data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{3\\pi}{4} < \\alpha < \\pi$, find the values for the following:\n1. $\\tan\\alpha$ given $\\tan\\alpha + \\frac{1}{\\tan\\alpha} = -\\frac{10}{3}$.\n2. The value of $$\\frac{5\\sin^2\\left(\\frac{\\alpha}{2}\\right) + 8\\sin\\left(\\frac{\\alpha}{2}\\right)\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{4}"
} | {
"split": "train",
"index": 31350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aria and Bianca walk at different, but constant speeds. They each begin at 8:00 a.m. from the opposite ends of a road and walk directly toward the other's starting point. They pass each other at 8:42 a.m. Aria arrives at Bianca's starting point at 9:10 a.m. When does Bianca arr... | math | {
"style": "rule",
"ground_truth": "9:45 a.m"
} | {
"split": "train",
"index": 31351,
"solution": "Let $A$ be Aria's starting point, $B$ be Bianca's starting point, and $M$ be their meeting point. It takes Aria 42 minutes to walk from $A$ to $M$ and 28 minutes from $M$ to $B$. Since Aria walks at a constant speed, then the ratio of the distance $A M$ to the dist... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{18}{8}=31824$, $\\binom{18}{9}=48620$, and $\\binom{18}{10}=43758$, calculate $\\binom{20}{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "172822"
} | {
"split": "train",
"index": 31352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the time in hours it will take to fill a 32,000 gallon swimming pool using three hoses that deliver 3 gallons of water per minute. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 31353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{0.\\overline{72}}{0.\\overline{27}}$? Express your answer as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 31354,
"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"
} | {
"split": "train",
"index": 31355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, it is known that the circle $C: x^{2} + y^{2} + 8x - m + 1 = 0$ intersects with the line $x + \\sqrt{2}y + 1 = 0$ at points $A$ and $B$. If $\\triangle ABC$ is an equilateral triangle, then the value of the real number $m$ is. Let's thi... | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 31356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $2.50 each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves a... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 31357,
"solution": "1. **Calculate the number of loaves sold in the morning and the revenue from these sales:**\n Bridget sells half of her 48 loaves in the morning. \n \\[\n \\frac{48}{2} = 24 \\text{ loaves}\n \\]\n Each of these loaves sells for $\\$2.50$, so the revenue ... |
deepscaler | [
{
"role": "user",
"content": "If \"For all $x \\in \\mathbb{R}, (a-2)x+1>0$\" is a true statement, then the set of values for the real number $a$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{2\\}"
} | {
"split": "train",
"index": 31358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \\cdot 7^y = z^3$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 1, 4)"
} | {
"split": "train",
"index": 31359,
"solution": "\nWe are tasked with finding all non-negative integer solutions \\((x, y, z)\\) to the equation:\n\n\\[\n2^x + 9 \\cdot 7^y = z^3\n\\]\n\nGiven that our solution must satisfy integer constraints and each variable is greater than or equal to zero, we will systematic... |
deepscaler | [
{
"role": "user",
"content": "Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 31360,
"solution": "To determine the greatest number of popsicles Pablo can buy with $8, we need to consider the cost-effectiveness of each purchasing option:\n\n1. **Single popsicle:** Costs $1 each, so the rate is $\\frac{1}{1} = 1$ dollar per popsicle.\n2. **3-popsicle box:** Costs... |
deepscaler | [
{
"role": "user",
"content": "Find $x+y+z$ when $$ a_1x+a_2y+a_3z= a $$ $$ b_1x+b_2y+b_3z=b $$ $$ c_1x+c_2y+c_3z=c $$ Given that $$ a_1\\left(b_2c_3-b_3c_2\\right)-a_2\\left(b_1c_3-b_3c_1\\right)+a_3\\left(b_1c_2-b_2c_1\\right)=9 $$ $$ a\\left(b_2c_3-b_3c_2\\right)-a_2\\left(bc_3-b_3c\\righ... | math | {
"style": "rule",
"ground_truth": "16/9"
} | {
"split": "train",
"index": 31361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table. \n\\[\n\\begin{tabular}{|l|c|}\n\\hline\nCyclone & 180 feet \\\\ \\hline\nGravity Rush & 120 feet \\\\ \\hline\nScreamer & 150 feet \\\\ \\hline\nSky High & 310 feet \\\\ \\hline\nTwister... | math | {
"style": "rule",
"ground_truth": "8.\\overline{3}"
} | {
"split": "train",
