data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\binom{12}{9}$ and then find the factorial of the result. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 25250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a transatlantic flight, three flight attendants are selected by lot from 20 girls competing for these positions. Seven of them are blondes, and the rest are brunettes. What is the probability that among the three chosen flight attendants there will be at least one blonde an... | math | {
"style": "rule",
"ground_truth": "0.718"
} | {
"split": "train",
"index": 25251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real solutions for $x$ to the equation $\\left(x^{2}+2 x+3\\right)^{\\left(x^{2}+2 x+3\\right)^{\\left(x^{2}+2 x+3\\right)}}=2012$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 25252,
"solution": "When $y=x^{2}+2 x+3$, note that there is a unique real number $y$ such that $y^{y^{y}}=2012$ because $y^{y^{y}}$ is increasing in $y$. The sum of the real distinct solutions of the equation $x^{2}+2 x+3=y$ is -2 by Vieta's Formulae as long as $2^{2}+4(y-3)>0$, whic... |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "152"
} | {
"split": "train",
"index": 25253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $n$ is $5 \\times 8 \\times 2 \\times n = 7!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 25254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=|x-2|+|x-a^{2}|$. \n$(1)$ If the inequality $f(x)\\leqslant a$ has solutions for $x$, find the range of the real number $a$; \n$(2)$ If the positive real numbers $m$, $n$ satisfy $m+2n=a$, when $a$ takes the maximum value from $(1)$, find the minimum ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3}{2}+ \\sqrt {2}"
} | {
"split": "train",
"index": 25255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many diagonals can be drawn for a hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 25256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $1+3+5+\\cdots+2017+2019-2-4-6-\\cdots-2016-2018$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 25257,
"solution": "We are given the expression:\n\\[ 1+3+5+\\cdots+2019 - (2+4+6+\\cdots+2018) \\]\n\n1. **Identify the sequences**:\n - The first sequence is the sum of all odd numbers from 1 to 2019.\n - The second sequence is the sum of all even numbers from 2 to 2018.\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "We have that $\\lceil x \\rceil - \\lfloor x \\rfloor = 0.$ Then, what is $\\lceil x \\rceil - x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 25258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Insert two numbers between 1 and 2 to form an arithmetic sequence. What is the common difference? 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": 25259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( (2^{a})(2^{b})=64 \\), what is the mean (average) of \\( a \\) and \\( b \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25260,
"solution": "Since \\( (2^{a})(2^{b})=64 \\), then \\( 2^{a+b}=64 \\), using an exponent law. Since \\( 64=2^{6} \\), then \\( 2^{a+b}=2^{6} \\) and so \\( a+b=6 \\). Therefore, the average of \\( a \\) and \\( b \\) is \\( \\frac{1}{2}(a+b)=3 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "(2014•Shanghai) In a certain game, the scores are 1, 2, 3, 4, 5. The random variable $\\xi$ represents Xiao Bai's score in this game. If $E(\\xi) = 4.2$, then the probability that Xiao Bai scores 5 points is at least ___. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"split": "train",
"index": 25261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the hypotenuse of a right triangle, a square is constructed externally. Find the distance from the center of this square to the vertex of the right angle, given the legs of the triangle are 3 and 5. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{8.5}"
} | {
"split": "train",
"index": 25262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=kx+1$, the maximum length of the chord intercepted by the ellipse $\\frac{x^{2}}{4}+y^{2}=1$ as $k$ varies, determine the maximum length of the chord. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 25263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a break, a fly entered the math classroom and began crawling on a poster depicting the graph of a quadratic function \\( y = f(x) \\) with a leading coefficient of -1. Initially, the fly moved exactly along the parabola to the point with an abscissa of 2, but then starte... | math | {
"style": "rule",
"ground_truth": "6070"
} | {
"split": "train",
"index": 25264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a non-zero common difference, the sum of the first $n$ terms is $S_n$, satisfying $S_3 = a_4 + 4$, and $a_2$, $a_6$, $a_{18}$ form a geometric sequence.\n1. Find the general formula for the sequence $\\{a_n\\}$.\n2. Let $b_n = \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 25265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The two lines $y = 4x - 19$ and $2x+y = 95$ intersect. What is the value of $x$ at the point of intersection? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 25266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the adjoining figure, $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$... | math | {
