data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A freight train was delayed on its route for 12 minutes. Then, over a distance of 60 km, it made up for the lost time by increasing its speed by 15 km/h. Find the original speed of the train."
}
] | math | {
"style": "rule",
"ground_truth": "39.375"
} | {
"index": 1800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\\%$-$69\\%$ range? \\begin{tabular}{|c|c|}\nTest Scores & Frequencies\\\\\n\\hline\n$90\\% - 100\\%$& IIII\\\\\n$80\\% - 89\\%$& IIII IIII\\\\\... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 1801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There ... | math | {
"style": "rule",
"ground_truth": "373"
} | {
"index": 1802,
"solution": "First of all, it is easy to see by induction that for any $P(x,y)$ in the frog's jump sequence, $x+y$ will be a multiple of $3$ and $x-y$ will be a multiple of $5.$ The base case $(x,y) = (0,0)$ obviously satisfies the constraints and if $x+y = 3n$ and $x-y = 5m,$ any of the four trans... |
DeepScaleR | [
{
"role": "user",
"content": "A frog is at the point $(0,0)$. Every second, he can jump one unit either up or right. He can only move to points $(x, y)$ where $x$ and $y$ are not both odd. How many ways can he get to the point $(8,14)$?"
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"index": 1803,
"solution": "When the frog is at a point $(x, y)$ where $x$ and $y$ are both even, then if that frog chooses to move right, his next move will also have to be a step right; similarly, if he moves up, his next move will have to be up. If we 'collapse' each double step into one step, the problem simp... |
DeepScaleR | [
{
"role": "user",
"content": "A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 1804,
"solution": "Another way would be to use a table representing the number of ways to reach a certain number\n$\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} 0 & 3 & 6 & 9 & 12 & 13 & 15 & 18 & 21 & 24 & 26 & 27 & 30 & 33 & 36 \\\\ \\hline 1 & 1 & 1 & 1 & 1 & 5 & 6 & 6 & 6 & 6 & 29 & 35 & 35 & 35 & ... |
DeepScaleR | [
{
"role": "user",
"content": "A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at... | math | {
"style": "rule",
"ground_truth": "556"
} | {
"index": 1805,
"solution": "Casework Solution: x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-1-1 (1 way to order) $\\dbinom{8}{4} = 70$ ways total\n x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-2 (3 ways to order) $\\dbinom{7}{3} \\times 3= 105$ ways total\n x-distribution: 1-1-1-1 ... |
DeepScaleR | [
{
"role": "user",
"content": "A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1806,
"solution": "1. **Understanding the Problem**: The frog starts at $(0,0)$ and can only jump to points where both coordinates are integers. Each jump has a length of $5$. We need to find the minimum number of jumps required for the frog to reach $(1,0)$.\n\n2. **Properties of Jumps**: Each jump of l... |
DeepScaleR | [
{
"role": "user",
"content": "A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 1807,
"solution": "To solve this problem, we consider the frog's position in a coordinate system after each jump. Each jump is of length 1 meter, and the direction is chosen randomly. We need to calculate the probability that after 3 jumps, the frog is no more than 1 meter away from its starting point.\n... |
DeepScaleR | [
{
"role": "user",
"content": "A frog sits at the point $(2, 3)$ on a grid within a larger square bounded by points $(0,0), (0,6), (6,6)$, and $(6,0)$. Each jump the frog makes is parallel to one of the coordinate axes and has a length $1$. The direction of each jump (up, down, left, right) is not necessaril... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 1808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 1809,
"solution": "We are given a problem where a frog jumps randomly in one of the four cardinal directions from a point $(1,2)$ within a square bounded by $(0,0), (0,4), (4,4),$ and $(4,0)$. We need to find the probability that the frog first reaches a vertical side of the square.\n\nLet $P_{(x,y)}$ de... |
DeepScaleR | [
{
"role": "user",
"content": "A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters f... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 1810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second t... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 1811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How man... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 1812,
"solution": "1. **Define Variables:**\n Let $b$ be the number of blueberries, $r$ be the number of raspberries, $g$ be the number of grapes, and $c$ be the number of cherries.\n\n2. **Set Up Equations Based on Conditions:**\n - Total number of fruits: \n \\[ b + r + g + c = 280 \\]\n - Ra... |
DeepScaleR | [
{
"role": "user",
"content": "A fruit store sells a type of fruit. The purchase price is $30$ yuan per kilogram. According to a survey, when the selling price is $40$ yuan per kilogram, the store can sell an average of $500$ kilograms per day. For every $1$ yuan increase in the selling price, the store sell... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fruit wholesaler sells apples at a cost price of 40 yuan per box. The price department stipulates that the selling price per box cannot exceed 55 yuan. Market research has found that if each box is sold for 50 yuan, an average of 90 boxes are sold per day, and for every 1 yua... | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"index": 1814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area."
