data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \\leqslant 4-a) = P(ξ \\geqslant 2+3a)$, solve for the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six people form a circle to play a coin-tossing game (the coin is fair). Each person tosses a coin once. If the coin shows tails, the person has to perform; if it shows heads, they do not have to perform. What is the probability that no two performers (tails) are adjacent? Let'... | math | {
"style": "rule",
"ground_truth": "9/32"
} | {
"split": "train",
"index": 20350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom has a red marble, a green marble, a blue marble, and three identical yellow marbles. How many different groups of two marbles can Tom choose? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 20351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?\n\n[asy]\nfill((0,0)--(21,0)--(21,21)--(0,2... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 20352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist $s$ unique nonnegative integers $m_1 > m_2 > \\cdots > m_s$ and $s$ unique integers $b_k$ ($1\\le k\\le s$) with each $b_k$ either $1$ or $-1$ such that \\[b_13^{m_1} + b_23^{m_2} + \\cdots + b_s3^{m_s} = 1729.\\]\nFind $m_1 + m_2 + \\cdots + m_s$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 20353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the area of triangle $ABC$ is 27 square units. What is the area of triangle $BCD$?\n\n[asy]\n\ndraw((0,0)--(32,0)--(9,15)--(0,0));\n\ndot((0,0));\nlabel(\"$A$\",(0,0),SW);\nlabel(\"6\",(3,0),S);\ndot((6,0));\nlabel(\"$C$\",(6,0),S);\nlabel(\"26\",(19,0),S);\ndot... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 20354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 20355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I won a VIP trip for five to a music festival. I can bring four of my friends. I have 10 friends to choose from: 4 are musicians, and 6 are non-musicians. In how many ways can I form my music festival group so that at least one musician is in the group? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 20356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in this display. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 20357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following construction is used for training astronauts:\nA circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls... | math | {
"style": "rule",
"ground_truth": "n - 1"
} | {
"split": "train",
"index": 20358,
"solution": "\nConsider the problem of determining the number of revolutions an astronaut performs relative to the ground while being fastened to a circle \\( C_3 \\) of radius \\( R \\). This circle rolls inside another circle \\( C_2 \\) of radius \\( 2R \\), which in turn ro... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt[3]{x}$, calculate the value of $\\lim\\limits_{\\Delta x \\to 0} \\frac{f(1-\\Delta x)-f(1)}{\\Delta x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 20359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomials $P\\left(x\\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\\left(x\\right)=4x^2+5x+2$ . If $a$ is a real number, what is the smallest possible value of $\\frac{P\\left(a\\right)}{Q\\left(a\\right)}$ ?\n\n*2016 CCA Math Bonanza Team #6* Let's think step b... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 20360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have a square with vertices at $(2,1)$, $(5,1)$, $(2,4)$, and $(5,4)$. A line joining $(2,1)$ and $(5,3)$ divides the square into two regions. What fraction of the area of the square is above this line? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 20361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $L(x) = x - \\frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by\n\\[\n a_n = L \\Bigl( L \\Bigl( L \\Bigl( \\cdots L \\Bigl( \\frac{17}{n} \\Bigr) \\cdots \\Bigr) \\Bigr) \\Bigr),\n\\]where there are $n$ iterations of $L$. For exam... | math | {
"style": "rule",
"ground_truth": "\\frac{34}{19}"
} | {
"split": "train",
"index": 20362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers satisfying the following equations: $$\\frac{a_{1}}{k^{2}+1}+\\frac{a_{2}}{k^{2}+2}+\\frac{a_{3}}{k^{2}+3}+\\frac{a_{4}}{k^{2}+4}+\\frac{a_{5}}{k^{2}+5}=\\frac{1}{k^{2}} \\text { for } k=1,2,3,4,5$$ Find the value of $\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{187465}{6744582}"
} | {
"split": "train",
"index": 20363,
"solution": "Let $R(x):=\\frac{a_{1}}{x^{2}+1}+\\frac{a_{2}}{x^{2}+2}+\\frac{a_{3}}{x^{2}+3}+\\frac{a_{4}}{x^{2}+4}+\\frac{a_{5}}{x^{2}+5}$. Then $R( \\pm 1)=1, R( \\pm 2)=\\frac{1}{4}, R( \\pm 3)=\\frac{1}{9}, R( \\pm 4)=\\frac{1}{16}, R( \\pm 5)=\\frac{1}{25}$ and $R(6)$ is t... |
