data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.\n\na) What is the minimum number of socks that can be in the box?\n\nb) What is the minimum number of socks that can be in the box, given t... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers \\(a\\) and \\(b\\) are such that the numbers \\(15a + 165\\) and \\(16a - 155\\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "481"
} | {
"split": "train",
"index": 23350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 23351,
"solution": "Note that an ascending number is exactly determined by its digits: for any set of digits (not including 0, since the only position for 0 is at the leftmost end of the number, i.e. a leading 0), there is exactly one ascending number with those digits.\n So, there ar... |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = \\frac{1}{\\sqrt[3]{1 - x^3}} \\). Find \\( f(f(f( \\ldots f(19)) \\ldots )) \\), calculated 95 times. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{1 - \\frac{1}{19^3}}"
} | {
"split": "train",
"index": 23352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A collection $\\mathcal{S}$ of 10000 points is formed by picking each point uniformly at random inside a circle of radius 1. Let $N$ be the expected number of points of $\\mathcal{S}$ which are vertices of the convex hull of the $\\mathcal{S}$. (The convex hull is the smallest ... | math | {
"style": "rule",
"ground_truth": "72.8"
} | {
"split": "train",
"index": 23353,
"solution": "Here is C++ code by Benjamin Qi to estimate the answer via simulation. It is known that the expected number of vertices of the convex hull of $n$ points chosen uniformly at random inside a circle is $O\\left(n^{1 / 3}\\right)$. See \"On the Expected Complexity of R... |
deepscaler | [
{
"role": "user",
"content": "A finite increasing sequence \\(a_{1}, a_{2}, \\ldots, a_{n}\\) of natural numbers is given, where \\(n \\geq 3\\), and for all \\(k \\leq n-2\\) the following equality holds: \\(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\\). The sequence must include \\(a_{k} = 2022\\). Determine the max... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 23354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations $x^{2} y+y^{2} z =1040$, $x^{2} z+z^{2} y =260$, $(x-y)(y-z)(z-x) =-540$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(16,4,1),(1,16,4)"
} | {
"split": "train",
"index": 23355,
"solution": "Call the three equations $(1),(2),(3) \\cdot(1) /(2)$ gives $y=4 z .(3)+(1)-(2)$ gives $\\left(y^{2}-z^{2}\\right) x=15 z^{2} x=240$ so $z^{2} x=16$. Therefore $z(x+2 z)^{2}=x^{2} z+z^{2} y+4 z^{2} x=\\frac{81}{5}$, $z(x-2 z)^{2}=x^{2} z+z^{2} y-4 z^{2} x=\\frac{49... |
deepscaler | [
{
"role": "user",
"content": "Find the value of $m$ for which the quadratic equation $3x^2 + mx + 36 = 0$ has exactly one solution in $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 23356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man went to a bank to cash a check. The cashier made a mistake and gave him the same number of cents instead of dollars and the same number of dollars instead of cents. The man did not count the money, put it in his pocket, and additionally dropped a 5-cent coin. When he got ... | math | {
"style": "rule",
"ground_truth": "31.63"
} | {
"split": "train",
"index": 23357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\cdots, 10\\} \\), and \\( A_1, A_2, \\cdots, A_n \\) be distinct non-empty subsets of \\( M \\). If \\(i \\neq j\\), then \\( A_i \\cap A_j \\) can have at most two elements. Find the maximum value of \\( n \\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 23358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle $\\triangle ABC$ satisfies $AB = 13$ , $BC = 14$ , and $AC = 15$ . Inside $\\triangle ABC$ are three points $X$ , $Y$ , and $Z$ such that:\n\n\n\n- $Y$ is the centroid of $\\triangle ABX$\n- $Z$ is the centroid of $\\triangle BCY$\n- $X$ is the centro... | math | {
"style": "rule",
"ground_truth": "84/13"
} | {
"split": "train",
"index": 23359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that:\n\\begin{align*} \ng(x, x) &= x, \\\\\ng(x, y) &= g(y, x), \\quad \\text{and} \\\\ \n(x + y) g(x, y) &= y g(x, x + y).\n\\end{align*}\nCalculate $g(15, 33)$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 23360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two concentric spheres of radii 3 units and 6 units. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "252\\pi"
} | {
"split": "train",
"index": 23361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and it is given that $S_4=20$, $S_{n-4}=60$, and $S_n=120$. Find the value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which positive integer values of $k$ does $kx^2+20x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6, 8\\text{, and }10"
} | {
"split": "train",
