data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 22049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The calculation result of the expression \\(143 \\times 21 \\times 4 \\times 37 \\times 2\\) is $\\qquad$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "888888"
} | {
"split": "train",
"index": 22050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?\n\n[asy]\ndraw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));\ndraw((0,0)--(12,1),dashed);\nlabel(\"8\",(2.5,2.5),NW);\nlabel(\"10\",(8.5,3),N... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 22051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the cosine value of the vertex angle of an isosceles triangle equals $\\dfrac{4}{5}$, calculate the sine value of the base angle of this triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{3}}{5}"
} | {
"split": "train",
"index": 22052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 22053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of six-digit palindromes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"split": "train",
"index": 22054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike eac... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 22055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the percentage of five-digit numbers that have at least one repeated digit (the repeated digits do not need to be adjacent). Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69.8"
} | {
"split": "train",
"index": 22056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\sqrt{113 + 56\\sqrt{2}}"
} | {
"split": "train",
"index": 22057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $100$ is chosen such that if $n\\le 60$, then the probability of choosing $n$ is $q$, and if $n > 60$, then the probability of choosing $n$ is $2q$. Find the probability that a perfect square is chosen.\n- **A)** $\\frac{1}{35}$\n- **B)** $\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{35}"
} | {
"split": "train",
"index": 22058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,3)$, $f(x)=\\begin{cases} -(x+2)^2, & -3 \\leqslant x < -1 \\\\ x, & -1 \\leqslant x < 3 \\end{cases}$. Find $f(4)$ and the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)+f(2017)$. Let's think st... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 22059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers \\( x \\) and \\( y \\) satisfy the equation \\( 26x^2 + 23xy - 3y^2 - 19 = 0 \\) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 22060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 cylindrical logs with a diameter of 5 decimeters each. They are tied together at two places using a rope. How many decimeters of rope are needed at least (excluding the rope length at the knots, and using $\\pi$ as 3.14)? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "91.4"
} | {
"split": "train",
"index": 22061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a\\_n\\}$ satisfies $\\frac{1}{a_{n+1}} - \\frac{1}{a_n} = d (n \\in \\mathbb{N}^*, d$ is a constant$)$, it is called a harmonic sequence. It is known that the sequence $\\{\\frac{1}{x\\_n}\\}$ is a harmonic sequence and $x\\_1 + x\\_2 + ... + x_{20}... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the solutions of the equation $(3x+5)(2x-9) = 0$? 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{17}{6}"
} | {
"split": "train",
"index": 22063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23456"
} | {
"split": "train",
"index": 22064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\sqrt{3}a=2b\\sin A$. \n$(1)$ Find angle $B$; \n$(2)$ If $b=\\sqrt{7}$, $c=3$, and $D$ is the midpoint of side $AC$, find $BD$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{19}}{2}"
} | {
"split": "train",
"index": 22065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the function $f(x)=4\\sin \\left(2x+\\frac{\\pi }{3}\\right)$, the following propositions are given:\n\n$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\\cos \\left(2... | math | {
"style": "rule",
"ground_truth": "(2)(3)"
} | {
"split": "train",
"index": 22066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Understanding and trying: When calculating $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, there are two methods; Method 1, please calculate directly: $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, Method 2, use letters to represent numb... | math | {
"style": "rule",
"ground_truth": "-1.35"
} | {
"split": "train",
