data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "There is only one set of five prime numbers that form an arithmetic sequence with a common difference of 6. What is the sum of those five prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 3800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 400 Newtons. What is the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 3801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\begin{vmatrix} 1 & x & y \\\\ 1 & x + y & y \\\\ 1 & x & x + y \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "xy"
} | {
"split": "train",
"index": 3802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a convex trapezoid such that $\\angle B A D=\\angle A D C=90^{\\circ}, A B=20, A D=21$, and $C D=28$. Point $P \\neq A$ is chosen on segment $A C$ such that $\\angle B P D=90^{\\circ}$. Compute $A P$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{143}{5}"
} | {
"split": "train",
"index": 3803,
"solution": "Construct the rectangle $A B X D$. Note that $$\\angle B A D=\\angle B P D=\\angle B X D=90^{\\circ}$$ so $A B X P D$ is cyclic with diameter $B D$. By Power of a Point, we have $C X \\cdot C D=C P \\cdot C A$. Note that $C X=C D-X D=C D-A B=8$ and $C A=\\sqrt{A D^{... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a, b, c$ , and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 3804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yan is between his home and the library. To get to the library, he can either walk directly to the library or walk home and then ride his bicycle to the library. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's d... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 3805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven people are seated together around a circular table. Each one will toss a fair coin. If the coin shows a head, then the person will stand. Otherwise, the person will remain seated. The probability that after all of the tosses, no two adjacent people are both standing, can ... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 3806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63_{10}"
} | {
"split": "train",
"index": 3807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\\parallel BC$, $|BC|=2$, and $\\text{Area}(ABC)=3\\sqrt{5}/8$, calculate $|AB|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"split": "train",
"index": 3809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 5 \\\\ -9 \\end{... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ -8 \\end{pmatrix}"
} | {
"split": "train",
"index": 3810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the digits \\(1, 2, 3, 4, 5, 6, 7, 8, 9\\) in some order to form a nine-digit number \\(\\overline{\\text{abcdefghi}}\\). If \\(A = \\overline{\\text{abc}} + \\overline{\\text{bcd}} + \\overline{\\text{cde}} + \\overline{\\text{def}} + \\overline{\\text{efg}} + \\overli... | math | {
"style": "rule",
"ground_truth": "4648"
} | {
"split": "train",
"index": 3811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 3812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon, and let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $FA$ , respectively. The intersection of lines $\\overline{AH}$ , $\\overline{BI}$ , $\\overline{CJ}$ , $\\overli... | math | {
"style": "rule",
"ground_truth": "4/7"
} | {
"split": "train",
"index": 3813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average (mean) number of hamburgers eaten per student if 12 students ate 0 hamburgers, 14 students ate 1 hamburger, 8 students ate 2 hamburgers, 4 students ate 3 hamburgers, and 2 students ate 4 hamburgers? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 3814,
"solution": "The mean number of hamburgers eaten per student equals the total number of hamburgers eaten divided by the total number of students. 12 students each eat 0 hamburgers. This is a total of 0 hamburgers eaten. 14 students each eat 1 hamburger. This is a total of 14 ham... |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$, and $z$ be positive real numbers such that $(x \\cdot y)+z=(x+z) \\cdot(y+z)$. What is the maximum possible value of $x y z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/27"
} | {
"split": "train",
"index": 3815,
"solution": "The condition is equivalent to $z^{2}+(x+y-1) z=0$. Since $z$ is positive, $z=1-x-y$, so $x+y+z=1$. By the AM-GM inequality, $$x y z \\leq\\left(\\frac{x+y+z}{3}\\right)^{3}=\\frac{1}{27}$$ with equality when $x=y=z=\\frac{1}{3}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $M = 36 \\cdot 36 \\cdot 85 \\cdot 128$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4094}"
} | {
"split": "train",
"index": 3816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ and $n$ be positive integers and let $$ S=\\left\\{\\left(a_{1}, \\ldots, a_{k}\\right) \\in \\mathbb{Z}^{k} \\mid 0 \\leq a_{k} \\leq \\cdots \\leq a_{1} \\leq n, a_{1}+\\cdots+a_{k}=k\\right\\} $$ Determine, with proof, the value of $$ \\sum_{\\left(a_{1}, \\ldots, a_... | math | {