"index": 23363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $1 = x_{1} < x_{2} < \\dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \\dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$ *Proposed by Justin Lee* ... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 23364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the insi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 23365,
"solution": "1. **Understanding the Problem:**\n Keiko walks around a track with straight sides and semicircular ends. The track has a width of 6 meters. It takes her 36 seconds longer to walk around the outside edge than the inside edge. We need to find her speed.\n\n2. **An... |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the graph of \\( |z - 5| = 3|z + 5| \\) intersects the graph of \\( |z| = k \\) in exactly one point. Find all possible values of \\( k \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 23366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\\dots,x_{2n}$ with $-1 < x_1 < x_2 < \\cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals \\[ [x_1^{2k-1}, x_2^{... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"split": "train",
"index": 23367,
"solution": "The largest such $m$ is $n$. To show that $m \\geq n$, we take \\[ x_j = \\cos \\frac{(2n+1-j)\\pi}{2n+1} \\qquad (j=1,\\dots,2n). \\] It is apparent that $-1 < x_1 < \\cdots < x_{2n} < 1$. The sum of the lengths of the intervals can be interpreted as \\begin{align... |
deepscaler | [
{
"role": "user",
"content": "Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives\n\\[{1000 \\choose 0}(0.2)^0+{1000 \\choose 1}(0.2)^1+{1000 \\choose 2}(0.2)^2+\\cdots+{1000 \\choose 1000}(0.2)^{1000}= A_0 + A_1 + A_2 + \\cdots + A_{1000},\\]where $A_k = {1000 \\choos... | math | {
"style": "rule",
"ground_truth": "166"
} | {
"split": "train",
"index": 23368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 4 students from 7 students, including 4 boys and 3 girls, to participate in an environmental knowledge contest, ensuring both boys and girls are among the selected students. Calculate the number of different selection methods. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 23369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are positive real numbers such that $a \\cdot 2^{b}=8$ and $a^{b}=2$, compute $a^{\\log _{2} a} 2^{b^{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 23370,
"solution": "Taking $\\log _{2}$ of both equations gives $\\log _{2} a+b=3$ and $b \\log _{2} a=1$. We wish to find $a^{\\log _{2} a} 2^{b^{2}}$; taking $\\log _{2}$ of that gives $\\left(\\log _{2} a\\right)^{2}+b^{2}$, which is equal to $\\left(\\log _{2} a+b\\right)^{2}-2 b ... |
deepscaler | [
{
"role": "user",
"content": "Given four real numbers $-9$, $a\\_1$, $a\\_2$, and $-1$ that form an arithmetic sequence, and five real numbers $-9$, $b\\_1$, $b\\_2$, $b\\_3$, and $-1$ that form a geometric sequence, find the value of $b\\_2(a\\_2-a\\_1)=$\\_\\_\\_\\_\\_\\_. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 23371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a four-digit positive integer $\\overline{abcd}$, if $a+c=b+d=11$, then this number is called a \"Shangmei number\". Let $f(\\overline{abcd})=\\frac{{b-d}}{{a-c}}$ and $G(\\overline{abcd})=\\overline{ab}-\\overline{cd}$. For example, for the four-digit positive integer $3... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 23372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "30\\pi - 9\\sqrt3"
} | {
"split": "train",
"index": 23373,
"solution": "1. **Identify the dimensions of the plot and path**: The circular grass plot has a diameter of 12 feet, so its radius \\( r \\) is \\( \\frac{12}{2} = 6 \\) feet. The gravel path is 3 feet wide.\n\n2. **Calculate the area of the entire circle**: The area \\( A \\) ... |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\frac{x}{a} + \\frac{y}{b} + \\frac{z}{c} = 3 \\quad \\text{and} \\quad \\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 0,\\]find $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 23374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a-4\\cos C=c\\cos B$. \n$(1)$ Find the value of $b$; \n$(2)$ If $a^{2}+b^{2}+c^{2}=2\\sqrt{3}ab\\sin C$, find the area of $\\triang... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 23375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain clothing factory produces jackets and $T$-shirts, with each jacket priced at $100$ yuan and each $T$-shirt priced at $60$ yuan. During a promotional period, the factory offers two discount options to customers:<br/>① Buy one jacket and get one $T$-shirt for free;<br/>... | math | {
"style": "rule",
"ground_truth": "3480"
} | {
"split": "train",
"index": 23376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180\\pi"
} | {
"split": "train",
"index": 23377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If Anna flips 8 coins, what is the probability that she gets more heads than tails? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{93}{256}"
} | {
"split": "train",
"index": 23378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $521_8$ and $146_8$ in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "667_8"