"index": 22067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $b_{0}, b_{1}, \\ldots, b_{59}$ by $$ b_{i}= \\begin{cases}1 & \\text { if } \\mathrm{i} \\text { is a multiple of } 3 \\\\ 0 & \\text { otherwise }\\end{cases} $$ Let \\left\\{a_{i}\\right\\} be a sequence of elements of \\{0,1\\} such that $$ b_{n} \\equiv... | math | {
"style": "rule",
"ground_truth": "0, 3, 5, 6"
} | {
"split": "train",
"index": 22068,
"solution": "Try the four possible combinations of values for $a_{0}$ and $a_{1}$. Since we can write $a_{n} \\equiv$ $b_{n-1}-a_{n-2}-a_{n-1}$, these two numbers completely determine the solution $\\left\\{a_{i}\\right\\}$ beginning with them (if there is one). For $a_{0}=a_{1... |
deepscaler | [
{
"role": "user",
"content": "Given that there is an extra $32.13 on the books due to a misplaced decimal point, determine the original amount of the sum of money. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.57"
} | {
"split": "train",
"index": 22069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $5273_{8} - 3614_{8}$. Express your answer in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1457_8"
} | {
"split": "train",
"index": 22070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 10.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{131}{11}"
} | {
"split": "train",
"index": 22071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(2,-1,1)$, $B(1,-2,1)$, $C(0,0,-1)$, the distance from $A$ to $BC$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{3}"
} | {
"split": "train",
"index": 22072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^3 + ax^2 = -30$ has only integer solutions for $x$. If $a$ is a positive integer, what is the greatest possible value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 22073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\(ABCD\\), \\(AB = 1\\), \\(BC = 4\\), and \\(\\angle ABC = 60^\\circ\\). Suppose that \\(AC\\) is extended from \\(A\\) to a point \\(E\\) beyond \\(C\\) so that triangle \\(ADE\\) has the same area as the parallelogram. Find the length of \\(DE\\). Let's thi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 22074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the two roots of the quadratic $7x^2+3x+k$ are $\\frac{-3\\pm i\\sqrt{299}}{14}$, what is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( P(x, y) \\) satisfies \\( |x| + |y| \\leq 2 \\), find the probability for point \\( P \\) to have a distance \\( d \\leq 1 \\) from the \\( x \\)-axis. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 22076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle $ABC$ with $AC=20$ is inscribed in a circle $\\omega$ . A tangent $t$ to $\\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not e... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.) Let'... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22078,
"solution": "If only two colors are used, there is only one possible arrangement up to rotation, so this gives 3 possibilities. If all three colors are used, then one is used twice. There are 3 ways to choose the color that is used twice. Say this color is red. Then the red ver... |
deepscaler | [
{
"role": "user",
"content": "Suppose in a right triangle $PQR$, $\\cos Q = 0.5$. Point $Q$ is at the origin, and $PQ = 15$ units along the positive x-axis. What is the length of $QR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has positive integer side lengths and an area of 24. What perimeter of the rectangle cannot be? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 22080,
"solution": "Since the rectangle has positive integer side lengths and an area of 24, its length and width must be a positive divisor pair of 24. Therefore, the length and width must be 24 and 1, or 12 and 2, or 8 and 3, or 6 and 4. Since the perimeter of a rectangle equals 2 t... |
deepscaler | [
{
"role": "user",
"content": "Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 22081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system \\(xOy\\), the set of points \\(K=\\{(x, y) \\mid x, y=-1,0,1\\}\\). Three points are randomly selected from \\(K\\). What is the probability that the distance between any two of these three points does not exceed 2? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "5/14"
} | {
"split": "train",
"index": 22082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arturo has an equal number of $\\$5$ bills, of $\\$10$ bills, and of $\\$20$ bills. The total value of these bills is $\\$700$. How many $\\$5$ bills does Arturo have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22083,
