data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "At the beginning of the school year, $50\\%$ of all students in Mr. Well's class answered \"Yes\" to the question \"Do you love math\", and $50\\%$ answered \"No.\" At the end of the school year, $70\\%$ answered \"Yes\" and $30\\%$ answered \"No.\" Altogether, $x\\%$ of the st... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39749,
"solution": "Let's analyze the problem step by step to determine the minimum and maximum possible values of $x$, the percentage of students who changed their answers from the beginning to the end of the school year.\n\n1. **Initial Setup**:\n - At the beginning of the year, $... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral, and let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively. Find $\\frac{[G_A G_B G_C G_D]}{[ABCD]}.$\n\n[asy]\nunitsize(0.6 cm);\n\npair A, B, C, D;\npair[] G;\n\nA = (0,0);\nB = (7,1... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 39750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two intersecting circles O: $x^2 + y^2 = 25$ and C: $x^2 + y^2 - 4x - 2y - 20 = 0$, which intersect at points A and B, find the length of the common chord AB. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{95}"
} | {
"split": "train",
"index": 39751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student used the \"five-point method\" to draw the graph of the function $f(x)=A\\sin(\\omega x+\\varphi)$ ($\\omega\\ \\gt 0$, $|\\varphi|＜\\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:\n\n| $\\omega x+\\varphi$ | $0$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{24}"
} | {
"split": "train",
"index": 39752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial equation \\[x^3 + bx + c = 0,\\]where $b$ and $c$ are rational numbers, has $5-\\sqrt{2}$ as a root. It also has an integer root. What is it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 39753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class has prepared 6 programs to participate in the Xiamen No.1 Middle School Music Square event. The order of the programs has the following requirements: Programs A and B must be adjacent, and Programs C and D cannot be adjacent. How many possible arrangements of the progra... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 39754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( m \\) and \\( n \\) (with \\( m > n \\)) be positive integers such that \\( 70^2 \\) divides \\( 2023^m - 2023^n \\). What is the smallest value of \\( m+n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even integers are there between $200$ and $700$ whose digits are all different and come from the set $\\{1,2,5,7,8,9\\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39756,
"solution": "To find the number of even integers between $200$ and $700$ with all different digits from the set $\\{1,2,5,7,8,9\\}$, we analyze the conditions:\n\n1. **The number must be even**: This implies that the last digit (units place) of the number must be an even digit.... |
deepscaler | [
{
"role": "user",
"content": "There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking pa... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39757,
"solution": "Let's define the variables for the Venn Diagram:\n- $a$: Number of students taking exactly Bridge and Yoga.\n- $b$: Number of students taking exactly Bridge and Painting.\n- $c$: Number of students taking all three classes (Yoga, Bridge, and Painting).\n- $d$: Numb... |
deepscaler | [
{
"role": "user",
"content": "In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this po... | math | {
"style": "rule",
"ground_truth": "5/6"
} | {
"split": "train",
"index": 39758,
"solution": "1. **Identify the shapes and their properties:**\n - $A$, $E$, and $F$ are isosceles right triangles.\n - $B$, $C$, and $D$ are squares with side length $1$.\n - $G$ is an equilateral triangle.\n\n2. **Analyze the geometric arrangement:**\n - The squares su... |
deepscaler | [
{
"role": "user",
"content": "Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the ... | math | {
"style": "rule",
"ground_truth": "4807"
} | {
"split": "train",
"index": 39759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four functions: $①y=-x$, $②y=- \\frac{1}{x}$, $③y=x^{3}$, $④y=x^{ \\frac{1}{2}}$, determine the probability of the event \"the graphs of the two selected functions have exactly one common point\" when two functions are selected from these four. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 39760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits from 1 to 9 are randomly arranged to make a 9-digit number. What is the probability that the resulting number is divisible by 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"split": "train",