deepscaler | [
{
"role": "user",
"content": "The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2444"
} | {
"split": "train",
"index": 20364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 20365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point ... | math | {
"style": "rule",
"ground_truth": "15/8"
} | {
"split": "train",
"index": 20366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$ . $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\\frac{AE}{EB} = 1$ and $\\frac{CF}{FD} = \\frac{1}{2}$ . Call $X$ the intersection of $AF$ and $DE$ . What is the area of pentagon $B... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 20367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the coefficients of all rational terms in the expansion of $$(2 \\sqrt {x}- \\frac {1}{x})^{6}$$ is \\_\\_\\_\\_\\_\\_ (answer with a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 20368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the power function $y=x^{m}$ is an even function and is a decreasing function when $x \\in (0,+\\infty)$, determine the possible value of the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 20369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. Let's think step b... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 20370,
"solution": "Proceed as in solution 2, until we reach \\[3x^2+2d^2=xd^2,\\] Write\n$d^2=\\frac{3x^2}{x-2}$, it follows that $x-2=3k^2$ for some (positive) integer k and $k \\mid x$.\n Taking both sides modulo $k$, $-2 \\equiv 0 \\pmod{k}$, so $k \\mid 2 \\rightarrow k=1,2$.\n W... |
deepscaler | [
{
"role": "user",
"content": "The value of \\(1 + 0.01 + 0.0001\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.0101"
} | {
"split": "train",
"index": 20371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEFGH$ be a regular octagon, and let $I, J, K$ be the midpoints of sides $AB, DE, GH$ respectively. If the area of $\\triangle IJK$ is $144$, what is the area of octagon $ABCDEFGH$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"split": "train",
"index": 20372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\cos x =0$ and $\\cos(x+z)= \\frac{1}{2}$. What is the smallest possible positive value of $z,$ in radians? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 20373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a parking lot with $10$ empty spaces. Three different cars, A, B, and C, are going to park in such a way that each car has empty spaces on both sides, and car A must be parked between cars B and C. How many different parking arrangements are there? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 20374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a polynomial $ p\\left(x\\right)$ with real coefficients such that\n\n$ \\left(x\\plus{}10\\right)p\\left(2x\\right)\\equal{}\\left(8x\\minus{}32\\right)p\\left(x\\plus{}6\\right)$\n\nfor all real $ x$ and $ p\\left(1\\right)\\equal{}210$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "2(x + 4)(x - 4)(x - 8)"
} | {
"split": "train",
"index": 20375,
"solution": "\nTo find the polynomial \\( p(x) \\) with real coefficients satisfying the functional equation:\n\n\\[\n(x + 10)p(2x) = (8x - 32)p(x + 6)\n\\]\n\nfor all real \\( x \\), and given the condition \\( p(1) = 210 \\), we start by analyzing the equation.\n\n### Step 1:... |
deepscaler | [
{
"role": "user",
"content": "In convex quadrilateral \\(ABCD\\) with \\(AB=11\\) and \\(CD=13\\), there is a point \\(P\\) for which \\(\\triangle ADP\\) and \\(\\triangle BCP\\) are congruent equilateral triangles. Compute the side length of these triangles. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 20376,
"solution": "Evidently \\(ABCD\\) is an isosceles trapezoid with \\(P\\) as its circumcenter. Now, construct isosceles trapezoid \\(ABB'C\\) (that is, \\(BB'\\) is parallel to \\(AC\\).) Then \\(AB'PD\\) is a rhombus, so \\(\\angle B'CD=\\frac{1}{2} \\angle B'PD=60^{\\circ}\\) ... |
deepscaler | [
{
"role": "user",
"content": "The hyperbola given by the equation \\[\\frac{y^2}{9}-\\frac{x^2}{4} = 1\\]has asymptotes $y = \\pm mx,$ where $m$ is positive. Find $m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac32"
} | {
"split": "train",
"index": 20377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 20378,