"index": 31362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.\n\nWhat is the probability that the second person's coin landed on heads more times than the first person's coin? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Pipkins said, \"I was walking along the road at a speed of $3 \\frac{1}{2}$ km/h when suddenly a car sped past me, almost knocking me off my feet.\"\n\n\"What was its speed?\" his friend asked.\n\n\"I can tell you now. From the moment it sped past me until it disappeared ar... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 31364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \\[g(x) = \\left\\{\\begin{aligned} \\log_3 x & \\quad \\text{ if } \\log_3 x \\text{ is an integer} \\\\ 1 + g(x + 1) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\]Compute $g(200).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 31365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the end of $1997$, Jason was one-third as old as his grandmother. The sum of the years in which they were born was $3852$. How old will Jason be at the end of $2004$?\nA) $41.5$\nB) $42.5$\nC) $43.5$\nD) $44.5$ Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "42.5"
} | {
"split": "train",
"index": 31366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $|AB|=18$ , $|AC|=24$ , and $m(\\widehat{BAC}) = 150^\\circ$ . Let $D$ , $E$ , $F$ be points on sides $[AB]$ , $[AC]$ , $[BC]$ , respectively, such that $|BD|=6$ , $|CE|=8$ , and $|CF|=2|BF|$ . Let $H_1$ , $H_2$ , $H_3$ be the ref... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 31367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\text{ inches}"
} | {
"split": "train",
"index": 31368,
"solution": "1. **Identify the formula for the area of the triangle and trapezoid:**\n - The area of a triangle is given by the formula:\n \\[\n \\text{Area}_{\\text{triangle}} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2}bh\n \\]\n -... |
deepscaler | [
{
"role": "user",
"content": "When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively.\n\nWhen the sum of the three integers is divided by $47$, what is the remainder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, ge... | math | {
"style": "rule",
"ground_truth": "4035"
} | {
"split": "train",
"index": 31370,
"solution": "The only possible way for Ritmo to get 2017 cubic centimeters is to have his measurements rounded to $1,1,2017$ centimeters respectively. Therefore the largest value of $V$ is achieved when the dimensions are $(1.5-\\epsilon)(1.5-\\epsilon)(2017.5-\\epsilon)=4539.3... |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of edges of a given square pyramid determine a plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the vectors $\\begin{pmatrix} -5 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}$ are both projected onto the same vector $\\mathbf{v},$ the result is $\\mathbf{p}$ in both cases. Find $\\mathbf{p}.$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -34/53 \\\\ 119/53 \\end{pmatrix}"
} | {
"split": "train",
"index": 31372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domain of the function $f(x)$ is $\\mathbf{R}$, and $f(x+2) - 2$ is an odd function, while $f(2x+1)$ is an even function. If $f(1) = 0$, determine the value of $f(1) + f(2) + \\cdots + f(2023)$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "4046"
} | {
"split": "train",
"index": 31373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ and $e$ denote the solutions of $2x^{2}+3x-5=0$. What is the value of $(d-1)(e-1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 31374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hairdresser moved from Vienna to Debrecen to continue his trade. Over the course of 3 years, he became impoverished despite having some money originally. In the first year, he had to spend half of his money. In the second year, he spent a third of what he initially took with ... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 31375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each real number $x$, let\n\\[\nf(x) = \\sum_{n\\in S_x} \\frac{1}{2^n},\n\\]\nwhere $S_x$ is the set of positive integers $n$ for which $\\lfloor nx \\rfloor$ is even. What is the largest real number $L$ such that $f(x) \\geq L$ for all $x \\in [0,1)$? (As usual, $\\lfloor... | math | {
"style": "rule",
"ground_truth": "4/7"
} | {
"split": "train",
"index": 31376,
"solution": "The answer is $L = 4/7$. For $S \\subset \\mathbb{N}$, let $F(S) = \\sum_{n\\in S} 1/2^n$, so that $f(x) = F(S_x)$. Note that for $T = \\{1,4,7,10,\\ldots\\}$, we have $F(T) = 4/7$.\n\nWe first show by contradiction that for any $x \\in [0,1)$, $f(x) \\geq 4/7$.\nS... |
deepscaler | [
{
"role": "user",
"content": "In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000. Let's think... | math | {
"style": "rule",
"ground_truth": "860"
} | {
"split": "train",
"index": 31377,
"solution": "We start by showing that every group of $6$ rows can be grouped into $3$ complementary pairs. We proceed with proof by contradiction. Without loss of generality, assume that the first row has columns $1$ and $2$ shaded. Note how if there is no complement to this, t... |
deepscaler | [
{
"role": "user",