"style": "rule",
"ground_truth": "15^\\circ"
} | {
"split": "train",
"index": 25267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified, the third term in the expansion of $(\\frac{a}{\\sqrt{x}} - \\frac{\\sqrt{x}}{a^2})^6$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{x}"
} | {
"split": "train",
"index": 25268,
"solution": "1. **Identify the terms in the binomial expansion**: We are given the expression \\((\\frac{a}{\\sqrt{x}} - \\frac{\\sqrt{x}}{a^2})^6\\). We need to find the third term in this expansion.\n\n2. **Apply the Binomial Theorem**: The Binomial Theorem states that \\((x+... |
deepscaler | [
{
"role": "user",
"content": "In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\\alpha$, $\\beta$, and $\\alpha + \\beta$ radians, respectively, where $\\alpha + \\beta < \\pi$. If $\\cos \\alpha$, which is a positive rational number, is expressed as a fraction in lowest term... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 25269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription \"MAME\" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a r... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 25270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 25271,
"solution": "We see that $f_{1}(11)=4$\n$f_2(11) = f_1(4)=16$\n$f_3(11) = f_1(16)=49$\n$f_4(11) = f_1(49)=169$\n$f_5(11) = f_1(169)=256$\n$f_6(11) = f_1(256)=169$\n Note that this revolves between the two numbers. Since $1988$ is even, we thus have $f_{1988}(11) = f_{4}(11) = \... |
deepscaler | [
{
"role": "user",
"content": "An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "48\\%"
} | {
"split": "train",
"index": 25272,
"solution": "1. **Identify the percentage of beads and coins:**\n Given that 20% of the objects in the urn are beads, the remaining objects must be coins. Therefore, the percentage of coins in the urn is:\n \\[\n 100\\% - 20\\% = 80\\%\n \\]\n\n2. **Determine the compos... |
deepscaler | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \\mathrm{~cm}^{2}$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25274,
"solution": "1. **Substitute the given values into the expression ignoring parentheses:** \n Larry ignored the parentheses, so he calculated the expression as:\n \\[\n a - b - c - d + e = 1 - 2 - 3 - 4 + e\n \\]\n Simplifying this, we get:\n \\[\n -8 + e\n \\]\... |
deepscaler | [
{
"role": "user",
"content": "Let $g(n)$ be the product of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \\le n \\le 50$? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 25275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Josef, Timothy, and Anna play a game. Josef picks an integer between 1 and 1440 inclusive. Timothy states whether the quotient of 1440 divided by Josef's integer is an integer. Finally, Anna adds a twist by stating whether Josef's integer is a multiple of 5. How many integers c... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 25276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marty wants to paint a box. He can choose to use either blue, green, yellow, or black paint. Also, he can style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the five-digit number abcde, \\(a, b, c, d, e\\) respectively represent the digits in the ten-thousands, thousands, hundreds, tens, and units places. Given that \\(d > e\\), \\(c > d + e\\), \\(b > c + d + e\\), and \\(a > b + c + d + e\\), what is the largest five-digit num... | math | {
"style": "rule",
"ground_truth": "95210"
} | {
"split": "train",
"index": 25278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at least once, but 430 is not.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 25279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least five-digit positive integer which is congruent to 6 (mod 17)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10,017"
} | {
"split": "train",
"index": 25280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, angles $\\alpha$ and $\\beta$ both start from $Ox$, and their terminal sides are symmetric about the $y$-axis. If the terminal side of angle $\\alpha$ passes through the point $(3,4)$, then $\\tan (\\alpha-\\beta)=$ ______. Let's think ... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {24}{7}"
} | {
"split": "train",
"index": 25281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a number less than -2, which of the following expressions has the least value: $x$, $x+2$, $\\frac{1}{2}x$, $x-2$, or $2x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x"
} | {
"split": "train",
"index": 25282,
"solution": "For any negative real number $x$, the value of $2x$ will be less than the value of $\\frac{1}{2}x$. Therefore, $\\frac{1}{2}x$ cannot be the least of the five values. Thus, the least of the five values is either $x-2$ or $2x$. When $x < -2$, we know that $2x - (x-2... |
deepscaler | [
{
"role": "user",
"content": "The arithmetic sequence \\( a, a+d, a+2d, a+3d, \\ldots, a+(n-1)d \\) has the following properties:\n\n- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.\n- When the first, fourth, seventh, and so on, terms are added, u... | math | {