}
] | math | {
"style": "rule",
"ground_truth": "(80 + 12\\sqrt{41})\\pi"
} | {
"index": 1815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower base radius of 6 inches, an upper base radius of 3 inches, and a height of 4 inches, what is its lateral surface area? (The lateral surface... | math | {
"style": "rule",
"ground_truth": "45\\pi"
} | {
"index": 1816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\\pi$ sq cm and the area of its upper base is $25\\pi$ sq cm, what is the altitud... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cyli... | math | {
"style": "rule",
"ground_truth": "39\\pi\\sqrt{5} + 20\\pi"
} | {
"index": 1818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum."
}
] | math | {
"style": "rule",
"ground_truth": "10\\pi \\sqrt{61} + 68\\pi"
} | {
"index": 1819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A full container holds 150 watermelons and melons with a total value of 24,000 rubles. The total value of all watermelons is equal to the total value of all melons. How much does one watermelon cost in rubles, given that the container can hold 120 melons (without watermelons) o... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 1820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f: \\N\\rightarrow\\N$ is circular if for every $p\\in\\N$ there exists $n\\in\\N,\\ n\\leq{p}$ such that $f^n(p)=p$ ( $f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\\in\\N$ $f^i(p)\\neq{p}$ for eve... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 1821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ from the integers to the integers is defined as follows:\n\\[f(n) = \\left\\{\n\\begin{array}{cl}\nn + 3 & \\text{if $n$ is odd}, \\\\\nn/2 & \\text{if $n$ is even}.\n\\end{array}\n\\right.\\]Suppose $k$ is odd and $f(f(f(k))) = 27.$ Find $k.$"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 1822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\\{x:a \\le x \\le b \\}$.) Let\n\\[g(x) = 1 - f(x + 1).\\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$"
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1,0,1)"
} | {
"index": 1823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 1824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \\alpha | + |\\gamma |... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 1825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined by $f(z) = i\\overline{z}$, where $i^2 = -1$ and $\\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 1826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$?"
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 1827,
"solution": "We notice that the function has reflectional symmetry across both $x=2$ and $x=7$. We also use the fact that $x=0$ is a root. This shows that $x=4$ and $x=14$ are also roots. We then apply the reflection across the other axis to form $x=\\pm 10$ as roots. Continuing this shows that the... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$?"
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 1828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \\leq x \\leq 500$."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 1829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{255}{4}"
} | {
"index": 1830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=... | math | {
"style": "rule",
"ground_truth": "259"
} | {
"index": 1831,
"solution": "Plugging in $z=1$ yields $f(1) = a+bi$. This implies that $a+bi$ must fall on the line $Re(z)=a=\\frac{1}{2}$, given the equidistant rule. By $|a+bi|=8$, we get $a^2 + b^2 = 64$, and plugging in $a=\\frac{1}{2}$ yields $b^2=\\frac{255}{4}$. The answer is thus $\\boxed{259}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ is defined recursively by $f(1)=f(2)=1$ and \\[f(n)=f(n-1)-f(n-2)+n\\]for all integers $n \\geq 3$. What is $f(2018)$?"
