10^k}} \\} > \\frac{n}{10^{10}} \\] \nfor all $k=1,2,\\ldots 9$.\nFind the number of interesting numbers."
}
] | math | {
"style": "rule",
"ground_truth": "999989991"
} | {
"index": 2739,
"solution": "\nA positive integer \\( n \\) is known as an interesting number if \\( n \\) satisfies\n\\[\n\\left\\{ \\frac{n}{10^k} \\right\\} > \\frac{n}{10^{10}}\n\\]\nfor all \\( k = 1, 2, \\ldots, 9 \\), where \\( \\{ x \\} \\) denotes the fractional part of \\( x \\).\n\nTo determine the numb... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \\geq 2 a$. Compute the sum of all loose positive integers less than 100."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 2740,
"solution": "Note that the condition in the problem implies that for any divisor $d$ of $n$, if $d$ is odd then all other divisors of $n$ cannot lie in the interval $\\left[\\left\\lceil\\frac{d}{2}\\right\\rceil, 2 d-1\\right]$. If $d$ is even, then all other divisors cannot lie in the interval $\... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is magical if $\\lfloor\\sqrt{\\lceil\\sqrt{n}\\rceil}\\rfloor=\\lceil\\sqrt{\\lfloor\\sqrt{n}\\rfloor}\\rceil$ where $\\lfloor\\cdot\\rfloor$ and $\\lceil\\cdot\\rceil$ represent the floor and ceiling function respectively. Find the number of magical int... | math | {
"style": "rule",
"ground_truth": "1330"
} | {
"index": 2741,
"solution": "First of all, we have $\\lfloor\\sqrt{n}\\rfloor=\\lceil\\sqrt{n}\\rceil$ when $n$ is a perfect square and $\\lfloor\\sqrt{n}\\rfloor=\\lceil\\sqrt{n}\\rceil-1$ otherwise. Therefore, in the first case, the original equation holds if and only if $\\sqrt{n}$ is a perfect square itself, i... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \\ldots, 2004$ are picante?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2742,
"solution": "The number of zeroes in base 7 is the total number of factors of 7 in $1 \\cdot 2 \\cdots n$, which is $$ \\lfloor n / 7\\rfloor+\\left\\lfloor n / 7^{2}\\right\\rfloor+\\left\\lfloor n / 7^{3}\\right\\rfloor+\\cdots $$ The number of zeroes in base 8 is $\\lfloor a\\rfloor$, where $$ a... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is"
}
] | math | {
"style": "rule",
"ground_truth": ".08"
} | {
"index": 2743,
"solution": "1. **Define the probabilities:** Given that if $n \\leq 50$, the probability of choosing $n$ is $p$, and if $n > 50$, the probability of choosing $n$ is $3p$. \n\n2. **Total probability must sum to 1:** Since there are 50 numbers in each category ($n \\leq 50$ and $n > 50$), the total ... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $100$ is chosen such that if $n\\le 60$, then the probability of choosing $n$ is $q$, and if $n > 60$, then the probability of choosing $n$ is $2q$. Find the probability that a perfect square is chosen.\n- **A)** $\\frac{1}{35}$\n- **B)** $\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{35}"
} | {
"index": 2744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $120$ is chosen such that if $n\\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $2p$. The probability that a perfect square is chosen is?\nA) $\\frac{1}{180}$\nB) $\\frac{7}{180}... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{180}"
} | {
"index": 2745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( A \\) divided by \\( 3! \\) gives a result where the number of factors is \\(\\frac{1}{3}\\) of the original number of factors. What is the smallest such \\( A \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( N \\) and \\( N^2 \\) end with the same sequence of digits \\(\\overline{abcd}\\), where \\( a \\) is a non-zero digit. Find \\(\\overline{abc}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "937"
} | {
"index": 2747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( m \\) has the property that when multiplied by 12, the result is a four-digit number \\( n \\) of the form \\( 20A2 \\) for some digit \\( A \\). What is the four-digit number \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2052"
} | {
"index": 2748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( n \\) cannot be divided by \\( 2 \\) or \\( 3 \\), and there do not exist non-negative integers \\( a \\) and \\( b \\) such that \\( |2^a - 3^b| = n \\). Find the smallest value of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 2749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be 'good' if \\( n^2 - 1 \\) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 2750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be good if \\( 3n \\) is a re-ordering of the digits of \\( n \\) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11."