"content": "Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to ... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 31378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Caos) A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he... | math | {
"style": "rule",
"ground_truth": "1416528"
} | {
"split": "train",
"index": 31379,
"solution": "```\nAnswer: 1416528\n# 1 = on ground, 0 = raised, 2 = back on ground\ncache = {}\ndef pangzi(legs):\n if legs == (2,2,2,2,2,2): return 1\n elif legs.count(0) > 3: return 0\n elif legs[0] + legs[1] + legs[2] == 0: return 0\n elif legs[3] + legs[4] + leg... |
deepscaler | [
{
"role": "user",
"content": "Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = b_0 + b_1x + b_2x^2 + \\dots + b_mx^m$ be a polynomial with integer coefficients, where $0 \\le b_i < 5$ for all $0 \\le i \\le m$.\n\nGiven that $P(\\sqrt{5})=23+19\\sqrt{5}$, compute $P(3)$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 31381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following flowchart: INPUT $\\rightarrow$ Subtract $8 \\rightarrow \\square \\rightarrow$ Divide by $2 \\rightarrow \\square$ Add $16 \\rightarrow$ OUTPUT. If the OUTPUT is 32, what was the INPUT? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 31382,
"solution": "We start from the OUTPUT and work back to the INPUT. Since the OUTPUT 32 is obtained from adding 16 to the previous number, then the previous number is $32 - 16 = 16$. Since 16 is obtained by dividing the previous number by 2, then the previous number is $2 \\times... |
deepscaler | [
{
"role": "user",
"content": "For each integer \\( n \\geq 2 \\), let \\( A(n) \\) be the area of the region in the coordinate plane defined by the inequalities \\( 1 \\leq x \\leq n \\) and \\( 0 \\leq y \\leq x \\left\\lfloor \\log_2{x} \\right\\rfloor \\), where \\( \\left\\lfloor \\log_2{x} \\right\\rfl... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 31383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all positive integers $n$, let $f(n)=\\log_{2002} n^2$. Find $f(11)+f(13)+f(14)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x$ when $3(x - 4) + 4(7 - 2x^2 + 5x) - 8(2x - 1)$ is simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 31385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $(-3 \\frac{3}{8})^{- \\frac{2}{3}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 31386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$, $B$, and $C$ be three sets such that:\n\n-A$|-|A|=100$, $|B|=50$, and $|C|=48$,\n\n-The number of elements belonging to exactly one of the three sets is twice the number of elements belonging to exactly two of the sets.\n\n-The number of elements belonging to exactly o... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 31387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F(0) = 0,$ $F(1) = \\frac{3}{2},$ and\n\\[F(n) = \\frac{5}{2} F(n - 1) - F(n - 2)\\]for $n \\ge 2.$ Find\n\\[\\sum_{n = 0}^\\infty \\frac{1}{F(2^n)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an $h$-meter race, Sunny is exactly $d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How m... | math | {
"style": "rule",
"ground_truth": "\\frac {d^2}{h}"
} | {
"split": "train",
"index": 31389,
"solution": "1. **Understanding the first race**: In the first race, Sunny finishes $d$ meters ahead of Windy in a race of $h$ meters. This implies that when Sunny has run $h$ meters, Windy has run $h-d$ meters. Let $s$ and $w$ be the speeds of Sunny and Windy, respectively. Si... |
deepscaler | [
{
"role": "user",
"content": "A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\\%$ chance of... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 31390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club hav... | math | {
"style": "rule",
"ground_truth": "980"
} | {
"split": "train",
"index": 31391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\\frac{a}{b}$, where $a, b$ are relatively prime positive integer... | math | {
"style": "rule",
"ground_truth": "1028"
} | {
"split": "train",
"index": 31392,
"solution": "Separate the sequence of coin flips into alternating blocks of heads and tails. Of the blocks of heads, exactly one block has length 2, and all other blocks have length 1. The same statement applies to blocks of tails. Thus, if there are $k$ blocks in total, there ... |
deepscaler | [
{
"role": "user",
"content": "Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$'s equation can be express... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 31393,