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 1832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \\leq x \\leq 4$. Find the smallest \\( x \\) for which \\( f(x) = f(2022) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 1833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f$ satisfies, for all nonnegative integers $x$ and $y$: - $f(0, x)=f(x, 0)=x$ - If $x \\geq y \\geq 0, f(x, y)=f(x-y, y)+1$ - If $y \\geq x \\geq 0, f(x, y)=f(x, y-x)+1$ Find the maximum value of $f$ over $0 \\leq x, y \\leq 100$."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 1834,
"solution": "Firstly, $f(100,100)=101$. To see this is maximal, note that $f(x, y) \\leq \\max \\{x, y\\}+1$, say by induction on $x+y$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x) = a \\cos ωx + b \\sin ωx (ω > 0)$ has a minimum positive period of $\\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \\frac{π}{6}$.\n\n1. Find the values of $a$, $b$, and $ω$.\n2. If $\\frac{π}{4} < x < \\frac{3π}{4}$ and $f(x + \\frac{π}{6... | math | {
"style": "rule",
"ground_truth": "-\\frac{4\\sqrt{6}}{3}"
} | {
"index": 1835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x)$ defined on $R$ satisfies $f(x+1) = 2f(x)$. When $x \\in (-1,0]$, $f(x) = x^{3}$. Find $f(\\frac{21}{2})$."
}
] | math | {
"style": "rule",
"ground_truth": "-256"
} | {
"index": 1836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x)$ defined on $\\mathbb{R}$ satisfies $f(0)=0$, $f(x)+f(1-x)=1$, $f(\\frac{x}{5})=\\frac{1}{2}f(x)$, and $f(x_1)\\leqslant f(x_2)$ when $0\\leqslant x_1 < x_2\\leqslant 1$. Find the value of $f(\\frac{1}{2007})$.\n\nA) $\\frac{1}{2}$\nB) $\\frac{1}{16}$\nC) $\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"index": 1837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$."
}
] | math | {
"style": "rule",
"ground_truth": "601"
} | {
"index": 1838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[3f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 6x + \\sin x + 3\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S... | math | {
"style": "rule",
"ground_truth": "445"
} | {
"index": 1839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x, y)$ is linear in $x$ and in $y . f(x, y)=\\frac{1}{x y}$ for $x, y \\in\\{3,4\\}$. What is $f(5,5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"index": 1840,
"solution": "The main fact that we will use in solving this problem is that $f(x+2, y)-f(x+1, y)=f(x+1, y)-f(x, y)$ whenever $f$ is linear in $x$ and $y$. Suppose that $f(x, y)=a x y+b y+c x+d=x(a y+c)+(b y+d)$ for some constants $a, b, c$, and $d$. Then it is easy to see that $$\\begin{aligned} f(... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f(x, y, z)$ is linear in $x, y$, and $z$ such that $f(x, y, z)=\\frac{1}{x y z}$ for $x, y, z \\in\\{3,4\\}$. What is $f(5,5,5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{216}"
} | {
"index": 1841,
"solution": "We use a similar method to the previous problem. Notice that $f(x, y, 5)=2 f(x, y, 4)-f(x, y, 3)$. Let $f_{2}$ denote the function from the previous problem and $f_{3}$ the function from this problem. Since $3 f_{3}(x, y, 3)$ is linear in $x$ and $y$, and $3 f_{3}(x, y, 3)=\\frac{1}{x ... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f: A \\rightarrow A$ is called idempotent if $f(f(x))=f(x)$ for all $x \\in A$. Let $I_{n}$ be the number of idempotent functions from $\\{1,2, \\ldots, n\\}$ to itself. Compute $\\sum_{n=1}^{\\infty} \\frac{I_{n}}{n!}$."
}
] | math | {
"style": "rule",
"ground_truth": "e^{e}-1"
} | {
"index": 1842,
"solution": "Let $A_{k, n}$ denote the number of idempotent functions on a set of size $n$ with $k$ fixed points. We have the formula $A_{k, n}=\\binom{n}{k} k^{n-k}$ for $1 \\leq k \\leq n$ because there are $\\binom{n}{k}$ ways to choose the fixed points and all $n-k$ remaining elements must map ... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfies: $f(0)=0$ and $$\\left|f\\left((n+1) 2^{k}\\right)-f\\left(n 2^{k}\\right)\\right| \\leq 1$$ for all integers $k \\geq 0$ and $n$. What is the maximum possible value of $f(2019)$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 1843,
"solution": "Consider a graph on $\\mathbb{Z}$ with an edge between $(n+1) 2^{k}$ and $n 2^{k}$ for all integers $k \\geq 0$ and $n$. Each vertex $m$ is given the value $f(m)$. The inequality $\\left|f\\left((n+1) 2^{k}\\right)-f\\left(n 2^{k}\\right)\\right| \\leq 1$ means that any two adjacent ve... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f:\\mathbb{Z} \\to \\mathbb{Z}$ satisfies\n\\begin{align*}\nf(x+4)-f(x) &= 8x+20, \\\\\nf(x^2-1) &= (f(x)-x)^2+x^2-2\n\\end{align*}for all integers $x.$ Enter the ordered pair $(f(0),f(1)).$"
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1)"
} | {
"index": 1844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $f:\\{1,2,3,4,5\\} \\rightarrow\\{1,2,3,4,5\\}$ is said to be nasty if there do not exist distinct $a, b \\in\\{1,2,3,4,5\\}$ satisfying $f(a)=b$ and $f(b)=a$. How many nasty functions are there?"