"style": "rule",
"ground_truth": "\\binom{k+n-1}{k} = \\binom{k+n-1}{n-1}"
} | {
"split": "train",
"index": 3817,
"solution": "Answer: $\\binom{k+n-1}{k}=\\binom{k+n-1}{n-1}$ Solution 1: Let $$ T=\\left\\{\\left(b_{1}, \\ldots, b_{n}\\right) \\mid 0 \\leq b_{1}, \\ldots, b_{n} \\leq k, b_{1}+\\cdots+b_{n}=k\\right\\} $$ The sum in question counts $|T|$, by letting $a_{i}$ be the number of $... |
deepscaler | [
{
"role": "user",
"content": "In coordinate space, $A = (1,2,3),$ $B = (5,3,1),$ and $C = (3,4,5).$ Find the orthocenter of triangle $ABC.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{5}{2}, 3, \\frac{7}{2} \\right)"
} | {
"split": "train",
"index": 3818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\\angle BAD + \\angle CDA$ if $B'C' = AB+CD$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 3819,
"solution": "\nConsider the convex quadrilateral \\(ABCD\\), and let equilateral triangles \\(ACB'\\) and \\(BDC'\\) be drawn on its diagonals such that points \\(B'\\) and \\(C'\\) are on specified sides of the lines, maintaining convexity. We are given that \\(B'C' = AB + CD\\... |
deepscaler | [
{
"role": "user",
"content": "What is the intersection of the lines given by $2y=-x+3$ and $-y=5x+1$? Enter the answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(-\\frac{5}{9}, \\frac{16}{9}\\right)"
} | {
"split": "train",
"index": 3820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, and so on. How many of the first 12 such sums are also prime? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\\frac{A}{B}$? Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"split": "train",
"index": 3822,
"solution": "Let's denote the rectangular array of numbers as $a_{ij}$ where $i$ ranges from $1$ to $40$ (rows) and $j$ ranges from $1$ to $75$ (columns).\n\n1. **Summing Rows and Columns:**\n - Andy sums each row, so for row $i$, the sum is $S_i = \\sum_{j=1}^{75} a_{ij}$.\n ... |
deepscaler | [
{
"role": "user",
"content": "Randomly select $3$ out of $6$ small balls with the numbers $1$, $2$, $3$, $4$, $5$, and $6$, which are of the same size and material. The probability that exactly $2$ of the selected balls have consecutive numbers is ____. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 3823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \\neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1026"
} | {
"split": "train",
"index": 3825,
"solution": "All 1 digit numbers, $0,1,2,3$, are solutions when, say, $b=5$. (Of course, $d \\in \\{0,1,2,3\\}$ works for any base $b$ of absolute value greater than $d$ but not equal to 4 .) Consider now positive integers $n=\\left(a_{d} \\ldots a_{1} a_{0}\\right)_{4}$ with mo... |
deepscaler | [
{
"role": "user",
"content": "In a particular year, the price of a commodity increased by $30\\%$ in January, decreased by $10\\%$ in February, increased by $20\\%$ in March, decreased by $y\\%$ in April, and finally increased by $15\\%$ in May. Given that the price of the commodity at the end of May was th... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 3826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \\frac{\\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.\n\n(1) Find the equation of t... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 3827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and it is known that $$2\\sin^{2}(B+C)= \\sqrt {3}\\sin2A.$$ \n(Ⅰ) Find the degree measure of $A$; \n(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\\triangle ABC$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "10 \\sqrt {3}"
} | {
"split": "train",
"index": 3828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The same amount of steel used to create eight solid steel balls, each with a radius of 1 inch, is used to create one larger steel ball. What is the radius of the larger ball?\n\n[asy]\nsize(150);\nfilldraw(circle((0,0),1),gray);\nfilldraw(circle((.9,-.8),1),gray);\nfilldraw(cir... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 3829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the cent... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 3830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$-plane, a triangle has vertices with coordinates $(x, y)$, where $x$ and $y$ are integers satisfying $1 \\leqslant x \\leqslant 4$ and $1 \\leqslant y \\leqslant 4$. How many such triangles are there?\n\n(Source: 44th American High School Mathematics Exam, 1993) Let'... | math | {
"style": "rule",
"ground_truth": "516"
} | {
"split": "train",
"index": 3831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest divisor of 372 that is smaller than 50 and also a factor of 72? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\\cos B=b\\cos C$, and the dot product $\\vec{AB}\\cdot \\vec{BC} = -3$,\n1. Find the area of $\\triangle ABC$;\n2. Find the minimum value of side $AC$.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 3833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 10^{10x}$, $g(x) = \\log_{10}\\left(\\frac{x}{10}\\right)$, $h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \\geq 2$. What is the sum of the digits of $h_{2011}(1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16089"