"solution": "Since Arturo has an equal number of $\\$5$ bills, of $\\$10$ bills, and of $\\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\\$5$ bill, one $\\$10$ bill, and one $\\$20$ bill. The value of the bills in each group is $\\$5 + ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression given by $$2+\\cfrac{3}{4+\\cfrac{5}{6+\\cfrac{7}{8}}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{137}{52}"
} | {
"split": "train",
"index": 22084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \\leq i < j \\leq 49$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 22085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student named Zhang has a set of 6 questions to choose from, with 4 categorized as type A and 2 as type B. Zhang randomly selects 2 questions to solve. \n(1) What is the probability that both selected questions are type A? \n(2) What is the probability that the selected que... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 22086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be non-zero real numbers such that \n\\[ \\frac{x \\sin \\frac{\\pi}{5} + y \\cos \\frac{\\pi}{5}}{x \\cos \\frac{\\pi}{5} - y \\sin \\frac{\\pi}{5}} = \\tan \\frac{9 \\pi}{20}. \\]\n\n(1) Find the value of \\(\\frac{y}{x}\\).\n\n(2) In triangle \\( ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 22087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, points $A$, $B$, $C$, and $P$ are situated so that $PA=2$, $PB=3$, $PC=4$, and $BC=5$. What is the maximum possible area of $\\triangle ABC$? [asy]\ndefaultpen(linewidth(0.8)); size(150);\npair B = (0,0), C = (5,0), A = (2,3), P = (2.2,2);\ndraw(A--B--C--c... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $(x, y)$ is randomly selected from the rectangular region defined by vertices $(0, 0), (3013, 0), (3013, 3014),$ and $(0, 3014)$. What is the probability that $x > 8y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3013}{48224}"
} | {
"split": "train",
"index": 22089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point on the line\n\\[y = -3x + 5\\]that is closest to the point $(-4,-2).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{17}{10}, -\\frac{1}{10} \\right)"
} | {
"split": "train",
"index": 22090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=x^{2}+bx+3$ (where $b$ is a real number), the range of $y$ is $\\left[0,+\\infty \\right)$. Find the value of the real number $c$ if the solution set of the inequality $x^{2}+bx+3 \\lt c$ is $m-8 \\lt x \\lt m$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the first day, Barry Sotter used his magic wand to make an object's length increase by $\\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \\frac{1}{3} x.$ On the second day, he increased the object's new length from t... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 22092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nOn a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exact... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 22093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "972538"
} | {
"split": "train",
"index": 22094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the edge \\(AD\\) and the diagonal \\(A_1C\\) of the parallelepiped \\(ABCDA_1B_1C_1D_1\\), points \\(M\\) and \\(N\\) are taken respectively, such that the line \\(MN\\) is parallel to the plane \\(BDC_1\\) and \\(AM:AD = 1:5\\). Find the ratio \\(CN:CA_1\\). Let's think st... | math | {
"style": "rule",
"ground_truth": "3/5"
} | {
"split": "train",
"index": 22095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the endpoints of an edge of the octahedron? Express your answer as a common fraction. [asy]\nsize(150);\npair A, B, C, D, E, F;\nA=(1,1);\nB=(-1,-1);\nC=(0,6);\nD=(... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 22096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1)"
} | {
"split": "train",
"index": 22097,
"solution": "\nTo find all positive integers \\( a, b \\) for which \\( a^4 + 4b^4 \\) is a prime number, we first analyze the expression:\n\n\\[\na^4 + 4b^4\n\\]\n\nThis can be rewritten using the Sophie Germain identity:\n\n\\[\na^4 + 4b^4 = (a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2a... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1=-2$, $a_2=2$, and $a_{n+2}-a_n=1+(-1)^n$, then $S_{50}=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 22098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\) with \\(BC \\parallel AD\\), it is known that \\(AD = 3 \\cdot BC\\). A line intersects the non-parallel sides of the trapezoid at points \\(M\\) and \\(N\\) such that \\(AM:MB = 3:5\\) and \\(CN:ND = 2:7\\). Find the ratio of the areas of quadrilaterals... | math | {