}
] | math | {
"style": "rule",
"ground_truth": "1950"
} | {
"index": 1845,
"solution": "We use complementary counting. There are $5^{5}=3125$ total functions. If there is at least one pair of numbers which map to each other, there are \\binom{5}{2}=10$ ways to choose the pair and $5^{3}=125$ ways to assign the other values of the function for a total of 1250 . But we over... |
DeepScaleR | [
{
"role": "user",
"content": "A function $f\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ is linear in each of the $x_{i}$ and $f\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)=\\frac{1}{x_{1} x_{2} \\cdots x_{n}}$ when $x_{i} \\in\\{3,4\\}$ for all $i$. In terms of $n$, what is $f(5,5, \\ldots, 5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6^{n}}"
} | {
"index": 1846,
"solution": "Let $f_{n}\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ denote the $n$-variable version of the function. We will prove that $f_{n}(5, \\ldots, 5)=\\frac{1}{6^{n}}$ by induction. The base case was done in the two previous problems. Suppose we know that $f_{n-1}(5,5, \\ldots, 5)=\\frac{1}... |
DeepScaleR | [
{
"role": "user",
"content": "A function $g$ from the integers to the integers is defined as follows:\n\\[g(n) = \\left\\{\n\\begin{array}{cl}\nn + 5 & \\text{if $n$ is odd}, \\\\\nn/2 & \\text{if $n$ is even}.\n\\end{array}\n\\right.\\]\nSuppose $m$ is odd and $g(g(g(m))) = 39.$ Find $m.$"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 1847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $g$ is defined by $g(z) = (3 + 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$?"
... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 1848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$?"
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 1849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function $g$ is ever more than a function $h$ if, for all real numbers $x$, we have $g(x) \\geq h(x)$. Consider all quadratic functions $f(x)$ such that $f(1)=16$ and $f(x)$ is ever more than both $(x+3)^{2}$ and $x^{2}+9$. Across all such quadratic functions $f$, compute the... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{2}"
} | {
"index": 1850,
"solution": "Let $g(x)=(x+3)^{2}$ and $h(x)=x^{2}+9$. Then $f(1)=g(1)=16$. Thus, $f(x)-g(x)$ has a root at $x=1$. Since $f$ is ever more than $g$, this means that in fact $$f(x)-g(x)=c(x-1)^{2}$$ for some constant $c$. Now $$f(x)-h(x)=\\left((f(x)-g(x))+(g(x)-h(x))=c(x-1)^{2}+6 x=c x^{2}-(2 c-6) x+... |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f \\) is defined on the complex numbers by \\( f(z) = (a + bi)z^2 \\), where \\( a \\) and \\( b \\) are real numbers. The function has the property that for each complex number \\( z \\), \\( f(z) \\) is equidistant from both \\( z \\) and the origin. Given that... | math | {
"style": "rule",
"ground_truth": "\\frac{99}{4}"
} | {
"index": 1851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f \\) satisfies the equation \\((n - 2019) f(n) - f(2019 - n) = 2019\\) for every integer \\( n \\).\nWhat is the value of \\( f(2019) \\)?\nA) 0\nB) 1\nC) \\(2018 \\times 2019\\)\nD) \\(2019^2\\)\nE) \\(2019 \\times 2020\\)"
}
] | math | {
"style": "rule",
"ground_truth": "2019 \\times 2018"
} | {
"index": 1852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f \\), defined on the set of integers, satisfies the following conditions:\n\n1) \\( f(1) + 1 > 0 \\)\n2) \\( f(x + y) - x f(y) - y f(x) = f(x)f(y) - x - y + xy \\) for any \\( x, y \\in \\mathbb{Z} \\)\n3) \\( 2f(x) = f(x + 1) - x + 1 \\) for any \\( x \\in \\ma... | math | {
"style": "rule",
"ground_truth": "1014"
} | {
"index": 1853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f(n) \\) defined for positive integers satisfies:\n\\[ f(n) = \\begin{cases} \nn - 3 & \\text{if } n \\geq 1000 \\\\ \nf[f(n + 7)] & \\text{if } n < 1000 \n\\end{cases} \\]\nDetermine \\( f(90) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 1854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f(x) \\) defined on the interval \\([1,2017]\\) satisfies \\( f(1)=f(2017) \\), and for any \\( x, y \\in [1,2017] \\), \\( |f(x) - f(y)| \\leqslant 2|x - y| \\). If the real number \\( m \\) satisfies \\( |f(x) - f(y)| \\leqslant m \\) for any \\( x, y \\in [1,2... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 1855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( f: \\{a, b, c, d\\} \\rightarrow \\{1, 2, 3\\} \\) is given. If \\( 10 < f(a) \\cdot f(b) \\) and \\( f(c) \\cdot f(d) < 20 \\), how many such mappings exist?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\( g(x) \\) is defined for all real numbers \\( x \\). For all non-zero values \\( x \\), we have\n\\[ 3g(x) + g\\left(\\frac{1}{x}\\right) = 7x + 6. \\]\nLet \\( T \\) denote the sum of all of the values of \\( x \\) for which \\( g(x) = 2005 \\). Compute the integ... | math | {