"index": 39761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(\\frac{{x-1}}{{x-3}}-\\frac{{x+1}}{x})÷\\frac{{{x^2}+3x}}{{{x^2}-6x+9}}$, where $x$ satisfies $x^{2}+2x-6=0$. 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": 39762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary can mow a lawn in four hours and Tom can mow the lawn in 5 hours. If Tom works for 2 hours alone, what fractional part of the lawn remains to be mowed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 39763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cátia leaves school every day at the same time and returns home by bicycle. When she pedals at $20 \\mathrm{~km/h}$, she arrives home at $4:30$ PM. If she pedals at $10 \\mathrm{~km/h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM? ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle and $k$ be a positive number such that altitudes $AD$, $BE$, and $CF$ are extended past $A$, $B$, and $C$ to points $A'$, $B'$, and $C'$ respectively, where $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$. Suppose further that $A''$ is a point such that the li... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 39765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When numbers are represented in \"base fourteen\", the digit before it becomes full is fourteen. If in \"base fourteen\", the fourteen digits are sequentially noted as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ten, J, Q, K, then convert the three-digit number JQK in \"base fourteen\" into ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cluster of circles ${C_n}:{({x-n})^2}+{({y-2n})^2}={n^2}(n≠0)$, the line $l:y=kx+b$ is a common tangent to them, then $k+b=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 39768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $8 \\times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39769,
"solution": "1. **Understanding the problem**: We are given an $8 \\times 18$ rectangle that is cut into two congruent hexagons. These hexagons are rearranged to form a square. We need to find the value of $y$, which is a dimension in the hexagon.\n\n2. **Area of the rectangle*... |
deepscaler | [
{
"role": "user",
"content": "Charlie is making a necklace with yellow beads and green beads. She has already used 4 green beads and 0 yellow beads. How many yellow beads will she have to add so that $\frac{4}{5}$ of the total number of beads are yellow? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39770,
"solution": "If $\frac{4}{5}$ of the beads are yellow, then $\frac{1}{5}$ are green. Since there are 4 green beads, the total number of beads must be $4 \times 5=20$. Thus, Charlie needs to add $20-4=16$ yellow beads."
} |
deepscaler | [
{
"role": "user",
"content": "Four circles are inscribed such that each circle touches the midpoint of each side of a square. The side of the square is 10 cm, and the radius of each circle is 5 cm. Determine the area of the square not covered by any circle. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "100 - 50\\pi"
} | {
"split": "train",
"index": 39771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sequence of positive integers starting from 1, some numbers are colored red according to the following rules: first color 1 red, then color the next 2 even numbers 2 and 4 red; next, color the 4 closest consecutive odd numbers after 4, which are 5, 7, 9, red; then color th... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 39772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 39773,
"solution": "First note that the integer clearly must be divisible by $9$ and $11$ since we can use the \"let the middle number be x\" trick. Let the number be $99k$ for some integer $k.$ Now let the $10$ numbers be $x,x+1, \\cdots x+9.$ We have $10x+45 = 99k.$ Taking mod $5$ y... |
deepscaler | [
{
"role": "user",
"content": "Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determi... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 39774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pirate finds three chests on the wrecked ship S.S. Triumph, recorded in base 7. The chests contain $3214_7$ dollars worth of silver, $1652_7$ dollars worth of precious stones, $2431_7$ dollars worth of pearls, and $654_7$ dollars worth of ancient coins. Calculate the total va... | math | {
"style": "rule",
"ground_truth": "3049"
} | {
"split": "train",
"index": 39775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) are \\( a \\), \\( b \\), and \\( c \\) respectively. Given that \\( a^2 - (b - c)^2 = (2 - \\sqrt{3})bc \\) and \\( \\sin A \\sin B = \\cos^2 \\frac{C}{2} \\), and the length o... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 39776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When someone tries to withdraw money from an ATM with a forgotten last digit of the password, calculate the probability that they will press the correct digit in no more than 2 attempts. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{5}"
} | {
"split": "train",
"index": 39777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\phi = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\text{(F)}"
} | {
"split": "train",
"index": 39778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: \\( 4.165 \\times 4.8 + 4.165 \\times 6.7 - 4.165 \\div \\frac{2}{3} = \\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41.65"