} | {
"split": "train",
"index": 23379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $θ$ is a real number, if the complex number $z=\\sin 2θ-1+i( \\sqrt {2}\\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, angle $C$ is trisected by $\\overline{CF}$ and $\\overline{CE}$, where $E$ is on $\\overline{AB}$, $F$ is on $\\overline{AD}$, $BE=6$, and $AF=2$. Find the area of $ABCD$.\n\n[asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); do... | math | {
"style": "rule",
"ground_truth": "108\\sqrt{3}-36"
} | {
"split": "train",
"index": 23381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive 3-digit numbers are divisible by 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 23382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 26 bricks, and two brothers are vying to take them. The younger brother arrives first and arranges the bricks. The elder brother arrives and thinks the younger brother has taken too many, so he takes half of the bricks from the younger brother. The younger brother, fe... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, given $BC=2$, $AC=\\sqrt{7}$, $B=\\dfrac{2\\pi}{3}$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 23384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the edges of the regular tetrahedron \\(P-ABC\\) have length \\(1\\). Let \\( L, M, N \\) be the midpoints of the edges \\( PA, PB, PC \\) respectively. Find the area of the cross-section of the circumsphere of the tetrahedron when cut by the plane \\( LMN \\). Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 23385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\triangle ABC\\) be equilateral, and let \\(D, E, F\\) be points on sides \\(BC, CA, AB\\) respectively, with \\(FA = 9\\), \\(AE = EC = 6\\), and \\(CD = 4\\). Determine the measure (in degrees) of \\(\\angle DEF\\). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid \\(ABCD\\) (\\(AD \\parallel BC\\)) and a rectangle \\(A_{1}B_{1}C_{1}D_{1}\\) are inscribed in a circle \\(\\Omega\\) with a radius of 13 such that \\(AC \\parallel B_{1}D_{1}\\) and \\(BD \\parallel A_{1}C_{1}\\). Find the ratio of the areas of \\(ABCD\\) and \\(A... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three real numbers \\( x, y, \\) and \\( z \\) are such that \\( (x+4) / 2=(y+9) /(z-3)=(x+5) /(z-5) \\). Determine the value of \\( x / y \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 23388,
"solution": "Because the first and third fractions are equal, adding their numerators and denominators produces another fraction equal to the others: \\( ((x+4)+(x+5)) /(2+(z-5))=(2 x+9) /(z-3) \\). Then \\( y+9=2 x+9 \\), etc."
} |
deepscaler | [
{
"role": "user",
"content": "On the board, two sums are written:\n\n\\[1+22+333+4444+55555+666666+7777777+88888888+999999999\\]\n\n\\[9+98+987+9876+98765+987654+9876543+98765432+987654321\\]\n\nDetermine which one is greater (or if they are equal). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "1097393685"
} | {
"split": "train",
"index": 23389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02, with at least one coin of each type. How many dimes must you have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 23390,
"solution": "1. **Identify the minimum number of coins and their values**: We know that there is at least one coin of each type (penny, nickel, dime, and quarter). The minimum value from one of each type is:\n \\[\n 1 \\text{ cent (penny)} + 5 \\text{ cents (nickel)} + 10 \... |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a\\_n\\}$, it is given that $a\\_3 + a\\_4 = 12$ and $S\\_7 = 49$.\n(I) Find the general term formula for the sequence $\\{a\\_n\\}$.\n(II) Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[0.9] = 0$ and $[2.6] = 2$. Define a... | math | {
"style": "rule",
"ground_truth": "5445"
} | {
"split": "train",
"index": 23391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$. [asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 23392,
"solution": "Call the midpoint of $\\overline{AB}$ $M$ and the midpoint of $\\overline{CD}$ $N$. $d$ is the median of triangle $\\triangle CDM$. The formula for the length of a median is $m=\\sqrt{\\frac{2a^2+2b^2-c^2}{4}}$, where $a$, $b$, and $c$ are the side lengths of trian... |
deepscaler | [
{
"role": "user",
"content": "Given a cubic polynomial \\( q(x) \\) that satisfies \\( q(3) = 2 \\), \\( q(8) = 20 \\), \\( q(18) = 12 \\), and \\( q(25) = 32 \\). Find the sum \\( q(4) + q(5) + \\ldots + q(26) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "391"
} | {
"split": "train",