} | {
"split": "train",
"index": 3834,
"solution": "1. **Define the functions and calculate $h_1(x)$:**\n - Given $f(x) = 10^{10x}$ and $g(x) = \\log_{10}\\left(\\frac{x}{10}\\right)$, we can simplify $g(x)$:\n \\[\n g(x) = \\log_{10}\\left(\\frac{x}{10}\\right) = \\log_{10}(x) - \\log_{10}(10) = \\log_{10}... |
deepscaler | [
{
"role": "user",
"content": "In a household, when someone is at home, the probability of the phone being answered at the 1st ring is 0.1, at the 2nd ring is 0.3, at the 3rd ring is 0.4, and at the 4th ring is 0.1. What is the probability that the phone is not answered within the first 4 rings? Let's think ... | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 3835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2B=A+C$ and $a+\\sqrt{2}b=2c$, find the value of $\\sin C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}+\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 3836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?\n \n[asy]\nimport three; size(2inch);\ncurrentprojection=orthographic(4,2,2);\ndraw((0,0,0)--(0,0,3),dashed);\ndraw((0,0,0)--(0,4,0),das... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rounded to 2 decimal places, what is $\\frac{7}{9}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.78"
} | {
"split": "train",
"index": 3838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 3839,
"solution": "1. **Assign Variables:**\n Let the shorter segment of the side of length $80$ be $x$, and let the altitude dropped to this side be $y$. The larger segment is then $80 - x$.\n\n2. **Apply the Pythagorean Theorem:**\n Since the altitude divides the triangle into t... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\\circ$ clockwise around vertex $D$, then it is rotated $90^\\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by po... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi(\\sqrt{34} + 5)}{2}"
} | {
"split": "train",
"index": 3840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\). On the extension of the hypotenuse \\(BC\\), a point \\(D\\) is chosen such that the line \\(AD\\) is tangent to the circumscribed circle \\(\\omega\\) of triangle \\(ABC\\). The line \\(AC\\) intersects the circumscribed circle of triangle \\(... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 3841,
"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": 3842,
"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": "Given the numbers \\(5^{1971}\\) and \\(2^{1971}\\) written consecutively, what is the number of digits in the resulting number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1972"
} | {
"split": "train",
"index": 3843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the game of *Winners Make Zeros*, a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:\n\n\n- If $m\\geq n$ , the player choose a positive integer $c$ such that $m-cn\\geq 0$ , and rep... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 3844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.\n\n1. $w$ consists of $n$ letters from the alphabet $\\{a,b,c,d\\}.$\n\n2. $w$ contains an even number of $a$'s\n\n3. $w$ contains an even number of $b$'s.\n\nFor example... | math | {
"style": "rule",
"ground_truth": "2^{n-1}(2^{n-1} + 1)"
} | {
"split": "train",
"index": 3845,
"solution": "\nWe are tasked with determining the number of words \\( w \\), consisting of \\( n \\) letters from the alphabet \\(\\{a, b, c, d\\}\\), that satisfy the following properties:\n1. The word \\( w \\) contains an even number of \\( a \\)'s.\n2. The word \\( w \\) con... |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 3846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer of the form <u>a</u> <u>b</u> <u>a</u> <u>a</u> <u>b</u> <u>a</u>, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "282282"
} | {
"split": "train",
"index": 3847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"split": "train",
"index": 3848,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 people standing in a row. If person A is not at the left end and not adjacent to person B, determine the number of possible arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 3849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{5}{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $d$ such that the remainder $r$ is the same when each of the numbers $1210, 1690$, and $2670$ is divided by $d$, and calculate the value of $d-4r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 3851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each permutation $a_1,a_2,a_3,\\cdots,a_{10}$ of the integers $1,2,3,\\cdots,10$, form the sum\n\\[|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.\\]\nThe average value of all such sums can be written in the form $\\dfrac{p}{q}$, where $p$ and $q$ are relatively prime... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 3852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $A B C$, a point $D$ is on hypotenuse $A C$ such that $B D \\perp A C$. Let $\\omega$ be a circle with center $O$, passing through $C$ and $D$ and tangent to line $A B$ at a point other than $B$. Point $X$ is chosen on $B C$ such that $A X \\perp B O$. If $A B... | math | {