} | {
"split": "train",
"index": 39779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle \\(ABC\\) with \\(AB = AC\\) and \\(\\angle ABC = 53^\\circ\\), find the measure of \\(\\angle BAM\\). Point \\(K\\) is such that \\(C\\) is the midpoint of segment \\(AK\\). Point \\(M\\) is chosen such that:\n\n- \\(B\\) and \\(M\\) are on the same... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 39780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $... | math | {
"style": "rule",
"ground_truth": "\\text { No }"
} | {
"split": "train",
"index": 39781,
"solution": "\nTo determine if it is possible to transform the triangle with vertices \\((0,0)\\), \\((1,0)\\), and \\((0,1)\\) into the triangle with vertices \\((0,0)\\), \\((1,0)\\), and \\((1,1)\\) through a sequence of 90° counterclockwise rotations about lattice points, w... |
deepscaler | [
{
"role": "user",
"content": "We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. 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": "4"
} | {
"split": "train",
"index": 39782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.\nThereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is: Let's think step by step... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 39783,
"solution": "1. **Understanding the Problem:**\n We are given that $49$ of the first $50$ balls are red, and thereafter, $7$ out of every $8$ balls counted are red. We need to find the maximum number of balls, $n$, such that at least $90\\%$ of them are red.\n\n2. **Setting U... |
deepscaler | [
{
"role": "user",
"content": "$|3-\\pi|=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi-3"
} | {
"split": "train",
"index": 39784,
"solution": "1. **Identify the Expression**: We are given the expression $|3-\\pi|$ and need to evaluate its absolute value.\n\n2. **Understanding $\\pi$**: The value of $\\pi$ (pi) is approximately 3.14159, which is greater than 3.\n\n3. **Calculate $3 - \\pi$**: Since $\\pi >... |
deepscaler | [
{
"role": "user",
"content": "For the set $\\{1,2,\\cdots,n\\}$ and each of its non-empty subsets, define a unique \"alternating sum\" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alte... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 39785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of points common to the graphs of\n$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 39786,
"solution": "1. **Identify the equations of the lines**: \n From the given equations, using the Zero Product Property, we have:\n - From $(x-y+2)(3x+y-4)=0$:\n - $x-y+2=0 \\Rightarrow y = x - 2$\n - $3x+y-4=0 \\Rightarrow y = -3x + 4$\n - From $(x+y-2)(2x-5y+7)=0$... |
deepscaler | [
{
"role": "user",
"content": "Given that $a-b=3$, find the value of $1+2b-(a+b)$.\nGiven that $2^x=3$, find the value of $2^{2x-3}$. 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": 39787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. 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 step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 39788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2m\\sin x-2\\cos ^{2}x+ \\frac{m^{2}}{2}-4m+3$, and the minimum value of the function $f(x)$ is $(-7)$, find the value of the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integral solution to the equation $\\frac{1+3+5+\\dots+(2n-1)}{2+4+6+\\dots+2n}=\\frac{115}{116}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 39790,
"solution": "1. **Identify the series in the numerator and denominator:**\n The numerator is the sum of the first $n$ odd numbers, which can be expressed as:\n \\[\n 1 + 3 + 5 + \\dots + (2n-1)\n \\]\n The denominator is the sum of the first $n$ even numbers, which ca... |
deepscaler | [
{
"role": "user",
"content": "A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, \"If you buy one more, you can get a 10% discount, which is $36 cheaper than now.\" Xiao Hua said, \"Then I'll buy one more, thank you.\"... | math | {
"style": "rule",
"ground_truth": "486"
} | {
"split": "train",
"index": 39791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles are centred at the origin. The point $P(8,6)$ is on the larger circle and the point $S(0, k)$ is on the smaller circle. If $Q R=3$, what is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39792,
"solution": "We can determine the distance from $O$ to $P$ by dropping a perpendicular from $P$ to $T$ on the $x$-axis. We have $O T=8$ and $P T=6$, so by the Pythagorean Theorem, $O P^{2}=O T^{2}+P T^{2}=8^{2}+6^{2}=64+36=100$. Since $O P>0$, then $O P=\\sqrt{100}=10$. Therefo... |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD,\\ BC=8,\\ CD=12,\\ AD=10,$ and $m\\angle A= m\\angle B = 60^\\circ.$ Given that $AB = p + \\sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 39793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $$\\lceil\\sqrt{10}\\rceil + \\lceil\\sqrt{11}\\rceil + \\lceil\\sqrt{12}\\rceil + \\cdots + \\lceil\\sqrt{34}\\rceil$$\nNote: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$ Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 39794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many squares can be formed by joining four of the twelve points marked on a rectangular grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum possible number of diagonals of equal length in a convex hexagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39796,