"style": "rule",
"ground_truth": "608"
} | {
"split": "train",
"index": 25283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three of the four endpoints of the axes of an ellipse are, in some order, \\[(10, -3), \\; (15, 7), \\; (25, -3).\\] Find the distance between the foci of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.18"
} | {
"split": "train",
"index": 25284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball team consists of 18 players, including a set of 3 triplets: Bob, Bill, and Ben; and a set of twins: Tim and Tom. In how many ways can we choose 7 starters if exactly two of the triplets and one of the twins must be in the starting lineup? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "4290"
} | {
"split": "train",
"index": 25285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that\n\\[\n\\cos^7 \\theta = a_1 \\cos \\theta + a_2 \\cos 2 \\theta + a_3 \\cos 3 \\theta + a_4 \\cos 4 \\theta + a_5 \\cos 5 \\theta + a_6 \\cos 6 \\theta + a_7 \\cos 7 \\theta\n\\]\nfor all angles $\\theta.$ Find... | math | {
"style": "rule",
"ground_truth": "\\frac{1716}{4096}"
} | {
"split": "train",
"index": 25286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\\left\\{1, a, a^{2}, \\ldots\\right\\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25288,
"solution": "Partition the odd residues mod 1024 into 10 classes: Class 1: $1(\\bmod 4)$. Class $n(2 \\leq n \\leq 9): 2^{n}-1\\left(\\bmod 2^{n+1}\\right)$. Class 10: $-1(\\bmod 1024)$. Let $S_{a}$ be the power cycle generated by $a$. If $a$ is in class 1, all of $S_{a}$ is in... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a $2n \\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{(2n)!}{4^n n!}"
} | {
"split": "train",
"index": 25289,
"solution": "The expected value equals \\[ \\frac{(2n)!}{4^n n!}. \\] Write the determinant of $A-A^t$ as the sum over permutations $\\sigma$ of $\\{1,\\dots,2n\\}$ of the product \\[ \\sgn(\\sigma) \\prod_{i=1}^{2n} (A-A^t)_{i \\sigma(i)} = \\sgn(\\sigma) \\prod_{i=1}^{2n} (A_... |
deepscaler | [
{
"role": "user",
"content": "Given $\\sqrt{15129}=123$ and $\\sqrt{x}=0.123$, calculate the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.015129"
} | {
"split": "train",
"index": 25290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45^\\circ"
} | {
"split": "train",
"index": 25291,
"solution": "\nGiven a square \\(ABCD\\) with side length \\(1\\), points \\(P\\) and \\(Q\\) are on sides \\(AB\\) and \\(AD\\) respectively. We are to find the angle \\( \\angle PCQ \\) given that the perimeter of \\( \\triangle APQ \\) is \\(2\\).\n\nLet \\( AP = x \\) and \... |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\heartsuit$ and the second card dealt is a $\\clubsuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"split": "train",
"index": 25292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $x$ that, when multiplied by $400$, produces a product that is a multiple of $576$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 25294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Within a cube with edge length 6, there is a regular tetrahedron with edge length \\( x \\) that can rotate freely inside the cube. What is the maximum value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 25295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $|z_{a} z_{b}+z_{c} z_{d}|$ where $\\{a, b, c, d\\}=\\{1,2,3,4\\}$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 25296,
"solution": "Note that $\\frac{1}{16} f(2 x)=x^{4}+7 x^{3}+13 x^{2}+7 x+1$. Because the coefficients of this polynomial are symmetric, if $r$ is a root of $f(x)$ then $\\frac{4}{r}$ is as well. Further, $f(-1)=-1$ and $f(-2)=16$ so $f(x)$ has two distinct roots on $(-2,0)$ and ... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\left(\\left((3+2)^{-1}-1\\right)^{-1}-1\\right)^{-1}-1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{13}{9}"
} | {
"split": "train",
"index": 25297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of \\(x^8\\) in the polynomial expansion of \\((1-x+2x^2)^5\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 25299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $I$ be the incenter of $\\triangle ABC$ , and $O$ be the excenter corresponding to $B$ . If $|BI|=12$ , $|IO|=18$ , and $|BC|=15$ , then what is $|AB|$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the letters in the English word \"good\" are written in the wrong order, then the total number of possible mistakes is _____ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, \\) and \\( z \\) be real numbers such that \\(\\frac{4}{x} + \\frac{2}{y} + \\frac{1}{z} = 1\\). Find the minimum of \\( x + 8y + 4z \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 25302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begi... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 25303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles \\(C_{1}\\) and \\(C_{2}\\) touch each other externally and the line \\(l\\) is a common tangent. The line \\(m\\) is parallel to \\(l\\) and touches the two circles \\(C_{1}\\) and \\(C_{3}\\). The three circles are mutually tangent. If the radius of \\(C_{2}\\) is... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square fee... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 25305,