"index": 23393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four non-coplanar points \\(A, B, C, D\\) in space where the distances between any two points are distinct, consider a plane \\(\\alpha\\) that satisfies the following properties: The distances from three of the points \\(A, B, C, D\\) to \\(\\alpha\\) are equal, while th... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 23394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a plane with a Cartesian coordinate system, there are 16 grid points \\((i, j)\\), where \\(0 \\leq i \\leq 3\\) and \\(0 \\leq j \\leq 3\\). If \\(n\\) points are selected from these 16 points, there will always exist 4 points among the \\(n\\) points that are the vertices ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A=\\{-3,-2,-1,0,1,2,3\\}$, $a$ and $b$ are elements of $A$. How many cases are there where $|a| \\lt |b|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line $x+ \\sqrt {3}y+c=0$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 23397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of a circle $x^2 + y^2 - 6x - 8y = 0$, if the longest chord AC and the shortest chord BD both pass through the point (-1, 4) on the circle, find the area of the quadrilateral ABCD. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers $x$, $y$ satisfying $xy= \\frac{x-y}{x+3y}$, find the maximum value of $y$. 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": 23399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$ ) such that there exists a convex $n$ -gon $A_{1}A_{2}\\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$ .) Le... | math | {
"style": "rule",
"ground_truth": "k = \\left\\lfloor \\frac{n}{2} \\right\\rfloor"
} | {
"split": "train",
"index": 23400,
"solution": "Lemma: If quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ and $A_{i+2}A_{i+3}A_{i+4}A_{i+5}$ in an equiangular $n$ -gon are tangential, and $A_iA_{i+3}$ is the longest side quadrilateral $A_iA_{i+1}A_{i+2}A_{i+3}$ for all $i$ , then quadrilateral $A_{i+1}A_{i+2}A_{i+3}A_... |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 23401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{13}"
} | {
"split": "train",
"index": 23402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $z$ in the following equation: $1-iz = -1 + iz$ (where $i^2 = -1$). Simplify your answer as much as possible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"split": "train",
"index": 23403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\log_{10}(\\tan 1^{\\circ})+\\log_{10}(\\tan 2^{\\circ})+\\log_{10}(\\tan 3^{\\circ})+\\cdots+\\log_{10}(\\tan 88^{\\circ})+\\log_{10}(\\tan 89^{\\circ}).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(C\\) be the circle with the equation \\(x^2 - 4y - 18 = -y^2 + 6x + 26\\). Find the center \\((a, b)\\) and radius \\(r\\) of the circle, and compute \\(a + b + r\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{57}"
} | {
"split": "train",
"index": 23405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the lateral area of a cylinder with a square cross-section is $4\\pi$, calculate the volume of the cylinder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 23406,
"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"
} | {
"split": "train",
"index": 23407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number of different integers that can be selected from the set $ \\{1,2,...,2013\\}$ so that no two exist that their difference equals to $17$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 23408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between 100 and 300 have both 11 and 8 as factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\\circ}$ right turn and sprints upwards for 3 meters. Calculate the average o... | math | {
"style": "rule",
"ground_truth": "5.75"
} | {
"split": "train",
"index": 23410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $ 2^{2004}$ is a $ 604$ -digit number whose first digit is $ 1$ , how many elements of the set $ S \\equal{} \\{2^0,2^1,2^2, \\ldots,2^{2003}\\}$ have a first digit of $ 4$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 23411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which integer $a$ does $x^2 - x + a$ divide $x^{13} + x + 90$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows.... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of points that are inside or within two units of a rectangular parallelepiped that measures 2 by 3 by 6 units. Calculate the total volume of this set, expressing your answer in the form $\\frac{m+n\\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n... | math | {
"style": "rule",
"ground_truth": "\\frac{540 + 164\\pi}{3}"
} | {
"split": "train",
"index": 23414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the two distinct square roots of a positive number $a$ are $2x-2$ and $6-3x$, find the cube root of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{36}"
} | {
"split": "train",
"index": 23415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \\( y = |x| \\) and \\( y = -x^2 + 8 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line \\(y = -\\frac{1}{2}x + 8\\) crosses the \\(x\\)-axis at \\(P\\) and the \\(y\\)-axis at \\(Q\\). Point \\(T(r, s)\\) is on line segment \\(PQ\\). If the area of \\(\\triangle POQ\\) is twice the area of \\(\\triangle TOP\\), then what is the value of \\(r+s\\)? Let's ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 23418,