"solution": "1. **Identify the highest and lowest prices:** From the problem, we know that the highest price of gasoline was $17 in Month 1, and the lowest price was $10 in Month 3.\n\n2. **Calculate the percentage increase from the lowest to the highest price:** To find by wha... |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 20379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $w$, $x$, $y$, and $z$ are real numbers satisfying: \\begin{align*}\nw+x+y &= -2, \\\\\nw+x+z &= 4, \\\\\nw+y+z &= 19, \\text{ and} \\\\\nx+y+z &= 12,\n\\end{align*} what is $wx + yz$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 20380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\\circ \\text{ N}$ latitude and $115^\\circ \\text{ W}$ longitude. Assume that Earth is a ... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 20381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that fewer than 3 of the flips come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{256}"
} | {
"split": "train",
"index": 20382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 20383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest $K$ satisfying the following:\n\nGiven any closed intervals $A_1,\\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\\ldots, A_N$ such that the inte... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 20384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?\n Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 20385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line with equation $y = x$ is an axis of symmetry of the curve with equation\n\\[y = \\frac{px + q}{rx + s},\\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?\n\n(A) $p + q = 0$\n\n(B) $p + r = 0$\n\n(C) $p + s = 0$\n\n(D) $q + r = 0$... | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"split": "train",
"index": 20386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9448"
} | {
"split": "train",
"index": 20387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the remainder when $$\\sum_{k=1}^{30303} k^{k}$$ is divided by 101. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 20388,
"solution": "The main idea is the following lemma: Lemma. For any non-negative integer $n$ and prime $p, \\sum_{k=n+1}^{n+p^{2}-p} k^{k} \\equiv 1(\\bmod p)$. Proof. Note that $a^{b}$ depends only on the value of $a(\\bmod p)$ and the value of $b(\\bmod p-1)$. Since $p$ and $p-... |
deepscaler | [
{
"role": "user",
"content": "A certain lottery has tickets labeled with the numbers $1,2,3, \\ldots, 1000$. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this... | math | {
"style": "rule",
"ground_truth": "1/501"
} | {
"split": "train",
"index": 20389,
"solution": "Notice that the outcome is the same as if the lottery instead draws all the tickets, in random order, and awards a prize to the holder of the odd ticket drawn earliest and each even ticket drawn before it. Thus, the probability of your winning is the probability th... |
deepscaler | [
{
"role": "user",
"content": "Keiko tosses one penny and Ephraim tosses two pennies. What is the probability that Ephraim gets the same number of heads that Keiko gets? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 20390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of 18 cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario hold... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 20391,
"solution": "Considering ordering the nine pairs by the time they are first complete. Since the pairs are treated equally by the drawing process, this ordering is a uniform ordering. Therefore the problem becomes the following: consider ordering 7 N's and 2 J's randomly. What i... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1089"
} | {
"split": "train",
"index": 20392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 20393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bev is driving from Waterloo, ON to Marathon, ON. She has driven 312 km and has 858 km still to drive. How much farther must she drive in order to be halfway from Waterloo to Marathon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "273 \\mathrm{~km}"
} | {
"split": "train",
"index": 20394,
"solution": "Since Bev has driven 312 km and still has 858 km left to drive, the distance from Waterloo to Marathon is $312 \\mathrm{~km} + 858 \\mathrm{~km} = 1170 \\mathrm{~km}$. The halfway point of the drive is $\\frac{1}{2}(1170 \\mathrm{~km}) = 585 \\mathrm{~km}$ from Wat... |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ,$ $\\angle X = 60^\\circ,$ $\\angle Y = 75^\\circ,$ and $YZ = 6.$ Find $XZ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2} + \\sqrt{6}"
} | {
"split": "train",