"solution": "1. **Calculate the total surface area of the cube**: \n The cube has 6 faces, and each face is a square with an edge length of 10 feet. The area of one face is:\n \\[\n 10 \\times 10 = 100 \\text{ square feet}\n \\]\n Therefore, the total surface area of ... |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\alpha = 3$, evaluate the expression $2\\sin ^{2}\\alpha + 4\\sin \\alpha \\cos \\alpha - 9\\cos ^{2}\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{21}{10}"
} | {
"split": "train",
"index": 25306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f_{1}(x)=-\\frac{2x+7}{x+3}, \\) and \\( f_{n+1}(x)=f_{1}(f_{n}(x)), \\) for \\( x \\neq -2, x \\neq -3 \\), find the value of \\( f_{2022}(2021) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 25307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can one fill a \\(4 \\times 4\\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 25308,
"solution": "First we name the elements of the square as follows: \\(a_{11}, a_{12}, a_{13}, a_{14}, a_{21}, a_{22}, a_{23}, a_{24}, a_{31}, a_{32}, a_{33}, a_{34}, a_{41}, a_{42}, a_{43}, a_{44}\\). We claim that for any given values of \\(a_{11}, a_{12}, a_{13}, a_{21}, a_{22... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the following expressions:\n$(1)(2 \\frac{7}{9})^{0.5}+0.1^{-2}+(2 \\frac{10}{27})\\,^{- \\frac{2}{3}}-3π^{0}+ \\frac{37}{48}$;\n$(2)(-3 \\frac{3}{8})\\,^{- \\frac{2}{3}}+(0.002)\\,^{- \\frac{1}{2}}-10( \\sqrt{5}-2)^{-1}+( \\sqrt{2}- \\sqrt{3})^{0}$. Let'... | math | {
"style": "rule",
"ground_truth": "- \\frac{167}{9}"
} | {
"split": "train",
"index": 25309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 25310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If A, B, and C stand in a row, calculate the probability that A and B are adjacent. 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": 25311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the tetrahedron \\( ABCD \\), points \\( X \\) and \\( Y \\) are given. The distances from point \\( X \\) to the faces \\( ABC, ABD, ACD, BCD \\) are \\( 14, 11, 29, 8 \\) respectively. The distances from point \\( Y \\) to the faces \\( ABC, ABD, ACD, BCD \\) are \\( 1... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 25312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $P = \\sqrt{1988 \\cdot 1989 \\cdot 1990 \\cdot 1991 + 1} + \\left(-1989^{2}\\right)$, then the value of $P$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"split": "train",
"index": 25313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest natural number \\( n \\) such that\n\n\\[ 4^{995} + 4^{1500} + 4^{n} \\]\n\nis a square number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 25314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are three balls, each labeled with the numbers $-1$, $0$, and $\\frac{1}{3}$, respectively. These balls are identical except for the numbers on them. Now, a ball is randomly drawn from the bag, and the number on it is denoted as $m$. After putting the ba... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 25315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y, n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^{n}-y^{n}=2^{100}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 25316,
"solution": "Break all possible values of $n$ into the four cases: $n=2, n=4, n>4$ and $n$ odd. By Fermat's theorem, no solutions exist for the $n=4$ case because we may write $y^{4}+\\left(2^{25}\\right)^{4}=x^{4}$. We show that for $n$ odd, no solutions exist to the more gene... |
deepscaler | [
{
"role": "user",
"content": "If $x - 2x + 3x = 100$, 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": "50"
} | {
"split": "train",
"index": 25317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a, b$, and $c$ are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 25318,
"solution": "Let $d$ be a fourth random variable, also chosen uniformly from $[0,1]$. For fixed $a, b$, and $c$, the probability that $d<\\min \\{a, b, c\\}$ is evidently equal to $\\min \\{a, b, c\\}$. Hence, if we average over all choices of $a, b, c$, the average value of $\... |
deepscaler | [
{
"role": "user",
"content": "There are 6 blue, 7 red, and 9 white light bulbs. In how many ways can you arrange them (using all the light bulbs) in a garland so that no two white light bulbs are consecutive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3435432"
} | {
"split": "train",
"index": 25319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the altern... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 25320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains 5 black ties, 7 gold ties, and 8 pink ties. What is the probability that Stephen randomly chooses a pink tie? 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": 25321,
"solution": "There are \\(5+7+8=20\\) ties in the box, 8 of which are pink. When Stephen removes a tie at random, the probability of choosing a pink tie is \\(\\frac{8}{20}\\) which is equivalent to \\(\\frac{2}{5}\\)."