"style": "rule",
"ground_truth": "8041"
} | {
"split": "train",
"index": 3853,
"solution": "Note that since $A D \\cdot A C=A B^{2}$, we have the tangency point of $\\omega$ and $A B$ is $B^{\\prime}$, the reflection of $B$ across $A$. Let $Y$ be the second intersection of $\\omega$ and $B C$. Note that by power of point, we have $B Y \\cdot B C=B B^{\\pri... |
deepscaler | [
{
"role": "user",
"content": "Given that $y=f(x)$ is an odd function, if $f(x)=g(x)+x^{2}$ and $g(1)=1$, then $g(-1)=$ _____ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 3854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many prime numbers have squares between 4000 and 7000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average student headcount for the spring terms of the `02-`03, `03-`04 and `04-`05 academic years? Express your answer to the nearest whole number.\n\n[asy]\nunitsize(0.35 cm);\n\nfill((1,0)--(1,11.7)--(4,11.7)--(4,0)--cycle,gray(.5));\nfill((4,0)--(4,10.9)--(7,10.9... | math | {
"style": "rule",
"ground_truth": "10700"
} | {
"split": "train",
"index": 3856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers? Let's think... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3857,
"solution": "Let's denote the six integers as $a, b, c, d, e, f$. We are given the following conditions:\n1. $a + b = 26$\n2. $a + b + c + d = 41$\n3. $a + b + c + d + e + f = 57$\n\nFrom these conditions, we can derive the sums of the additional integers:\n- From 1 and 2, $c + ... |
deepscaler | [
{
"role": "user",
"content": "29 boys and 15 girls came to the ball. Some boys danced with some girls (no more than once per pair). After the ball, each person told their parents how many times they danced. What is the greatest number of distinct counts that the children could report? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 3858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of roots satisfying the equation $\\sqrt{5 - x} = x\\sqrt{5 - x}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 3859,
"solution": "1. Start by solving the equation given:\n \\[\\sqrt{5 - x} = x\\sqrt{5 - x}\\]\n\n2. Factor out the common term $\\sqrt{5 - x}$:\n \\[\\sqrt{5 - x}(x - 1) = 0\\]\n\n3. From the factored equation, we have two cases to consider:\n - Case 1: $\\sqrt{5 - x} = 0$\n... |
deepscaler | [
{
"role": "user",
"content": "A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}-\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 3860,
"solution": "1. **Understanding the Geometry**: The problem states that six rectangular place mats are placed on a round table with radius $4$. Each place mat has width $1$ and length $x$. The mats are positioned such that two corners of each mat (on the same side) touch the edg... |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of $302^2 - 298^2$ and $152^2 - 148^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"split": "train",
"index": 3861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 12-digit positive integers have all digits either 1 or 2, and have numbers ending in 12, but have no two consecutive 1's? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 3862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $1011001_2$ to base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1121_4"
} | {
"split": "train",
"index": 3863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quantities $r$ and $s$ vary inversely. When $r$ is $1200,$ $s$ is $0.35.$ What is the value of $s$ when $r$ is $2400$? Express your answer as a decimal to the nearest thousandths. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".175"
} | {
"split": "train",
"index": 3864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $2.5252525\\ldots$ can be written as a fraction.\nWhen reduced to lowest terms the sum of the numerator and denominator of this fraction is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "349"
} | {
"split": "train",
"index": 3865,
"solution": "\n#### Step 1: Represent the repeating decimal as a fraction\nLet $x = 2.5252525\\ldots$. We can express $x$ in terms of itself shifted by two decimal places:\n\\[ 100x = 252.5252525\\ldots \\]\n\n#### Step 2: Set up the equation\nSubtract the original $x$ from $100... |
deepscaler | [
{
"role": "user",