"solution": "We can obtain $n=7$ in the following way: Consider a rhombus $A B C D$ made up of two equilateral triangles of side length 1 , where $\\angle D A B=60^{\\circ}$. Rotate the rhombus clockwise about $A$ to obtain a new rhombus $A B^{\\prime} C^{\\prime} D^{\\prime}$ ... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 23419,
"solution": "Using Simon's Favorite Factoring Trick, we rewrite as $[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$. Both binomial expansions will contain $n+1$ non-like terms; their product will contain $(n+1)^2$ terms, as each term will have an unique power of $x$ or $y$ and so none of the ... |
deepscaler | [
{
"role": "user",
"content": "An six-digit integer is formed by repeating a positive three-digit integer. For example, 123,123 or 456,456 are integers of this form. What is the greatest common divisor of all six-digit integers of this form? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 23420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 23421,
"solution": "1. **Calculate the volume of the first box**: \n The volume $V$ of a rectangular prism is calculated by the formula:\n \\[\n V = \\text{height} \\times \\text{width} \\times \\text{length}\n \\]\n For the first box, the dimensions are given as height = $... |
deepscaler | [
{
"role": "user",
"content": "A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23422,
"solution": "1. **Identify the total width and picture width**: The wall is $19$ feet wide and the picture is $3$ feet wide.\n\n2. **Set up the equation**: The picture is centered, so the distances from each end of the wall to the nearest edge of the picture are equal. Let $x$ ... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} -5 & 3 \\\\ 4 & -4 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\\{a, b, c\\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice ca... | math | {
"style": "rule",
"ground_truth": "\\frac{3-3^{-999}}{4}"
} | {
"split": "train",
"index": 23424,
"solution": "Let $n_{a}, n_{b}$, and $n_{c}$ be the number of $a$ 's, $b$ 's, and $c$ 's on the board, respectively. The key observation is that each move always changes the parity of all three of $n_{a}, n_{b}$, and $n_{c}$. Since the final configuration must have $n_{a}, n_{b... |
deepscaler | [
{
"role": "user",
"content": "The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13} : 2\\sqrt{10} : 3\\sqrt{5}"
} | {
"split": "train",
"index": 23425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that a triangle formed by randomly selecting 3 vertices from a regular octagon is a right triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 23426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange the numbers \\(\\{1,2,3,4,5,6,7,8\\}\\) in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23427,
"solution": "Note that 6 can only be adjacent to 1, 5, and 7, so there are \\(\\binom{3}{2}=3\\) ways to pick its neighbors. Since each of 1, 5, and 7 is relatively prime to every number in \\(\\{1,2,3,4,5,6,7,8\\}\\) but itself (and hence can have arbitrary neighbors), without... |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\) and \\( b \\) be positive integers. The quotient of \\( a^{2} + b^{2} \\) divided by \\( a + b \\) is \\( q \\), and the remainder is \\( r \\), such that \\( q^{2} + r = 2010 \\). Find the value of \\( ab \\). Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "1643"
} | {
"split": "train",
"index": 23428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $0 \\leq x_0 < 1$, for all integers $n > 0$, let\n$$\nx_n = \\begin{cases}\n2x_{n-1}, & \\text{if } 2x_{n-1} < 1,\\\\\n2x_{n-1} - 1, & \\text{if } 2x_{n-1} \\geq 1.\n\\end{cases}\n$$\nFind the number of initial values of $x_0$ such that $x_0 = x_6$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 23429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed b... | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"split": "train",
"index": 23430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23431,
"solution": "1. **Identify Symmetry in Function**: Given the function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. This implies that $f(x)$ is symmetric about $x = 2$.\n\n2. **Roots of the Function**: We know that $f(x) = 0$ has exactly four distinct real roots.... |
deepscaler | [
{
"role": "user",
"content": "\nCompute the definite integral:\n\n$$\n\\int_{-\\pi}^{0} 2^{8} \\sin ^{6} x \\cos ^{2} x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"split": "train",