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\), point \\(D\\) is located on the extension of hypotenuse \\(BC\\) such that line \\(AD\\) is tangent to the circumcircle \\(\\omega\\) of triangle \\(ABC\\). Line \\(AC\\) intersects the circumcircle of triangle \\(ABD\\) at point \\(E\\). It tu... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 25322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A departs from location A to location B, while persons B and C depart from location B to location A. After A has traveled 50 kilometers, B and C start simultaneously from location B. A and B meet at location C, and A and C meet at location D. It is known that A's speed i... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 25323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of \\(\\frac{5}{4\\sqrt{7} - 3\\sqrt{2}}\\) and write your answer in the form \\(\\displaystyle \\frac{A\\sqrt{B} + C\\sqrt{D}}{E}\\), where \\(B < D\\), the fraction is in lowest terms, and all radicals are in simplest radical form. What is \\(A+B+C... | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 25324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \\mathrm{~cm}$, and the son's step length is $60 \\mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What dis... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 25325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{(31)(30)(29)(28)+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "869"
} | {
"split": "train",
"index": 25326,
"solution": "Note that the four numbers to multiply are symmetric with the center at $29.5$. Multiply the symmetric pairs to get $31\\cdot 28=868$ and $30\\cdot 29=870$. $\\sqrt{868\\cdot 870 + 1} = \\sqrt{(869-1)(869+1) + 1} = \\sqrt{869^2 - 1^2 + 1} = \\sqrt{869^2} = \\boxed{... |
deepscaler | [
{
"role": "user",
"content": "A bar of chocolate is made of 10 distinguishable triangles as shown below. How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1689"
} | {
"split": "train",
"index": 25327,
"solution": "Every way to divide the bar can be described as a nonempty set of edges to break, with the condition that every endpoint of a broken edge is either on the boundary of the bar or connects to another broken edge. Let the center edge have endpoints $X$ and $Y$. We do ... |
deepscaler | [
{
"role": "user",
"content": "Two distinct positive integers $x$ and $y$ are factors of 36. If $x\\cdot y$ is not a factor of 36, what is the smallest possible value of $x\\cdot y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 25328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the proportion 3:5 = 6:10, if 3 is changed to 12, determine the new value of 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(0 \\leq a_{k} \\leq 1 \\) for \\(k=1,2, \\ldots, 2020\\), and defining \\(a_{2021} = a_{1}\\), \\(a_{2022} = a_{2}\\), find the maximum value of \\(\\sum_{k=1}^{2020}\\left(a_{k} - a_{k+1}a_{k+2}\\right)\\). Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 25330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $\\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \\ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\\pm 1$-sequence, there exists an integer $k$ and indices $1 \\le t_1 < \\ldots < t_k \\le 2022$ so that $t_{i+1} ... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"split": "train",
"index": 25331,
"solution": "\nTo solve the given problem, we first need to understand the requirements for a \\(\\pm 1\\)-sequence. We are looking for the largest integer \\( C \\) such that, for any sequence of numbers \\( a_1, a_2, \\ldots, a_{2022} \\) where each \\( a_i \\) is either \\( ... |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{40}"
} | {
"split": "train",
"index": 25332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider positive integers $a \\leq b \\leq c \\leq d \\leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 25333,
"solution": "Since the median of the list $a, b, c, d, e$ is 2023 and $a \\leq b \\leq c \\leq d \\leq e$, then $c=2023$. Since 2023 appears more than once in the list, then it appears 5,4,3, or 2 times. Case 1: 2023 appears 5 times Here, the list is 2023, 2023, 2023, 2023, 202... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\( S_{1} = 1, S_{2} = 1 - 2, S_{3} = 1 - 2 + 3, S_{4} = 1 - 2 + 3 - 4, S_{5} = 1 - 2 + 3 - 4 + 5, \\cdots \\), find the value of \\( S_{1} + S_{2} + S_{3} + \\cdots + S_{299} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 25334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equations of curve $C$ are \n$$\n\\begin{cases} \nx=3+ \\sqrt {5}\\cos \\alpha \\\\\ny=1+ \\sqrt {5}\\sin \\alpha\n\\end{cases}\n(\\alpha \\text{ is the parameter}),\n$$\nwith the origin of the Cartesian coordinate system as the pole and the positive half-a... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 25335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, $C$, and $D$ lie on a line, in that order. If $AB=2$ units, $BC=5$ units and $AD=14$ units, what is the ratio of $AC$ to $BD$? 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{7}{12}"