"content": "Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\\equal{}\\minus{}x^4\\plus{}2x^2$ and the line $ y\\equal{}k$ are all equal. 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": 3866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yu Semo and Yu Sejmo have created sequences of symbols $\\mathcal{U} = (\\text{U}_1, \\ldots, \\text{U}_6)$ and $\\mathcal{J} = (\\text{J}_1, \\ldots, \\text{J}_6)$ . These sequences satisfy the following properties.\n\n\n- Each of the twelve symbols must be $\\Sigma$ , $\... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least integer value of $x$ for which $2|x| + 7 < 17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 3868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the c... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 3869,
"solution": "1. **Understanding the problem**: Carl has a rectangular garden and uses 20 fence posts, including one on each corner. The posts are spaced 4 yards apart. The longer side has twice as many posts as the shorter side.\n\n2. **Setting up the equations**:\n - Let the ... |
deepscaler | [
{
"role": "user",
"content": "A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is $1: 2: 4$. The ratio of green to yellow to orange tickets is $1: 3: 6$. What is the smallest number of tickets that must be drawn t... | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 3870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M \\) be the centroid of \\( \\triangle ABC \\), and \\( AM = 3 \\), \\( BM = 4 \\), \\( CM = 5 \\). 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": "18"
} | {
"split": "train",
"index": 3871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle JKL, where $\\angle J$ is $90^\\circ$, side JL is known to be 12 units, and the hypotenuse KL is 13 units. Calculate $\\tan K$ and $\\cos L$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{13}"
} | {
"split": "train",
"index": 3872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0}, a_{1}, \\ldots$ and $b_{0}, b_{1}, \\ldots$ be geometric sequences with common ratios $r_{a}$ and $r_{b}$, respectively, such that $$\\sum_{i=0}^{\\infty} a_{i}=\\sum_{i=0}^{\\infty} b_{i}=1 \\quad \\text { and } \\quad\\left(\\sum_{i=0}^{\\infty} a_{i}^{2}\\right)\... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 3873,
"solution": "Let $a_{0}=a$ and $b_{0}=b$. From \\sum_{i=0}^{\\infty} a_{i}=\\frac{a_{0}}{1-r_{a}}=1$ we have $a_{0}=1-r_{a}$ and similarly $b_{0}=1-r_{b}$. This means \\sum_{i=0}^{\\infty} a_{i}^{2}=\\frac{a_{0}^{2}}{1-r_{a}^{2}}=\\frac{a^{2}}{\\left(1-r_{a}\\right)\\left(1+r_{a... |
deepscaler | [
{
"role": "user",
"content": "The points $(x, y)$ represented in this table lie on a straight line. The point $(28, t)$ lies on the same line. What is the value of $t?$ \\begin{tabular}{c|c}\n$x$ & $y$ \\\\ \\hline\n1 & 7 \\\\\n3 & 13 \\\\\n5 & 19 \\\\\n\\end{tabular} Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 3874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ilya Muromets encounters the three-headed Dragon, Gorynych. Each minute, Ilya chops off one head of the dragon. Let $x$ be the dragon's resilience ($x > 0$). The probability $p_{s}$ that $s$ new heads will grow in place of a chopped-off one ($s=0,1,2$) is given by $\\frac{x^{s}... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{97}}{8}"
} | {
"split": "train",
"index": 3875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid below so that a path is formed from the upper left-hand corner $A$ to the lower righthand corner $B$. In a path, consecutive dominoes must touch at their sides and not just their co... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 3876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)$ be a differentiable function, and $\\lim_{\\Delta x \\to 0} \\frac{{f(1)-f(1-2\\Delta x)}}{{\\Delta x}}=-1$, determine the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 3877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 3878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\\overline{AC}$ and points $D$ and $F$ are on $\\overline{AB}$ such that both $\\overline{DE}$ and $\\overline{FG}$ are parallel to $\\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBC... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{13}"
} | {
"split": "train",
"index": 3879,
"solution": "1. **Assign Variables:**\n Let $AD = x$ and $AG = y$. Since $DE$ and $FG$ are parallel to $BC$, and $ABC$ is equilateral, $DE$ and $FG$ are also equal to $x$ and $y$ respectively. We need to find $DE + FG = x + y$.\n\n2. **Use Perimeter Conditions:**\n Given tha... |
deepscaler | [
{
"role": "user",
"content": "A stock investment went up $25\\%$ in 2006. Starting at this increased value, what percent would it have to go down in 2007 to be back to its original price at the beginning of 2006? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 3880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When is it advisable to swap the tires so that they wear out equally? (Assume that the tires are swapped only once, although in practice drivers do this more frequently.) Let's think... | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"split": "train",