"solution": "First, we will prove that 7 is possible. Consider the following hexagon \\(A B C D E F\\) whose vertices are located at \\(A(0,0), B\\left(\\frac{1}{2}, 1-\\frac{\\sqrt{3}}{2}\\right), C\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right), D(0,1), E\\left(-\\frac{1}{2... |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$ , $AB = 10$ , and $AB = 2BC$ . Let $M$ be the midpoint of $CD$ , and suppose that $BM = 2AM$ . Compute $AM$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 39797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer $N$ is selected at random in the range $1 \\leq N \\leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 39798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(\\alpha\\\\) be an acute angle. If \\\\(\\cos (\\alpha+ \\dfrac {\\pi}{6})= \\dfrac {3}{5}\\\\), then \\\\(\\sin (\\alpha- \\dfrac {\\pi}{6})=\\\\) \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {4-3 \\sqrt {3}}{10}"
} | {
"split": "train",
"index": 39799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A block of wood has the shape of a right circular cylinder with a radius of $8$ and a height of $10$. The entire surface of the block is painted red. Points $P$ and $Q$ are chosen on the edge of one of the circular faces such that the arc $\\overarc{PQ}$ measures $180^\\text{o}... | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 39800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 39801,
"solution": "1. **Identify the total number of girls and their characteristics:**\n - Total number of girls = $50$\n - Characteristics: blonde or brunette, blue-eyed or brown-eyed\n\n2. **Calculate the number of blondes:**\n - Given that there are $31$ brunettes, the numb... |
deepscaler | [
{
"role": "user",
"content": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 39802,
"solution": "1. **Identify the number of students in each class:**\n - Mrs. Germain's class: 11 students\n - Mr. Newton's class: 8 students\n - Mrs. Young's class: 9 students\n\n2. **Assumption of no overlap in students:**\n Since the problem does not mention any studen... |
deepscaler | [
{
"role": "user",
"content": "The measures of the three interior angles of a triangle are $50^\\circ$, $55^\\circ$ and $x^\\circ$. What is the degree measure of the largest interior angle of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 39803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve $\\log_4 x + \\log_2 x^2 = 10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya plans to spend all 90 days of his vacation in the village, swimming in the lake every second day (i.e., every other day), going shopping for groceries every third day, and solving math problems every fifth day. (On the first day, Petya did all three tasks and got very tir... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $2, 7, 12, a, b, 27$ is arithmetic. What is the value of $a + b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 39806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 16 positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 39807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $AB$ is perpendicular to $BC$, and $CD$ is perpendicular to $AD$. Also, $AC = 625$ and $AD = 600$. If $\\angle BAC = 2 \\angle DAC$, what is the length of $BC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 39808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(\\{x_{1},x_{2},x_{3},x_{4}\\} \\subseteq \\{x | (x-3) \\cdot \\sin \\pi x = 1, x > 0\\}\\\\), find the minimum value of \\\\(x_{1}+x_{2}+x_{3}+x_{4}\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive integers. All the roots of each of the quadratics \\(a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c\\) are integers. Over all triples \\((a, b, c)\\), find the triple with the third smallest value of \\(a+b+c\\). Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "(1,10,24)"
} | {
"split": "train",
"index": 39810,
"solution": "The quadratic formula yields that the answers to these four quadratics are \\(\\frac{ \\pm b \\pm \\sqrt{b^{2} \\pm 4 a c}}{2 a}\\). Given that all eight of these expressions are integers, we can add or subtract appropriate pairs to get that \\(\\frac{b}{a}\\) and ... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $mx^{2}+3my^{2}=1$ ($m > 0$) with a major axis length of $2\\sqrt{6}$, and $O$ is the origin.\n\n$(1)$ Find the equation of the ellipse $C$.\n\n$(2)$ Let point $A(3,0)$, point $B$ be on the $y$-axis, and point $P$ be on the ellipse $C$ and to the right of... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 39811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\\delta=0.3$, given that the standard deviation $\\sigma=1.2$ of the normally distributed population is known. Let's thi... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 39812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive real numbers \\(a\\) and \\(b\\), define \\(a \\circ b=a+b+2 \\sqrt{a b}\\). Find all positive real numbers \\(x\\) such that \\(x^{2} \\circ 9x=121\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31-3\\sqrt{53}}{2}"