"style": "rule",
"ground_truth": "763"
} | {
"index": 1857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function \\(f(x)\\) is defined for all real numbers \\(x\\). For all non-zero values \\(x\\), we have\n\\[3f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 15x + 8.\\]\nLet \\(S\\) denote the sum of all of the values of \\(x\\) for which \\(f(x) = 2004\\). Compute the integ... | math | {
"style": "rule",
"ground_truth": "356"
} | {
"index": 1858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function is given by\n$$\nf(x)=\\ln (a x+b)+x^{2} \\quad (a \\neq 0).\n$$\n(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.\n\n(2) If $f(x) \\leqslant x^{2}+x$ always holds, find the maximum value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{e}{2}"
} | {
"index": 1859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A function, $f$, has $f(2)=5$ and $f(3)=7$. In addition, $f$ has the property that $f(m)+f(n)=f(mn)$ for all positive integers $m$ and $n$. What is the value of $f(12)$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 1860,
"solution": "Since $f(2)=5$ and $f(mn)=f(m)+f(n)$, then $f(4)=f(2 \\cdot 2)=f(2)+f(2)=10$. Since $f(3)=7$, then $f(12)=f(4 \\cdot 3)=f(4)+f(3)=10+7=17$. While this answers the question, is there actually a function that satisfies the requirements? The answer is yes. One function that satisfies the ... |
DeepScaleR | [
{
"role": "user",
"content": "A function, defined on the set of positive integers, is such that \\( f(x y) = f(x) + f(y) \\) for all \\( x \\) and \\( y \\). It is known that \\( f(10) = 14 \\) and \\( f(40) = 20 \\).\n\nWhat is the value of \\( f(500) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 1861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A furniture store received a batch of office chairs that were identical except for their colors: 15 chairs were black and 18 were brown. The chairs were in demand and were being bought in a random order. At some point, a customer on the store's website discovered that only two ... | math | {
"style": "rule",
"ground_truth": "0.489"
} | {
"index": 1862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 1863,
"solution": "1. **Calculate the amount of paint remaining after the first day:**\n Initially, there is 1 gallon of paint. On the first day, one third of the paint is used. The amount of paint used on the first day is:\n \\[\n \\frac{1}{3} \\times 1 = \\frac{1}{3} \\text{ gallons}\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A gambler starts with $100. During a series of 4 rounds, they bet each time one-third of the amount they have. They win twice and lose twice, but now the wins are twice the staked amount, and losses mean losing the staked amount. Determine the final amount of money the gambler ... | math | {
"style": "rule",
"ground_truth": "\\frac{8000}{81}"
} | {
"index": 1864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining white squares in t... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 1865,
"solution": "1. **Understanding the Problem:**\n The problem involves finding the number of valid 7-step paths from square $P$ to square $Q$ on a chessboard, where each step must move to an adjoining white square in the row above.\n\n2. **Visualizing the Board and Paths:**\n The board is a stan... |
DeepScaleR | [
{
"role": "user",
"content": "A game board is constructed by shading three of the regions formed by the diagonals of a regular pentagon. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land in any region with equal likelihood."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 1866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A game board is constructed by shading two of the regions formed by the altitudes of an equilateral triangle as shown. What is the probability that the tip of the spinner will come to rest in a shaded region? Express your answer as a common fraction. [asy]\nimport olympiad; siz... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 1867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers\nand $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers\n$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?"