}
] | math | {
"style": "rule",
"ground_truth": "2475"
} | {
"index": 2751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be increasing if, by reversing the digits of \\( n \\), we get an integer larger than \\( n \\). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positiv... | math | {
"style": "rule",
"ground_truth": "4005"
} | {
"index": 2752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\( n \\) with \\( n \\) digits is called an \"auspicious number\" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \\( x \\). For example, 6 is an \"auspicious number,\" but 16 is not, because \\( 116 \\... | math | {
"style": "rule",
"ground_truth": "1114"
} | {
"index": 2753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer \\overline{A B C}, where $A, B, C$ are digits, satisfies $\\overline{A B C}=B^{C}-A$. Find $\\overline{A B C}$."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 2754,
"solution": "The equation is equivalent to $100 A+10 B+C=B^{C}-A$. Suppose $A=0$, so that we get $10 B+C=B^{C}$. Reducing $\\bmod B$, we find that $C$ must be divisible by $B$. $C \\neq 0$, since otherwise $10 B=1$, contradiction, so $C \\geq B$. Thus $10 B+C \\geq B^{B}$ for digits $B, C$. For $B ... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \\(a\\) and \\(b\\) such that \\(|2^a - 3^b| = n\\). Find the smallest value of \\(n\\)."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 2755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\\frac{m}{n}$, and find the sum of the numerator and denominator."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 2758,
"solution": "1. **Prime Factorization of $12!$:**\n The prime factorization of $12!$ (12 factorial) is necessary to determine the number of divisors and the condition for these divisors to be perfect squares. We start by finding the prime factorization of $12!$:\n \\[\n 12! = 1 \\times 2 \\ti... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"index": 2759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"index": 2760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold ... | math | {
"style": "rule",
"ground_truth": "1614"
} | {
"index": 2761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is *happy* if:\n\n1. All its digits are different and not $0$ ,\n2. One of its digits is equal to the sum of the other digits.\n\nFor example, 253 is a *happy* number. How many *happy* numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called *oneic* if it consists of only $1$ 's. For example, the smallest three oneic numbers are $1$ , $11$ , and $111$ . Find the number of $1$ 's in the smallest oneic number that is divisible by $63$ ."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 2763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called a perfect power if it can be written in the form \\(a^b\\), where \\(a\\) and \\(b\\) are positive integers with \\(b \\geq 2\\). The increasing sequence \\(2, 3, 5, 6, 7, 10, \\ldots\\) consists of all positive integers which are not perfect powers... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 2764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"index": 2765,
"solution": "Note that an ascending number is exactly determined by its digits: for any set of digits (not including 0, since the only position for 0 is at the leftmost end of the number, i.e. a leading 0), there is exactly one ascending number with those digits.\n So, there are nine digits that ma... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called jubilant if the number of 1 's in its binary representation is even. For example, $6=110_{2}$ is a jubilant number. What is the 2009 th smallest jubilant number?"
}
] | math | {
"style": "rule",
"ground_truth": "4018"
} | {
"index": 2766,
"solution": "Notice that for each pair of consecutive positive integers $2 k$ and $2 k+1$, their binary representation differs by exactly one 1 (in the units digit), so exactly one of 2 and 3 is jubilant, exactly one of 4 and 5 is jubilant, etc. It follows that there are exactly 2009 jubilant numbe... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. A positive integer is called prime-minister if it has a primest number of distinct primest factors.... | math | {
"style": "rule",
"ground_truth": "378000"
} | {
"index": 2767,
"solution": "An estimate of $E>0$ earns \\left\\lfloor 20 \\min \\left(\\frac{N}{E}, \\frac{E}{N}\\right)\\right\\rfloor$ points. One heuristic for estimating the answer is that numbers of the form $p^{q} r^{s}$ for primes $p, q, r, s$ with $p \\neq r, q \\neq s$ are primest. Thus, primest numbers ... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. Find the smallest primest number."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 2768,
"solution": "We claim the answer is 72 , as it has 6 primer factors: $6,12,24,18,36,72$, and 6 is a primer. We now prove that there is no smaller primer. Suppose there were a smaller primer $r<72$. We do casework on the number of distinct prime factors of $r$. - $r$ has \\geq 4$ distinct prime fact... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is said to be \"nefelibata\" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possib... | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"index": 2770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is said to be a \"palindrome\" if it reads the same from left to right as from right to left. For example, 2002 is a palindrome. Find the sum of all 4-digit palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "495000"
} | {
"index": 2771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer \\( b \\), the number of bi-digital positive int... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"index": 2772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 2773,
"solution": "Two opposite vertices are relatively prime, but they both share a factor with their common neighbor. So that common neighbor must have two prime factors. So each of the 4 numbers has two prime factors, which are not shared with the opposite vertex. Moreover, it suffices to choose the v... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer n is called *primary divisor* if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$ , $2$ , $4$ , and $8$ each differ by $1$ f... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer will be called \"sparkly\" if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers $2,3, \\ldots, 2003$ are sparkly?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2777,
"solution": "Suppose $n$ is sparkly; then its smallest divisor other than 1 is some prime $p$. Hence, $n$ has $p$ divisors. However, if the full prime factorization of $n$ is $p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{r}^{e_{r}}$, the number of divisors is $\\left(e_{1}+1\\right)\\left(e_{2}+1\\right)... |
DeepScaleR | [
{
"role": "user",
"content": "A positive integer with 3 digits $\\overline{ABC}$ is $Lusophon$ if $\\overline{ABC}+\\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers."