"style": "rule",
"ground_truth": "9/23"
} | {
"split": "train",
"index": 22099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_n\\}$ be a decreasing geometric sequence, where $q$ is the common ratio, and $S_n$ is the sum of the first $n$ terms. Given that $\\{a_1, a_2, a_3\\} \\subseteq \\{-4, -3, -2, 0, 1, 2, 3, 4\\}$, find the value of $$\\frac {S_{8}}{1-q^{4}}$$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac {17}{2}"
} | {
"split": "train",
"index": 22100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when the same integer is divided by 3 and by 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 22101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c=2$, $2\\sin A= \\sqrt {3}a\\cos C$.\n(1) Find the measure of angle $C$;\n(2) If $2\\sin 2A+ \\sin (2B+C)= \\sin C$, find the area of $\\triangle ABC$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\dfrac {2 \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 22102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a square with side length 10, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square. What is the side length of the largest square that can be inscribed in the space inside the square and outside of... | math | {
"style": "rule",
"ground_truth": "5-\\frac{5\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 22103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\\triangle ABC, \\triangle BCD,$ and $\\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "12+10\\sqrt{3}"
} | {
"split": "train",
"index": 22104,
"solution": "1. **Assigning Position Vectors:**\n Let's place the origin at $A$ and assign position vectors such that $B = \\vec{p}$ and $D = \\vec{q}$. Since $AB$ is not parallel to $AD$, vectors $\\vec{p}$ and $\\vec{q}$ are linearly independent. We can express $C$ as a lin... |
deepscaler | [
{
"role": "user",
"content": "Convert $813_9$ to base 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220110_3"
} | {
"split": "train",
"index": 22105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 22106,
"solution": "1. **Understanding the Problem**: Tom, Dick, and Harry each flip a fair coin repeatedly until they get a head. We need to find the probability that all three stop flipping their coins after the same number of flips.\n\n2. **Modeling the Coin Flips**: The probabilit... |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{2} \\right)^x\\]on the interval $(0,100 \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 22107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest trian... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 22108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23 , only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible s... | math | {
"style": "rule",
"ground_truth": "321"
} | {
"split": "train",
"index": 22109,
"solution": "Let the original numbers be $m<n$. We know $$m n=(m-20)(n+23)=m n-20 n+23 m-460 \\Longrightarrow 23 m-20 n=460$$ Furthermore, $23 m<23 n$ hence $460<3 n \\Longrightarrow n \\geq 154$. Furthermore, we must have $23 \\mid n$ hence the least possible value of $n$ is 1... |
deepscaler | [
{
"role": "user",
"content": "A sequence of twelve \\(0\\)s and/or \\(1\\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \\(1\\)s can be written in the form \\(\\dfrac{m}{n}\\), where \\(m,n\\) are relatively prime positive integers,... | math | {
"style": "rule",
"ground_truth": "2281"
} | {
"split": "train",
"index": 22110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=1$, and $\\overrightarrow{a}\\cdot \\overrightarrow{b}=-\\sqrt{2}$, calculate the angle between vector $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\pi}{4}"
} | {
"split": "train",
"index": 22111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$ satisfies $f(x+6)=f(x)$. When $-3<x\\leq-1$, $f(x)=-(x+2)^2$, and when $-1\\leq x\\leq 3$, $f(x)=x$. Calculate the value of $f(1) + f(2) + \\dots + f(2015)$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 22112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $CAT$, we have $\\angle{ACT}=\\angle{ATC}$ and $\\angle{CAT}=36^\\circ$. If $\\overline{TR}$ bisects $\\angle{ATC}$, then how many degrees is $\\angle{CRT}$? [asy]\n/* AMC8 2000 #13 Problem */\ndraw((0,0)--(.5,1.75)--(1,0)--cycle);\ndraw((1,0)--(.15,.5));\nlabel(\"$... | math | {
"style": "rule",
"ground_truth": "72^\\circ"
} | {
"split": "train",
"index": 22113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) betw... | math | {
"style": "rule",