"index": 20395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)=2\\cos x-3\\sin x$ reaches its minimum value when $x=\\theta$, calculate the value of $\\tan \\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 20396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid wooden rectangular prism measures $3 \\times 5 \\times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 20397,
"solution": "Consider the triangular-based prism on the front of the rectangular prism. This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back. The rectangle on the front measure... |
deepscaler | [
{
"role": "user",
"content": "Let $\\theta$ be an angle such that $\\sin 2 \\theta = \\frac{1}{3}.$ Compute $\\sin^6 \\theta + \\cos^6 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"split": "train",
"index": 20398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 20399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a function for which $f\\left(\\dfrac{x}{3}\\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1/9"
} | {
"split": "train",
"index": 20401,
"solution": "1. **Identify the function and equation:** Given the function $f\\left(\\frac{x}{3}\\right) = x^2 + x + 1$, we need to find the sum of all values of $z$ for which $f(3z) = 7$.\n\n2. **Relate $f(3z)$ to the given function:** Since $f\\left(\\frac{x}{3}\\right) = x^2... |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral \\(ABCD\\), the lengths of the sides \\(BC\\) and \\(CD\\) are 2 and 6, respectively. The points of intersection of the medians of triangles \\(ABC\\), \\(BCD\\), and \\(ACD\\) form an equilateral triangle. What is the maximum possible area of quadrilateral... | math | {
"style": "rule",
"ground_truth": "29.32"
} | {
"split": "train",
"index": 20402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:\n\n$\\bullet$ Carolyn always has the first turn.\n\n$\\bullet$ Carolyn and Paul alternate turns.\n\n$\\bullet$ On each of her turns, Carolyn must remove one number... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\\frac{4}{7}$, then the number common to both sets of four numbers is Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20404,
"solution": "1. **Calculate the total of the first four numbers**: \n Given that the average of the first four numbers is $5$, the sum of these numbers is:\n \\[\n 4 \\times 5 = 20\n \\]\n\n2. **Calculate the total of the last four numbers**: \n Given that the avera... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all roots of the equation:\n\n$$\n\\begin{gathered}\n\\sqrt{2 x^{2}-2024 x+1023131} + \\sqrt{3 x^{2}-2025 x+1023132} + \\sqrt{4 x^{2}-2026 x+1023133} = \\\\\n= \\sqrt{x^{2}-x+1} + \\sqrt{2 x^{2}-2 x+2} + \\sqrt{3 x^{2}-3 x+3}\n\\end{gathered}\n$$ Let's think ste... | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 20405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\left(\\sqrt[4]{(\\sqrt{5})^5}\\right)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt[4]{5}"
} | {
"split": "train",
"index": 20406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We are approaching a 120-meter high skyscraper on a horizontal road. After traveling 300 meters, we see the building at an angle of elevation that is $45^\\circ$ greater than at the start of our journey. How close have we approached the skyscraper? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 20407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A paperboy delivers newspapers to 11 houses along Main Street. He may skip delivering to any house, but he must not skip three consecutive houses to avoid losing his job. Determine the number of possible delivery sequences available to the paperboy under these conditions. Let's... | math | {
"style": "rule",
"ground_truth": "927"
} | {
"split": "train",
"index": 20408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( 5^{\\log 30} \\times \\left(\\frac{1}{3}\\right)^{\\log 0.5} = d \\), find the value of \\( d \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 20409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cuboid has a diagonal $A A^{\\prime}$. In what ratio does the plane passing through the endpoints $B, C, D$ of the edges originating from vertex $A$ divide the $A A^{\\prime}$ diagonal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 20410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1, a_2, a_3, \\ldots, a_{15})$ be a permutation of $(1, 2, 3, \\ldots, 15)$ for which\n$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \\mathrm{\\ and \\ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$\nFind the number of such permutations. Let's thin... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 20411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial \\( P \\) is of the form \\( \\pm x^{6} \\pm x^{5} \\pm x^{4} \\pm x^{3} \\pm x^{2} \\pm x \\pm 1 \\). Given that \\( P(2)=27 \\), what is \\( P(3) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "439"