}... | math | {
"style": "rule",
"ground_truth": "1/3 < c \\leq 1"
} | {
"index": 1868,
"solution": "The desired real numbers $c$ are precisely those for which $1/3 < c \\leq 1$.\nFor any positive integer $m$ and any\nsequence $0 = x_0 < x_1 < \\cdots < x_m = 1$,\nthe cost of jumping along this sequence is\n$\\sum_{i=1}^m (x_i - x_{i-1})x_i^2$. Since\n\\begin{align*}\n1 = \\sum_{i=1}^... |
DeepScaleR | [
{
"role": "user",
"content": "A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 1869,
"solution": "\nWe will analyze the game by observing the token distribution and the rules of the game. The key observation is that in each round, the player with the most tokens gives one token to each of the other two players and one token to the discard pile, effectively losing three tokens, whil... |
DeepScaleR | [
{
"role": "user",
"content": "A game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box befor... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 1870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three pric... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 1871,
"solution": "[Clarification: You are supposed to find the number of all possible tuples of prices, $(A, B, C)$, that could have been on that day.]\n Since we have three numbers, consider the number of ways we can put these three numbers together in a string of 7 digits. For example, if $A=113, B=13... |
DeepScaleR | [
{
"role": "user",
"content": "A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$\n ~ pi_is_3.14"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 1872,
"solution": "The number of ways to draw six cards from $n$ is given by the binomial coefficient ${n \\choose 6} = \\frac{n\\cdot(n-1)\\cdot(n-2)\\cdot(n-3)\\cdot(n-4)\\cdot(n-5)}{6\\cdot5\\cdot4\\cdot3\\cdot2\\cdot1}$.\n The number of ways to choose three cards from $n$ is ${n\\choose 3} = \\frac{n... |
DeepScaleR | [
{
"role": "user",
"content": "A game wheel is divided into six regions labeled $A$, $B$, $C$, $D$, $E$, and $F$. The probability of the wheel stopping on region $A$ is $\\frac{1}{3}$, the probability it stops on region $B$ is $\\frac{1}{6}$, and the probability of it stopping on regions $C$ and $D$ are equa... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 1873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A garden can be watered by any of three sprinklers X, Y, or Z. Sprinklers X and Y together take 5 hours to water the garden. Sprinklers X and Z together take 6 hours to water the garden. Sprinklers Y and Z together take 7 hours to water the garden. How many hours does it take s... | math | {
"style": "rule",
"ground_truth": "3.93"
} | {
"index": 1874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from en... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 1875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gardener plans to enclose a rectangular garden with 480 feet of fencing. However, one side of the garden will be twice as long as another side. What is the maximum area of this garden?"
}
] | math | {
"style": "rule",
"ground_truth": "12800"
} | {
"index": 1876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gardener plans to place potted plants along both sides of a 150-meter-long path (including at both ends), with one pot every 2 meters. In total, \\_\\_\\_\\_\\_\\_ pots are needed."
}
] | math | {
"style": "rule",
"ground_truth": "152"
} | {
"index": 1877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 1878,
"solution": "Let $b$, $n$ denote birch tree and not-birch tree, respectively. Notice that we only need $4$ $n$s to separate the $5$ $b$s. Specifically, \\[b,n,b,n,b,n,b,n,b\\] Since we have $7$ $n$s, we are placing the extra $3$ $n$s into the $6$ intervals beside the $b$s.\n Now doing simple casewo... |
DeepScaleR | [
{
"role": "user",
"content": "A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent?"