"ground_truth": "3705"
} | {
"split": "train",
"index": 22114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3059"
} | {
"split": "train",
"index": 22115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In my office, there are two digital 24-hour clocks. One clock gains one minute every hour and the other loses two minutes every hour. Yesterday, I set both of them to the same time, but when I looked at them today, I saw that the time shown on one was 11:00 and the time on the ... | math | {
"style": "rule",
"ground_truth": "15:40"
} | {
"split": "train",
"index": 22116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an arithmetic sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. When $k \\geqslant 2$, if $S\\_{k-1}=8$, $S\\_k=0$, and $S\\_{k+1}=-10$, what is the maximum value of $S\\_n$? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has a right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n\n[asy]\nunitsize(0.2 cm);\n\npair A, B, C, P;\n\nA = (0,14);\nB = (0,0);\nC = (21*sqrt(3),0);\nP = intersectionpoint(arc(... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 22118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of inducing cerebrovascular disease by smoking 5 cigarettes in one hour is 0.02, and the probability of inducing cerebrovascular disease by smoking 10 cigarettes in one hour is 0.16. An employee of a certain company smoked 5 cigarettes in one hour without induci... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 22119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 22120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega = \\cos\\frac{2\\pi}{7} + i \\cdot \\sin\\frac{2\\pi}{7},$ where $i = \\sqrt{-1}.$ Find the value of the product\\[\\prod_{k=0}^6 \\left(\\omega^{3k} + \\omega^k + 1\\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "024"
} | {
"split": "train",
"index": 22121,
"solution": "The product can be factored into $-(r-1)(s-1)(t-1)(r-w)(s-w)(t-w)(r-w^2)(s-w^2)(t-w^2)....(r-w^6)(s-w^6)(t-w^6)$,\n where $r,s,t$ are the roots of the polynomial $x^3+x+1=0$.\n This is then $-(r^7-1)(s^7-1)(t^7-1)$ because $(r^7-1)$ and $(r-1)(r-w)(r-w^2)...(r-w^6)... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the lengths of sides opposite to angles A, B, C are denoted by a, b, c respectively. If \\( c = \\sqrt{3} \\), \\( b = 1 \\), and \\( B = 30^\\circ \\),\n(1) find angles A and C;\n(2) find the area of \\( \\triangle ABC \\). Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 22122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is labeled with one of the numbers 1, 2, 3, 4. Four identical regular tetrahedrons are simultaneously rolled onto a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"split": "train",
"index": 22123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. Exactly one of the numbers must be even. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 22124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gustave has 15 steel bars of masses $1 \\mathrm{~kg}, 2 \\mathrm{~kg}, 3 \\mathrm{~kg}, \\ldots, 14 \\mathrm{~kg}, 15 \\mathrm{~kg}$. He also has 3 bags labelled $A, B, C$. He places two steel bars in each bag so that the total mass in each bag is equal to $M \\mathrm{~kg}$. Ho... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 22125,
"solution": "The total mass of the six steel bars in the bags is at least $1+2+3+4+5+6=21 \\mathrm{~kg}$ and at most $10+11+12+13+14+15=75 \\mathrm{~kg}$. This is because the masses of the 15 given bars are $1 \\mathrm{~kg}, 2 \\mathrm{~kg}, 3 \\mathrm{~kg}, \\ldots, 14 \\mathr... |
deepscaler | [
{
"role": "user",
"content": "\nOn the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum numbe... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 22126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $b_1, b_2, \\ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\\log_4 b_1+\\log_4 b_2+\\cdots+\\log_4 b_{10} = 2010,$ find the number of possible ordered pairs $(b,s).$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 22127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle \\( \\triangle AMC \\), \\( AM = AC \\), the median \\( MV = CU = 12 \\), and \\( MV \\perp CU \\) at point \\( P \\). What is the area of \\( \\triangle AMC \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 22128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\\mathcal{S}.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 22129,
"solution": "By symmetry, the average over all $720=(10)(9)(8)$ numbers is $\\frac{1}{2}$. Then, their sum is $\\frac{1}{2}(720)=\\boxed{360}$."