"index": 23432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the length of side $y$ in the following diagram?\n\n[asy]\nimport olympiad;\ndraw((0,0)--(2,0)--(0,2*sqrt(3))--cycle); // modified triangle lengths\ndraw((0,0)--(-2,0)--(0,2*sqrt(3))--cycle);\nlabel(\"10\",(-1,2*sqrt(3)/2),NW); // changed label\nlabel(\"$y$\",(2/2,2*sqr... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 23433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain school, 3 teachers are chosen from a group of 6 to give support teaching in 3 remote areas, with each area receiving one teacher. There are restrictions such that teacher A and teacher B cannot go together, and teacher A can only go with teacher C or not go at all.... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 23434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ where $a_1 = \\frac{1}{2}$ and $a_{n+1} = \\frac{1+a_n}{1-a_n}$ for $n \\in N^*$, find the smallest value of $n$ such that $a_1+a_2+a_3+…+a_n \\geqslant 72$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "238"
} | {
"split": "train",
"index": 23435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A system of inequalities defines a region on a coordinate plane as follows:\n$$\n\\begin{cases}\nx+y \\leq 5 \\\\\n3x+2y \\geq 3 \\\\\nx \\geq 1 \\\\\ny \\geq 1\n\\end{cases}\n$$\nDetermine the number of units in the length of the longest side of the quadrilateral formed by the... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 23436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers greater than 6 but less than 60 are relatively prime to 15? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 23437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose two distinct competitors of the HMMT 2021 November contest are chosen uniformly at random. Let $p$ be the probability that they can be labelled $A$ and $B$ so that $A$ 's score on the General round is strictly greater than $B$ 's, and $B$ 's score on the theme round is ... | math | {
"style": "rule",
"ground_truth": "2443"
} | {
"split": "train",
"index": 23438,
"solution": "If competitors' scores on the General and Theme rounds were completely uncorrelated, we would expect the answer to be approximately $\\frac{1}{2}$. If they were maximally correlated, we would expect the answer to be exactly 0 . It turns out that guessing $\\frac{1}... |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a natural number. Find the smallest constant \\( C \\) such that for all \\( x_{1}, \\ldots, x_{n} \\geqslant 0 \\),\n\n\\[\n\\sum_{1 \\leqslant i<j \\leqslant n} x_{i} x_{j}\\left(x_{i}^{2}+x_{j}^{2}\\right) \\leqslant C\\left(\\sum_{i=1}^{n} x_{i}\\right)^{4}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 23439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$ . Define $\\{x \\} = x - \\lfloor x \\rfloor$ . For example, $\\{ 3 \\} = 3-3 = 0$ , $\\{ \\pi \\} = \\pi - 3$ , and $\\{ - \\pi \\} = 4-\\pi$ . If $\\{n\\} + \\{ 3n\\} = 1.4$ , then find the ... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 23440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are two interior angles of an oblique triangle, if $\\frac{{\\cos \\alpha - \\sin \\alpha}}{{\\cos \\alpha + \\sin \\alpha}} = \\cos 2\\beta$, then the minimum value of $\\tan \\alpha + \\tan \\beta$ is ______. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 23441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, sides $\\overline{AB}$ and $\\overline{CD}$ are parallel, $\\angle A = 2\\angle D$, and $\\angle C = 3\\angle B$. Find $\\angle B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45^\\circ"
} | {
"split": "train",
"index": 23442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many terms are in the expansion of \\[(a+b+c)(d+e+f+g)?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "9.80"
} | {
"split": "train",
"index": 23444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\cos 2x + \\cos 6x + \\cos 10x + \\cos 14x$ can be written in the equivalent form\n\\[a \\cos bx \\cos cx \\cos dx\\] for some positive integers $a$, $b$, $c$, and $d$. Find $a+b+c+d$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two buildings facing each other, each 5 stories high. How many ways can Kevin string ziplines between the buildings so that: (a) each zipline starts and ends in the middle of a floor. (b) ziplines can go up, stay flat, or go down, but can't touch each other (this incl... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 23446,
"solution": "Associate with each configuration of ziplines a path in the plane as follows: Suppose there are $k$ ziplines. Let $a_{0}, \\ldots, a_{k}$ be the distances between consecutive ziplines on the left building ($a_{0}$ is the floor on which the first zipline starts, and... |
deepscaler | [
{
"role": "user",
"content": "A plan is to transport 1240 tons of goods A and 880 tons of goods B using a fleet of trucks to a certain location. The fleet consists of two different types of truck carriages, A and B, with a total of 40 carriages. The cost of using each type A carriage is 6000 yuan, and the c... | math | {
"style": "rule",
"ground_truth": "26.8"
} | {
"split": "train",
"index": 23447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest root of the equation $|\\sin (2 \\pi x) - \\cos (\\pi x)| = ||\\sin (2 \\pi x)| - |\\cos (\\pi x)|$, which belongs to the interval $\\left(\\frac{1}{4}, 2\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 23448,
"solution": ""
} |
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