}
] | math | {
"style": "rule",
"ground_truth": "7/99"
} | {
"index": 1879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gasoline tank is $\\frac78$ full. After $12$ gallons have been used, it is half full. How many gallons does this tank hold when it is full?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 1880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 1881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric progression \\( b_{1}, b_{2}, \\ldots \\) is such that \\( b_{25} = 2 \\tan \\alpha \\) and \\( b_{31} = 2 \\sin \\alpha \\) for some acute angle \\( \\alpha \\). Find the term number \\( n \\) for which \\( b_{n} = \\sin 2\\alpha \\)."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 1882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric progression with 10 terms starts with the first term as 2 and has a common ratio of 3. Calculate the sum of the new geometric progression formed by taking the reciprocal of each term in the original progression.\nA) $\\frac{29523}{59049}$\nB) $\\frac{29524}{59049}$\... | math | {
"style": "rule",
"ground_truth": "\\frac{29524}{59049}"
} | {
"index": 1883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence $(a_n)$ has $a_1=\\sin x$, $a_2=\\cos x$, and $a_3= \\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\\cos x$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 1884,
"solution": "1. **Identify the common ratio**: Given a geometric sequence $(a_n)$ with $a_1 = \\sin x$, $a_2 = \\cos x$, and $a_3 = \\tan x$, we know that the common ratio $r$ satisfies $a_2 = r a_1$ and $a_3 = r a_2$. Thus, we have:\n \\[\n \\cos x = r \\sin x \\quad \\text{and} \\quad \\tan x... |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence $\\left\\{a_{n}\\right\\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\\frac{1}{2}$. Let $\\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\\Pi_{n}$ maximized?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 1885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 1886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers is formed for which the first term is 2 and the fifth term is 162. What is the sixth term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "486"
} | {
"index": 1888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers is formed for which the first term is 3 and the fifth term is 375. What is the sixth term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"index": 1889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers is formed for which the first term is 3 and the fourth term is 192. What is the third term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 1890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers starts with a first term of 4 and the fourth term is 324. What is the fifth term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 1891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A geometric sequence starts $16$, $-24$, $36$, $-54$. What is the common ratio of this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 1893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A gift shop offers free gift wrapping. They have eight varieties of wrapping paper, three colors of ribbon, and four types of gift cards. Using one variety of wrapping paper, one color of ribbon and one type of gift card, how many combinations are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 1894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines... | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"index": 1895,
"solution": "\nConsider a finite number of lines in the plane, none of which are parallel, and no three of which are concurrent. These lines divide the plane into several regions—both finite and infinite.\n\n### Problem Analysis\n\nIn this scenario, we assign the number \\( 1 \\) or \\( -1 \\) to e... |
DeepScaleR | [
{
"role": "user",
"content": "A given integer Fahrenheit temperature $F$ is first converted to Kelvin using the formula $K = \\frac{5}{9}(F - 32) + 273.15$, rounded to the nearest integer, then converted back to Fahrenheit using the inverse formula $F' = \\frac{9}{5}(K - 273.15) + 32$, and rounded to the ne... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 1896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A given odd function $f(x)$, defined on $\\mathbb{R}$, is symmetric about the line $x=1$, and $f(-1) = 1$. Find the value of $f(1) + f(2) + f(3) + \\ldots + f(2009)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 1897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A given sequence $r_1, r_2, \\dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more \"bubble passes\". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the se... | math | {
"style": "rule",
"ground_truth": "931"
} | {
"index": 1898,
"solution": "If any of $r_1, \\ldots, r_{19}$ is larger than $r_{20}$, one of these numbers will be compared with $r_{20}$ on the 19th step of the first bubble pass and $r_{20}$ will be moved back to the 19th position. Thus, $r_{20}$ must be the largest of the first 20 terms. In addition, $r_{20}$ ... |
DeepScaleR | [
{
"role": "user",
"content": "A given sequence $r_1, r_2, \\dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more \"bubble passes\". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the se... | math | {
"style": "rule",
"ground_truth": "931"
} | {
"index": 1899,
"solution": ""
} |
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