"index": 3881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sophie has written three tests. Her marks were $73\\%$, $82\\%$, and $85\\%$. She still has two tests to write. All tests are equally weighted. Her goal is an average of $80\\%$ or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach h... | math | {
"style": "rule",
"ground_truth": "73\\% and 83\\%"
} | {
"split": "train",
"index": 3882,
"solution": "For Sophie's average over 5 tests to be $80\\%$, the sum of her marks on the 5 tests must be $5 \\times 80\\% = 400\\%$. After the first 3 tests, the sum of her marks is $73\\% + 82\\% + 85\\% = 240\\%$. Therefore, she will reach her goal as long as the sum of her m... |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{2}{5}$ divided by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"split": "train",
"index": 3883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 3884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Machines A, B, and C operate independently and are supervised by a single worker, who cannot attend to two or more machines simultaneously. Given that the probabilities of these machines operating without needing supervision are 0.9, 0.8, and 0.85 respectively, calculate the pr... | math | {
"style": "rule",
"ground_truth": "0.059"
} | {
"split": "train",
"index": 3885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function \\( y = g(x) = \\frac{x^2}{Ax^2 + Bx + C} \\), where \\( A, B, \\) and \\( C \\) are integers. The function has vertical asymptotes at \\( x = -1 \\) and \\( x = 2 \\), and for all \\( x > 4 \\), it is true that \\( g(x) > 0.5 \\). Determine the value of \... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 3886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1... | math | {
"style": "rule",
"ground_truth": "\\frac{1155}{4}"
} | {
"split": "train",
"index": 3887,
"solution": "We first compute the circumradius of $A B C$ : Since $\\cos A=\\frac{9^{2}-17^{2}-10^{2}}{2 \\cdot 9 \\cdot 17}=-\\frac{15}{17}$, we have $\\sin A=\\frac{8}{17}$ and $R=\\frac{a}{2 \\sin A}=\\frac{170}{16}$. Moreover, we get that the area of triangle $A B C$ is $\\f... |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(x^2 + yf(z)) = xf(x) + zf(y)\\]for all real numbers $x,$ $y,$ and $z.$\n\nLet $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \\times s.$ Let's... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers between 1 and 2013 with the property that the sum of its digits equals 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 3889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the domain of the real-valued function $$f(x)=\\sqrt{3-\\sqrt{5-\\sqrt{x}}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[0, 25]"
} | {
"split": "train",
"index": 3890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Non-negative numbers \\(a\\) and \\(b\\) satisfy the equations \\(a^2 + b^2 = 74\\) and \\(ab = 35\\). What is the value of the expression \\(a^2 - 12a + 54\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 3892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest root of $f(x) = 15x^4-13x^2+2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 3893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit positive integers have at least one 7 as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 3894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ and $\\mathbf{d}$ are unit vectors, find the largest possible value of\n\\[\n\\|\\mathbf{a} - \\mathbf{b}\\|^2 + \\|\\mathbf{a} - \\mathbf{c}\\|^2 + \\|\\mathbf{a} - \\mathbf{d}\\|^2 + \\|\\mathbf{b} - \\mathbf{c}\\|^2 + \\|\\math... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 3895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$ , where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$ , compute the area of the concave quadrilate... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 3896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 9 digits: 0, 1, 2, …, 8. Using five cards, with the two sides respectively marked as 0/8, 1/7, 2/5, 3/4, 6/9; and 6 can be used as 9. How many different four-digit numbers can be formed with these five cards? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 3897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Q(x) = x^2 - 4x - 16$. A real number $x$ is chosen at random from the interval $6 \\le x \\le 20$. The probability that $\\lfloor\\sqrt{Q(x)}\\rfloor = \\sqrt{Q(\\lfloor x \\rfloor)}$ is equal to $\\frac{\\sqrt{a} + \\sqrt{b} + \\sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$,... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 3898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces? Le... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 3899,
"solution": ""
} |
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