} |
deepscaler | [
{
"role": "user",
"content": "795. Calculate the double integral \\(\\iint_{D} x y \\, dx \\, dy\\), where region \\(D\\) is:\n\n1) A rectangle bounded by the lines \\(x=0, x=a\\), \\(y=0, y=b\\);\n2) An ellipse \\(4x^2 + y^2 \\leq 4\\);\n3) Bounded by the line \\(y=x-4\\) and the parabola \\(y^2=2x\\). Let... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 22130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Faces $ABC^{}_{}$ and $BCD^{}_{}$ of tetrahedron $ABCD^{}_{}$ meet at an angle of $30^\\circ$. The area of face $ABC^{}_{}$ is $120^{}_{}$, the area of face $BCD^{}_{}$ is $80^{}_{}$, and $BC=10^{}_{}$. Find the volume of the tetrahedron. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 22131,
"solution": "Since the area $BCD=80=\\frac{1}{2}\\cdot10\\cdot16$, the perpendicular from $D$ to $BC$ has length $16$.\n The perpendicular from $D$ to $ABC$ is $16 \\cdot \\sin 30^\\circ=8$. Therefore, the volume is $\\frac{8\\cdot120}{3}=\\boxed{320}$."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 22132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x, y)$ is selected uniformly at random from the unit square $S=\\{(x, y) \\mid 0 \\leq x \\leq 1,0 \\leq y \\leq 1\\}$. If the probability that $(3x+2y, x+4y)$ is in $S$ is $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$. Let's th... | math | {
"style": "rule",
"ground_truth": "820"
} | {
"split": "train",
"index": 22133,
"solution": "Under the transformation $(x, y) \\mapsto(3x+2y, x+4y), S$ is mapped to a parallelogram with vertices $(0,0),(3,1),(5,5)$, and $(2,4)$. Using the shoelace formula, the area of this parallelogram is 10. The intersection of the image parallelogram and $S$ is the quad... |
deepscaler | [
{
"role": "user",
"content": "A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\n are drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\\tfrac{2}{\\sqrt{3}}.\\,$ How many such... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 22135,
"solution": "Picturing the diagram in your head should give you an illustration similar to the one above. The distance from parallel sides of the center hexagon is 20 units, and by extending horizontal lines to the sides of the hexagon. This shows that for every side of the hex... |
deepscaler | [
{
"role": "user",
"content": "The line $y = \\frac{3}{2} x - 25$ is parameterized by $(x,y) = (f(t),15t - 7).$ Enter the function $f(t).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10t + 12"
} | {
"split": "train",
"index": 22136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{2 \\sqrt{2}} \\frac{x^{4} \\, dx}{\\left(16-x^{2}\\right) \\sqrt{16-x^{2}}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20 - 6\\pi"
} | {
"split": "train",
"index": 22137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least possible sum of two bases, $c$ and $d$, such that the numeral $29$ in base $c$ represents the same number as $92$ in base $d$, where $c$ and $d$ are positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 22138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles right triangle $ABC$, with $AB = AC$, a circle is inscribed touching $AB$ at point $D$, $AC$ at point $E$, and $BC$ at point $F$. If line $DF$ is extended to meet $AC$ at point $G$ and $\\angle BAC = 90^\\circ$, and the length of segment $DF$ equals $6$ cm, calc... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 22139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all non-negative integers \\( m \\) and \\( n \\) (with \\( m \\geq n \\)), the sequence \\( a_0, a_1, \\cdots \\) satisfies\n\\[ a_{m+n} + a_{m-n} + 2m - 2n - 1 = \\frac{1}{2}(a_{2m} + a_{2n}). \\]\n\nIf \\( a_1 = 0 \\) and \\( a_p = 2019^2 \\), find the value of \\( p \\)... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 22140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the Fibonacci sequence $\\{a_n\\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 22141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student rolls two dice simultaneously, and the scores obtained are a and b respectively. The eccentricity e of the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1$ (a > b > 0) is greater than $\\frac{\\sqrt{3}}{2}$. What is the probability of this happening? Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 22142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable X follows a normal distribution N(2, σ²) and P(X≤4)=0.88, find P(0<X<4). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.76"
} | {
"split": "train",
"index": 22143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Lucas sequence is the sequence 1, 3, 4, 7, 11, $\\ldots$ where the first term is 1, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the $100^{\\mathrm{th}}$ term of the sequence is divided by 8? Let's think step... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 22144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $x > 1,$ find the minimum value of\n\\[\\frac{x + 8}{\\sqrt{x - 1}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$ and $D$ are not necessarily distinct. If each of the two-digit primes is less than 40, how many such house numb... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 22146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the larger of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{48}"
} | {
"split": "train",
"index": 22147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac13"
} | {
"split": "train",
"index": 22148,
"solution": ""
} |
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