"solution": "The line passing through the tangency point of the bottom left circle and the one to its right and through the tangency of the top circle in the middle column and the one beneath it is the line we are looking for: a line passing through the tangency of two circles ... |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, we have $\\angle X = 90^\\circ$, $YZ = 20$, and $\\tan Z = 3 \\sin Y$. Calculate $XY$.\n\n--- Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 31394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two more than three times $B$ is equal to 20. What is the value of $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 31395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $\\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 31396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mia is designing a rectangular flower bed against her house on one side, using 450 feet of fencing to enclose the remaining three sides. If 150 feet of it is planned to be used along the house, what is the maximum area she can achieve for her garden? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "22500"
} | {
"split": "train",
"index": 31397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"split": "train",
"index": 31398,
"solution": "There are $\\left\\lfloor\\frac{999}{10}\\right\\rfloor = 99$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when $a$ or $b$ have a 0 in the tens digit, and since the equation is symmetric, we will just count when $... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the digits in the number $\\underbrace{44 \\ldots 4}_{2012 \\text{ times}} \\cdot \\underbrace{99 \\ldots 9}_{2012 \\text{ times}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18108"
} | {
"split": "train",
"index": 31399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the square of an integer $x$ is 1521, what is the value of $(x+1)(x-1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1520"
} | {
"split": "train",
"index": 31400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\\frac{a}{b}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\\overline{AB}$, and $OA=2\\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\\overline{EF}$ is a common tangent. What is the are... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}-4-\\pi"
} | {
"split": "train",
"index": 31402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $$\\begin{aligned} & A=(1+2 \\sqrt{2}+3 \\sqrt{3}+6 \\sqrt{6})(2+6 \\sqrt{2}+\\sqrt{3}+3 \\sqrt{6})(3+\\sqrt{2}+6 \\sqrt{3}+2 \\sqrt{6})(6+3 \\sqrt{2}+2 \\sqrt{3}+\\sqrt{6}) \\\\ & B=(1+3 \\sqrt{2}+2 \\sqrt{3}+6 \\sqrt{6})(2+\\sqrt{2}+6 \\sqrt{3}+3 \\sqrt{6})(3+6 \\sqrt{2}+... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31403,
"solution": "Note that $$\\begin{aligned} & A=((1+2 \\sqrt{2})(1+3 \\sqrt{3}))((2+\\sqrt{3})(1+3 \\sqrt{2}))((3+\\sqrt{2})(1+2 \\sqrt{3}))((3+\\sqrt{3})(2+\\sqrt{2})) \\\\ & B=((1+3 \\sqrt{2})(1+2 \\sqrt{3}))((2+\\sqrt{2})(1+3 \\sqrt{3}))((3+\\sqrt{3})(1+2 \\sqrt{2}))((2+\\sqrt... |
deepscaler | [
{
"role": "user",
"content": "The sum of n terms of an arithmetic progression is 180, and the common difference is 3. If the first term must be a positive integer, and n > 1, then find the number of possible values for n. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 31404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let QR = x, PR = y, and PQ = z. Given that the area of the square on side QR is 144 = x^2 and the area of the square on side PR is 169 = y^2, find the area of the square on side PQ. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A racer departs from point \\( A \\) along the highway, maintaining a constant speed of \\( a \\) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \\( 1.25a \\) km/h. How many minutes after the start of the first racer was a third racer... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 31406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. If the slant height is 30 meters, what is the length of the side of its base, in meters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.\n\n(Ⅰ) If $\\overrightarrow{PQ}=\\lambda \\overrightarrow{PR}$, find the value of $\\lambda$;\n\n(Ⅱ) I... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{9}"
} | {
"split": "train",
"index": 31408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pools $A$ and $B$ are both rectangular cuboids with a length of 3 meters, a width of 2 meters, and a depth of 1.2 meters. Valve 1 is used to fill pool $A$ with water and can fill an empty pool $A$ in 18 minutes. Valve 2 is used to transfer water from pool $A$ to pool $B$, takin... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"split": "train",
"index": 31409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\left( a > b > 0 \\right)$ has an eccentricity of $\\frac{\\sqrt{2}}{2}$, and the distance from one endpoint of the minor axis to the right focus is $\\sqrt{2}$. The line $y = x + m$ intersects the ellipse $C$ at po... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 31410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A nonempty set $S$ is called well-filled if for every $m \\in S$, there are fewer than $\\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\\{1,2, \\ldots, 42\\}$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\binom{43}{21}-1"
} | {
"split": "train",
"index": 31411,