}
] | math | {
"style": "rule",
"ground_truth": "110,143,242,341,440,164,263,362,461,560,198,297,396,495,594,693,792,891,990"
} | {
"index": 2778,
"solution": "To find all three-digit Lusophon numbers \\( \\overline{ABC} \\), we first need to establish the conditions under which a number meets the Lusophon criteria. A number is defined as Lusophon if the sum of the number and its digit reversal is a perfect square. Therefore, we need to consi... |
DeepScaleR | [
{
"role": "user",
"content": "A positive number $x$ has the property that $x\\%$ of $x$ is $4$. What is $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2779,
"solution": "\nWe are given that $x\\%$ of $x$ is $4$. We need to interpret and solve this equation mathematically.\n\n1. **Understanding the percentage operation**: The expression \"$x\\%$ of $x$\" translates to $x\\% \\times x$. Since $x\\%$ means $x$ percent, which is $\\frac{x}{100}$, we can re... |
DeepScaleR | [
{
"role": "user",
"content": "A positive number $x$ satisfies the inequality $\\sqrt{x} < 2x$ if and only if"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2780,
"solution": "To solve the inequality $\\sqrt{x} < 2x$, we start by squaring both sides, assuming $x \\geq 0$ since we are dealing with a square root and $x$ is given as positive:\n\n1. Squaring both sides:\n \\[\n (\\sqrt{x})^2 < (2x)^2\n \\]\n \\[\n x < 4x^2\n \\]\n\n2. Rearrange the i... |
DeepScaleR | [
{
"role": "user",
"content": "A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 500?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 2782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive number is increased by $60\\%$. By what percentage should the result be decreased to return to the original value?"
}
] | math | {
"style": "rule",
"ground_truth": "37.5\\%"
} | {
"index": 2783,
"solution": "Solution 1: Suppose that the original number is 100. When 100 is increased by $60\\%$, the result is 160. To return to the original value of 100, 160 must be decreased by 60. This percentage is $\\frac{60}{160} \\times 100\\%=\\frac{3}{8} \\times 100\\%=37.5\\%$. Solution 2: Suppose th... |
DeepScaleR | [
{
"role": "user",
"content": "A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is"
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"index": 2784,
"solution": "Let the positive number be $x$. According to the problem, the number $x$ should have been multiplied by $6$, which would give the correct result as $6x$. However, it was mistakenly divided by $6$, giving the erroneous result $\\frac{x}{6}$.\n\nTo find the error committed, we calculate ... |
DeepScaleR | [
{
"role": "user",
"content": "A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^3} + \\sqrt[3]{1+x^3} = 1.\n\\]Find $x^6.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28}{27}"
} | {
"index": 2785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^4} + \\sqrt[3]{1+x^4} = 1.\n\\]Find $x^8.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28}{27}"
} | {
"index": 2786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive real number \\( x \\) is such that\n$$\n\\sqrt[3]{1-x^{3}} + \\sqrt[3]{1+x^{3}} = 1.\n$$\n\nFind \\( x^{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt[3]{28}}{3}"
} | {
"index": 2787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive two-digit number is even and is a multiple of 11. The product of its digits is a perfect cube. What is this two-digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 2788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 2789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the... | math | {
"style": "rule",
"ground_truth": "998"
} | {
"index": 2790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:"
}
] | math | {
"style": "rule",
"ground_truth": "245 yd."
} | {
"index": 2791,
"solution": "1. **Define the problem in terms of equations:**\n - Let $p(t)$ be the distance in feet that the powderman is from the starting point at time $t$ seconds after the fuse is lit. Since he runs at $8$ yards per second, and there are $3$ feet in a yard, he runs $8 \\times 3 = 24$ feet pe... |
DeepScaleR | [
{
"role": "user",
"content": "A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 2792,
"solution": "\nLet's analyze the problem step by step:\n\n1. **Define Variables:**\n - Let $t$ be the time it takes for the power boat to travel from dock $A$ to dock $B$.\n - Let $r$ be the speed of the river current (and also the speed of the raft, since it drifts with the current).\n - Let... |
DeepScaleR | [
{
"role": "user",
"content": "A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the lo... | math | {
"style": "rule",
"ground_truth": "20 \\sqrt[3]{5}"
} | {
"index": 2793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of stud... | math | {
"style": "rule",
"ground_truth": "621"
} | {
"index": 2794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 2795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A prime number of the form \\( 2^{p} - 1 \\) is called a Mersenne prime, where \\( p \\) is also a prime number. To date, the largest known Mersenne prime is \\( 2^{82589933} - 1 \\). Find the last two digits of this number."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 2796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A printer prints text pages at a rate of 17 pages per minute and graphic pages at a rate of 10 pages per minute. If a document consists of 250 text pages and 90 graphic pages, how many minutes will it take to print the entire document? Express your answer to the nearest whole n... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word \"JANUARY\", 10,000 posters with the word \"FEBRUARY\", 10,000 posters with the word \"MAR... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 2799,
"solution": ""
} |
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