} | {
"split": "train",
"index": 39813,
"solution": "Since \\(a \\circ b=(\\sqrt{a}+\\sqrt{b})^{2}\\), we have \\(x^{2} \\circ 9x=(x+3\\sqrt{x})^{2}\\). Moreover, since \\(x\\) is positive, we have \\(x+3\\sqrt{x}=11\\), and the only possible solution is that \\(\\sqrt{x}=\\frac{-3+\\sqrt{53}}{2}\\), so \\(x=\\frac{3... |
deepscaler | [
{
"role": "user",
"content": "In response to the national call for \"entrepreneurship and innovation for everyone,\" Xiao Wang decided to start a business in his field after graduating from college. After market research, Xiao Wang found that the annual fixed cost for producing a certain small electronic pr... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the squares of the roots of the equation \\[x^{2018} +44x^{2015} + 3x^3 + 404 = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 39816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39817,
"solution": "1. **Define the variables:**\n Let $M$ represent the temperature in Minneapolis at noon, and $L$ represent the temperature in St. Louis at the same time. Given that Minneapolis is $N$ degrees warmer than St. Louis, we can express this relationship as:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that $y$ is a multiple of $42522$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+17)$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 39818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor completely over the set of polynomials with integer coefficients:\n\\[4(x + 5)(x + 6)(x + 10)(x + 12) - 3x^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2x^2 + 35x + 120)(x + 8)(2x + 15)"
} | {
"split": "train",
"index": 39819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Bob was instructed to subtract 5 from a certain number and then divide the result by 7, but instead subtracted 7 and then divided by 5, yielding an answer of 47, determine what his answer would have been had he worked the problem correctly. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 39820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $e^{i \\theta} = \\frac{3 + i \\sqrt{8}}{5}$, find $\\cos 4 \\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{287}{625}"
} | {
"split": "train",
"index": 39821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( x_{1}, x_{2}, \\ldots, x_{13} \\) are such that \\( \\frac{1}{x_{1}} + \\frac{1}{x_{2}} + \\ldots + \\frac{1}{x_{13}} = 2 \\). What is the minimum value of the sum of these numbers? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 39822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add 3 digits after 325 to make a six-digit number such that it is divisible by 3, 4, and 5, and make this number as small as possible. What is the new six-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "325020"
} | {
"split": "train",
"index": 39823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the linear function $y=-3x+m$ (where $m$ is a constant) passes through the second, third, and fourth quadrants, then the possible values of $m$ are _____. (Write down one possible value) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{4} + \\frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $\\frac{\\sqrt{3}}{2}$, find the slope of the line that intersects the ellipse at points $A$ and $B$, where the midpoint of segment $AB$ is $M(-2, 1)$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $a = (2, -1, 3)$, $b = (-1, 4, -2)$, and $c = (7, 5, \\lambda)$, if vectors $a$, $b$, and $c$ are coplanar, the real number $\\lambda$ equals ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{9}"
} | {
"split": "train",
"index": 39826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent sides of a rhombus form a $60$-degree angle. If each side of the rhombus measures $2$ cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 39827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33\\frac {1}{3}\\%"
} | {
"split": "train",
"index": 39828,
"solution": "1. **Define Variables:**\n Let $C$ be the cost of the product and $S$ be the selling price. According to the problem, Adams plans a profit of 10% on the selling price, and his expenses are 15% of the selling price.\n\n2. **Express the Relationships:**\n The pro... |
deepscaler | [
{
"role": "user",
"content": "Digits are placed in the two boxes of $2 \\square \\square$, with one digit in each box, to create a three-digit positive integer. In how many ways can this be done so that the three-digit positive integer is larger than 217? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 39829,
"solution": "The question is equivalent to asking how many three-digit positive integers beginning with 2 are larger than 217. These integers are 218 through 299 inclusive. There are $299 - 217 = 82$ such integers."