"solution": "Let $a_{n}$ be the number of well-filled subsets whose maximum element is $n$ (setting $a_{0}=1$). Then it's easy to see that $a_{2k+1}=a_{2k}+a_{2k-1}+\\cdots+a_{0}$ and $a_{2k+2}=(a_{2k+1}-C_{k})+a_{2k}+\\cdots+a_{0}$ where $C_{k}$ is the number of well-filled su... |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{2+\\sqrt{5}}{3-\\sqrt{5}}$. When you write your answer in the form $A+B\\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "275"
} | {
"split": "train",
"index": 31412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\\leq k \\leq n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the square root of \\( x \\) is cubed, the result is 100. What is the value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10^{\\frac{4}{3}}"
} | {
"split": "train",
"index": 31414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the arithmetic sequence $\\{a_n\\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 31415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $24.50$. How much would $8$ adult tickets and $6$ child tickets cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38.50"
} | {
"split": "train",
"index": 31416,
"solution": "1. **Define the variables:**\n Let $x$ be the price of an adult ticket. Then, the price of a child ticket is $\\frac{x}{2}$.\n\n2. **Set up the equation based on the given information:**\n The total cost for $5$ adult tickets and $4$ child tickets is $24.50. Th... |
deepscaler | [
{
"role": "user",
"content": "Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number \\( k \\), she calls a placement of nonzero real numbers on the \\( 2^{2019} \\) vertices of the hypercube \\( k \\)-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this ve... | math | {
"style": "rule",
"ground_truth": "2040200"
} | {
"split": "train",
"index": 31417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{281}{13}"
} | {
"split": "train",
"index": 31418,
"solution": "1. **Identify the triangle formed by the line and the axes**: The line $12x + 5y = 60$ intersects the x-axis and y-axis, forming a right triangle with the axes. To find the intercepts:\n - **x-intercept**: Set $y = 0$ in the equation $12x + 5y = 60$:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "In each cell of a $5 \\times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\frac{\\cos\\alpha + \\sin\\alpha}{\\cos\\alpha - \\sin\\alpha} = 2$$, find the value of $$\\frac{1 + \\sin4\\alpha - \\cos4\\alpha}{1 + \\sin4\\alpha + \\cos4\\alpha}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 31420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"split": "train",
"index": 31421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C\\_1$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\\_1$ and $F\\_2$, respectively, and hyperbola $C\\_2$: $\\frac{x^2}{16} - \\frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\\_1$ given that p... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of positive real numbers. Let $g : S \\to \\mathbb{R}$ be a function such that\n\\[g(x) g(y) = g(xy) + 3003 \\left( \\frac{1}{x} + \\frac{1}{y} + 3002 \\right)\\]for all $x,$ $y > 0.$\n\nLet $m$ be the number of possible values of $g(2),$ and let $t$ be the ... | math | {
"style": "rule",
"ground_truth": "\\frac{6007}{2}"
} | {
"split": "train",
"index": 31423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_n\\}$ with $2017$ terms, and all terms are non-zero, $a_{2017}=1$. If two terms $a_i$, $a_j$ are arbitrarily chosen from $\\{a_n\\}$, when $i < j$, $a_j-a_i$ is still a term in the sequence $\\{a_n\\}$. Then, the sum of all terms in the sequen... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 31424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{32}"
} | {
"split": "train",
"index": 31425,
"solution": "1. **Assume the common perimeter**: Let the common perimeter of the square and the equilateral triangle be $P$.\n\n2. **Calculate side lengths**:\n - For the square, each side length is $\\frac{P}{4}$.\n - For the equilateral triangle, each side length is $\\fr... |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31426,
"solution": "\nTo determine the smallest positive real number \\( k \\) such that for any convex quadrilateral \\( ABCD \\) with points \\( A_1 \\), \\( B_1 \\), \\( C_1 \\), and \\( D_1 \\) on sides \\( AB \\), \\( BC \\), \\( CD \\), and \\( DA \\) respectively, the inequalit... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "284160"
} | {
"split": "train",
"index": 31427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least five-digit positive integer which is congruent to 7 (mod 12)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10,003"
} | {
"split": "train",