} | {
"split": "train",
"index": 25336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 25337,
"solution": "1. **Define Variables and Known Values:**\n Let $n$ be the number of students in the class. We know that five students scored $100$ points each, every student scored at least $60$ points, and the mean score of the class was $76$ points.\n\n2. **Set Up the Equatio... |
deepscaler | [
{
"role": "user",
"content": "The circumcircle of acute $\\triangle ABC$ has center $O$ . The line passing through point $O$ perpendicular to $\\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$ , respectively. Also $AB=5$ , $BC=4$ , $BQ=4.5$ , and $BP=\\frac{m}{n}$ , where $m$ and $n$ are relativ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 25338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{{x}^{2}}{3{{m}^{2}}}+\\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\\frac{{{x}^{2}}}{2{{m}^{2}}}-\\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{19}}{4}"
} | {
"split": "train",
"index": 25339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\pi / 4} \\frac{7+3 \\operatorname{tg} x}{(\\sin x+2 \\cos x)^{2}} d x\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\ln \\left(\\frac{3}{2}\\right) + \\frac{1}{6}"
} | {
"split": "train",
"index": 25340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point where the line passing through $(3,4,1)$ and $(5,1,6)$ intersects the $xy$-plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{13}{5}, \\frac{23}{5}, 0 \\right)"
} | {
"split": "train",
"index": 25341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the line segment with end-points $-11 + 3i$ and $3 - 7i$ is plotted in the complex plane. Find the complex number corresponding to the mid-point of this line segment. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4 - 2i"
} | {
"split": "train",
"index": 25342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several white rabbits and gray rabbits. When 6 white rabbits and 4 gray rabbits are placed in a cage, there are still 9 more white rabbits remaining, and all the gray rabbits are placed. When 9 white rabbits and 4 gray rabbits are placed in a cage, all the white rabbi... | math | {
"style": "rule",
"ground_truth": "159"
} | {
"split": "train",
"index": 25343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangem... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25344,
"solution": "1. **Positioning the Boys**: The problem specifies that a boy must sit at each end of the row. Since there are two boys, we can arrange them in $2!$ ways. This is because the first position can be filled by either of the two boys, and the second position is automat... |
deepscaler | [
{
"role": "user",
"content": "The shaded design shown in the diagram is made by drawing eight circular arcs, all with the same radius. The centers of four arcs are the vertices of the square; the centers of the four touching arcs are the midpoints of the sides of the square. The diagonals of the square have... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} \\pi"
} | {
"split": "train",
"index": 25345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let a line passing through the origin \\\\(O\\\\) intersect a circle \\\\((x-4)^{2}+y^{2}=16\\\\) at point \\\\(P\\\\), and let \\\\(M\\\\) be the midpoint of segment \\\\(OP\\\\). Establish a polar coordinate system with the origin \\\\(O\\\\) as the pole and the positive half... | math | {
"style": "rule",
"ground_truth": "3+ \\dfrac {3}{2} \\sqrt {3}"
} | {
"split": "train",
"index": 25346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S = 1 - 2 + 3 - 4 + \\cdots + 2009 - 2010$. What is the residue of $S$, modulo 2010? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 25347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $\\angle B=90^{\\circ}$. Given that there exists a point $D$ on $A C$ such that $A D=D C$ and $B D=B C$, compute the value of the ratio $\\frac{A B}{B C}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 25348,
"solution": "$D$ is the circumcenter of $A B C$ because it is the midpoint of the hypotenuse. Therefore, $D B=D A=D C$ because they are all radii of the circumcircle, so $D B C$ is an equilateral triangle, and $\\angle C=60^{\\circ}$. This means that $A B C$ is a $30^{\\circ}-6... |
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