} | {
"split": "train",
"index": 20412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $\\sin A = \\sin B = -\\cos C$.\n\n$(1)$ Find the sizes of angles $A$, $B$, and $C$;\n\n$(2)$ If the length of the median $AM$ on side $BC$ is $\\sqrt{7}$, find the are... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 20413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, \"At least half of you guessed too low, two of you are off by one, and my age is a prime number.\" How old is Norb? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 20414,
"solution": "1. **Identify the conditions given by Norb:**\n - At least half of the guesses are too low.\n - Two guesses are off by one.\n - Norb's age is a prime number.\n\n2. **List of guesses:** $24, 28, 30, 32, 36, 38, 41, 44, 47, 49$.\n\n3. **Determine possible value... |
deepscaler | [
{
"role": "user",
"content": "Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and *bigness* $N$ and another one with integer side ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 20415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $18x^2 - mx + 252 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 20416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt[4]{12960000}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 20417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{1}^{e} \\sqrt{x} \\cdot \\ln^{2} x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10e\\sqrt{e} - 16}{27}"
} | {
"split": "train",
"index": 20418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Lauren has 4 sisters and 7 brothers, and her brother Lucas has S sisters and B brothers. Find the product of S and B. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 20419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing? Le... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20420,
"solution": "Since 20 students went skating and 5 students went both skating and skiing, then \\( 20-5=15 \\) students went skating only. Since 9 students went skiing and 5 students went both skating and skiing, then \\( 9-5=4 \\) students went skiing only. The number of studen... |
deepscaler | [
{
"role": "user",
"content": "If two circles $(x-m)^2+y^2=4$ and $(x+1)^2+(y-2m)^2=9$ are tangent internally, then the real number $m=$ ______ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 20421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \\mathrm{~km} / \\mathrm{h}$.\n... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the tr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{3}"
} | {
"split": "train",
"index": 20423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?\n\n[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20424,
"solution": "1. **Understanding the Grid**: The grid is a $4 \\times 4$ square grid, which means there are 16 small squares in total.\n\n2. **Objective**: We need to place the maximum number of $\\text{X}$'s such that no three $\\text{X}$'s are aligned vertically, horizontally,... |
deepscaler | [
{
"role": "user",
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, such that the first outcome is the thousand digit, the second outcome is the hundreds digit, the third outcome is the tens digit, and... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 20425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD = 60^\\circ$. \nThe area of the figure is Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 20426,
"solution": "1. **Identify the Shape and Symmetry**: The given figure can be visualized as composed of several triangles. Notably, the problem states that sides $AF$ and $CD$ are parallel, as are $AB$ and $EF$, and $BC$ and $ED$. Each side has a length of $1$, and the angles $\... |
deepscaler | [
{
"role": "user",
"content": "Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 20427,
"solution": "We see that for every 2 adjacent spots on the table, there is exactly one way for an ambassador and his or her partner to sit. There are 2 cases:\nThere is an ambassador at the 12th chair and his or her partner at the 1st chair\nThere is no pair that has their chai... |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n,$ let\n\\[f(n) = \\frac{1}{2^n} + \\frac{1}{3^n} + \\frac{1}{4^n} + \\dotsb.\\]Find\n\\[\\sum_{n = 2}^\\infty f(n).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 20428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.<br/>① Grouping method:<br/>For example, $x^{2}-2xy+y^{2}-25... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 20429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $5 \\times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 20430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer \\(k\\), denote by \\(S_{k}\\) the set of all visible lattice points \\((x, y)\\) such that \\(x^{2}+y^{2}=k^{2}\\). Let \\... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\\triangle ABD?$\n\n[asy]\ndraw((0,0)--(1,4)--(9,4)--(18,0)--cycle);\ndraw((9,4)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(1,4),NW);\nlabel(\"$B$\",(... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 20432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word TARTAR. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 20433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an international sprinting event, there are 10 sprinters, including 4 Americans. The medals awarded are gold, silver, and bronze. Determine the number of ways the medals can be awarded if exactly two Americans must win medals. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 20434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 4^\\circ + \\sin^2 8^\\circ + \\sin^2 12^\\circ + \\dots + \\sin^2 176^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"split": "train",
"index": 20435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $A = 3009 \\div 3$, $B = A \\div 3$, and $Y = A - 2B$, then what is the value of $Y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 20436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(\\frac{x}{x-1}-1\\right) \\div \\frac{{x}^{2}-1}{{x}^{2}-2x+1}$, where $x=\\sqrt{5}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 20437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "264"
} | {
"split": "train",
"index": 20438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer \\( n \\) such that \\( 5n \\equiv 105 \\pmod{24} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 20439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the expression\n\\[\\log_2 (27 + \\log_2 (27 + \\log_2 (27 + \\cdots))),\\]assuming it is positive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 20440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\\circ}$. After the cuts, the paper is folded up along t... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 20441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$, and it satisfies $S_{2016} > 0$, $S_{2017} < 0$. For any positive integer $n$, we have $|a_n| \\geqslant |a_k|$. Determine the value of $k$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 20442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all real values of $r$ for which $\\frac{1}{2x}=\\frac{r-x}{7}$ has exactly one real solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 20443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral with $AC=7$ and $BD=17$. Let $M, P, N, Q$ be the midpoints of sides $AB, BC, CD, DA$ respectively. Compute $MN^{2}+PQ^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 20444,
"solution": "$MPNQ$ is a parallelogram whose side lengths are 3.5 and 8.5 so the sum of squares of its diagonals is $\\frac{7^{2}+17^{2}}{2}=169$."
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, the curve $y=x^{2}-6x+1$ intersects the coordinate axes at points that lie on circle $C$.\n(1) Find the equation of circle $C$;\n(2) Given point $A(3,0)$, and point $B$ is a moving point on circle $C$, find the maximum value of $\\over... | math | {
"style": "rule",
"ground_truth": "\\frac{36\\sqrt{37}}{37}"
} | {
"split": "train",
"index": 20445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \\leq n \\leq 50$ such that $n$ divides $\\phi^{!}(n)+1$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 20446,
"solution": "Note that, if $k$ is relatively prime to $n$, there exists a unique $0<k^{-1}<n$ such that $k k^{-1} \\equiv 1(\\bmod n)$. Hence, if $k^{2} \\not \\equiv 1(\\bmod n)$, we can pair $k$ with its inverse to get a product of 1. If $k^{2} \\equiv 1(\\bmod n)$, then $(n-... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\sum_{n=1}^{\\infty} \\frac{4n + 1}{(4n - 1)^3 (4n + 3)^3}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{432}"
} | {
"split": "train",
"index": 20447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC, AB = 13, BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. \nThe length of $HM$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 20448,
"solution": "1. **Calculate the semi-perimeter and area of $\\triangle ABC$**:\n - The semi-perimeter, $s$, is given by:\n \\[\n s = \\frac{AB + BC + CA}{2} = \\frac{13 + 14 + 15}{2} = 21\n \\]\n - Using Heron's formula, the area, $[ABC]$, is:\n \\[\n [A... |
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