"index": 31428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ages of Jo, her daughter, and her grandson are all even numbers. The product of their three ages is 2024. How old is Jo? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 31429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the coordinates of the vertices of triangle $\\triangle O A B$ are $O(0,0), A(4,4 \\sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 31430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers have at least one 7 or at least one 9 as digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 31431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 31432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation wi... | math | {
"style": "rule",
"ground_truth": "294"
} | {
"split": "train",
"index": 31433,
"solution": "Let there be $m$ members and $n$ members for the square and $c$ for the number of columns of the other formation. We have $n^2 +5 = c(c+7) \\implies n^2+5 = \\left(c+\\frac{7}{2}\\right)^2 -\\frac{49}{4} \\implies n^2 - \\left(c+\\frac{7}{2}\\right)^2 = -\\frac{69}... |
deepscaler | [
{
"role": "user",
"content": "Jenna rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number she obtains? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.55"
} | {
"split": "train",
"index": 31434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right triangle \\( \\triangle ABC \\), \\( \\angle B = 90^\\circ \\). Point \\( P \\) is on the angle bisector of \\( \\angle A \\) within \\( \\triangle ABC \\). Point \\( M \\) (distinct from \\( A \\) and \\( B \\)) is a point on side \\( AB \\). The lines \\( AP \\),... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 31435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two 2's, \"plus\" can be changed to \"times\" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While standing in line to buy concert tickets, Kit moved 60 feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 31437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If α is in the interval (0, π) and $\\frac{1}{2}\\cos2α = \\sin\\left(\\frac{π}{4} + α\\right)$, then find the value of $\\sin2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 31438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABCD$ is a square with side length $6,$ and $WXYZ$ is a rectangle with $ZY=10$ and $XY=6.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to half of the area of $WXYZ,$ what is the length of $AP?$\n\n[asy]\ndraw((0,0)--(10,0)--(10,6)--(0,6)-... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"split": "train",
"index": 31440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given cos($$α+ \\frac {π}{6}$$)= $$\\frac {1}{3}$$, find the value of sin($$ \\frac {5π}{6}+2α$$). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-$$\\frac {7}{9}$$"
} | {
"split": "train",
"index": 31441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram ABCD, AB=2AD=4, ∠BAD=60°, E is the midpoint of BC, calculate the dot product of vectors BD and AE. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 31442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 31443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the method of random simulation experiments to estimate the probability of having exactly two days of rain in three days: First, use a calculator to generate random integers between \\\\(0\\\\) and \\\\(9\\\\), with \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(4\\\\) represe... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 31444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets $S$ of $\\{1,2,3,\\ldots,20\\}$ have the following three properties?\n$(1)$ No two consecutive integers belong to $S$.\n$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.\n$(3)$ $S$ contains at least one number greater than... | math | {
"style": "rule",
"ground_truth": "2526"
} | {
"split": "train",
"index": 31445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:<br/>$(1)x^{2}-4x-1=0$;<br/>$(2)\\left(x+3\\right)^{2}=x+3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 31446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two 18-24-30 triangles in the plane share the same circumcircle as well as the same incircle. What's the area of the region common to both the triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 31447,
"solution": "Notice, first of all, that $18-24-30$ is 6 times $3-4-5$, so the triangles are right. Thus, the midpoint of the hypotenuse of each is the center of their common circumcircle, and the inradius is $\\frac{1}{2}(18+24-30)=6$. Let one of the triangles be $A B C$, where... |
deepscaler | [
{
"role": "user",
"content": "When \\(2x^2\\) is added to a quadratic polynomial \\(f(x)\\), its maximum value increases by 10, and when \\(5x^2\\) is subtracted from it, its maximum value decreases by \\(\\frac{15}{2}\\). By how much will the maximum value of \\(f(x)\\) change if \\(3x^2\\) is added to it?... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"split": "train",
"index": 31448,
"solution": ""
} |
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