} |
deepscaler | [
{
"role": "user",
"content": "A keen archaeologist is holding a competition where participants must guess the age of a rare artifact. The age of the artifact is formed using the six digits: 2, 2, 3, 3, 7, and 9, and it must begin with an odd digit.\n\nHow many different ages can be there for the artifact? L... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 39830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "619"
} | {
"split": "train",
"index": 39831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated t... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 39832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 39833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of $a?$ Let's think step by... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a group of 10 basketball teams which includes 2 strong teams, the teams are randomly split into two equal groups for a competition. What is the probability that the 2 strong teams do not end up in the same group? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 39835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_{n}\\}$, $a_{1}=1$, $\\sqrt{{a}_{n+1}}-\\sqrt{{a}_{n}}=1$ ($n\\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\\{b_{n}\\}$ is $S_{n}=2^{n}-m$. For $n\\in N^{*}$, the smallest value of the real number $\\lambda$ that satisfies $\\lambda... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 39836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arcsin 1.$ Express your answer in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 39837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{148}{590}"
} | {
"split": "train",
"index": 39838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"split": "train",
"index": 39839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\cos ^{8}\\left(\\frac{x}{2}\\right) dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35\\pi}{8}"
} | {
"split": "train",
"index": 39840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of positive odd numbers $\\{1, 3, 5, \\cdots\\}$ is arranged in ascending order and grouped by the $n$th group having $(2n-1)$ odd numbers as follows:\n$$\n\\begin{array}{l}\n\\{1\\}, \\quad \\{3,5,7\\}, \\quad \\{9,11,13,15,17\\}, \\cdots \\\\\n\\text{ (First group) (S... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 39841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four distinct points P, A, B, C on a plane satisfying $\\overrightarrow{PA} + \\overrightarrow{PB} + \\overrightarrow{PC} = \\overrightarrow{0}$ and $\\overrightarrow{AB} + \\overrightarrow{AC} + m\\overrightarrow{AP} = \\overrightarrow{0}$, determine the value of the rea... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 39842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In regular pentagon $ABCDE$, diagonal $AC$ is drawn, as shown. Given that each interior angle of a regular pentagon measures 108 degrees, what is the measure of angle $CAB$?\n\n[asy]\nsize(4cm,4cm);\ndefaultpen(linewidth(1pt)+fontsize(10pt));\n\npair A,B,C,D,E;\nA = (0,0);\nB =... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 39843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ . Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 39844,
"solution": "[asy] size(15cm,0); draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); draw((-1,2)--(9,2)); draw((0,0)--(2,2)); draw((2,2)--(1,1)); draw((0,0)--(4,2)); draw((0,2)--(1,1)); draw(circle((0,1),1)); draw(circle((4,-3),5)); dot((0,0)); dot((0,2)); dot((2,2)); dot((4,2)); dot((4,... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of $m$ such that any $m$ -element subset of the set of integers $\\{1,2,...,2016\\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\\le 3$ . Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 39845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of positive integers less than 10! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 39846,
"solution": "Since $0!=1!=1$, we ignore any possible 0!'s in our sums. Call a sum of factorials reduced if for all positive integers $k$, the term $k$! appears at most $k$ times. It is straightforward to show that every positive integer can be written uniquely as a reduced sum ... |
deepscaler | [
{
"role": "user",
"content": "There exist $s$ unique nonnegative integers $m_1 > m_2 > \\cdots > m_s$ and $s$ integers $b_k$ ($1\\le k\\le s$), with each $b_k$ either $1$ or $-1$, such that \\[b_13^{m_1} + b_23^{m_2} + \\cdots + b_s3^{m_s} = 1007.\\] Find $m_1 + m_2 + \\cdots + m_s$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle A = 100^\\circ$, $\\angle B = 50^\\circ$, $\\angle C = 30^\\circ$, $\\overline{AH}$ is an altitude, and $\\overline{BM}$ is a median. Then $\\angle MHC=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30^\\circ"
} | {
"split": "train",
"index": 39848,
"solution": "1. **Identify Given Information**: In $\\triangle ABC$, we have $\\angle A = 100^\\circ$, $\\angle B = 50^\\circ$, and $\\angle C = 30^\\circ$. $\\overline{AH}$ is an altitude, and $\\overline{BM}$ is a median.\n\n2. **Properties of Median**: Since $\\overline{BM}$... |
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