data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $\\frac{1}{x} + \\frac{1}{2y} = \\frac{1}{7}$, what is the least possible value of $xy$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 2800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 2801,
"solution": "1. **Identify the initial amount of water**: The pool initially contains 200 gallons of water.\n\n2. **Determine the rate of evaporation**: Water evaporates at a rate of 0.5 gallons per day.\n\n3. **Calculate the total evaporation over 30 days**: Multiply the daily ... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest \\( n > 4 \\) for which we can find a graph on \\( n \\) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the fractions that have the value $\\frac{45}{56}$ and have a numerator that is a perfect square and a denominator that is a perfect cube? Among these fractions, which one is in the simplest form? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{525^2}{70^3}"
} | {
"split": "train",
"index": 2803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 2804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that the distance between any two points selected from the four vertices and the center of a square is not less than the side length of the square is $\\boxed{\\text{answer}}$. 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": 2805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $*$ is defined by\n\\[a * b = \\frac{a - b}{1 - ab}.\\]Compute\n\\[1 * (2 * (3 * (\\dotsb (999 * 1000) \\dotsb))).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right tetrahedron \\(ABCD\\), \\(DA\\), \\(DB\\), and \\(DC\\) are mutually perpendicular. Let \\(S\\) and \\(R\\) represent its surface area and the radius of the circumscribed sphere, respectively. What is the maximum value of \\(\\frac{S}{R^2}\\)? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}(3+\\sqrt{3})"
} | {
"split": "train",
"index": 2807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Let there be $P_{n}$ ways for Pretty Penny to make exactly $n$ dollars out of quarters, dimes, nickels, and pennies. Also, let there be $B_{n}$ ways for Beautiful Bill to make exactly $n$ dollars out of one dollar bills, quarters, dimes, and nicke... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2808,
"solution": "Let $d_{x}$ be the number ways to make exactly $x$ cents using only dimes and nickels. It is easy to see that when $x$ is a multiple of 5 , $$d_{x}=\\left\\lfloor\\frac{x}{10}\\right\\rfloor+1$$ Now, let $c_{x}$ be the number of ways to make exactly $x$ cents using ... |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $$81^{2x} = 27^{3x - 4}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x = 12"
} | {
"split": "train",
"index": 2809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 8\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 2810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $1-2+3-4+ \\dots -98+99$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 2811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular painting that measures $12$ inches high and $16$ inches wide, and is placed in a frame that forms a border three inches wide on all sides, find the area of the border, in square inches. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 2812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular $17$ -gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$ . A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ ... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 2813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos x\\cdot\\sin \\left(x+ \\frac {\\pi}{3}\\right)- \\sqrt {3}\\cos ^{2}x+ \\frac { \\sqrt {3}}{4}$, $x\\in\\mathbb{R}$. \n(I) Find the smallest positive period of $f(x)$. \n(II) Find the maximum and minimum values of $f(x)$ on the closed interval ... | math | {
"style": "rule",
"ground_truth": "- \\frac {1}{2}"
} | {
"split": "train",
"index": 2814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that $f(0)\\neq 0$ and for all $x,y\\in\\mathbb{R}$,\n \\[ f(x+y)^2 = 2f(x)f(y) + \\max \\left\\{ f(x^2+y^2), f(x^2)+f(y^2) \\right\\}. \\] Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "f(x) = -1 \\text{ and } f(x) = x - 1."
} | {
"split": "train",
"index": 2815,
"solution": "\nTo find all functions \\( f:\\mathbb{R}\\rightarrow\\mathbb{R} \\) satisfying the given functional equation, we start with the condition:\n\\[ f(0)\\neq 0, \\]\nand the functional equation for all \\( x, y \\in \\mathbb{R} \\):\n\\[ f(x+y)^2 = 2f(x)f(y) + \\max \\... |
deepscaler | [
{
"role": "user",
"content": "Let \\( m \\) be the largest positive integer such that for every positive integer \\( n \\leqslant m \\), the following inequalities hold:\n\\[ \n\\frac{2n + 1}{3n + 8} < \\frac{\\sqrt{5} - 1}{2} < \\frac{n + 7}{2n + 1} \n\\]\nWhat is the value of the positive integer \\( m \\... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 2816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral \\(ABCD\\),\n\\[\n\\angle BAD = \\angle BCD = 120^\\circ, \\quad BC = CD = 10.\n\\]\n\nFind \\(AC.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the function\n\n$$\nf(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\\ldots+(x-104)^{2}\n$$\n\nIf the result is a non-integer, round it to the nearest integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49608"
} | {
"split": "train",
"index": 2818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest value of $n$ such that $5x^2 + nx + 90$ can be factored as the product of two linear factors with integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "451"
} | {
"split": "train",
"index": 2819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Handy Aaron helped a neighbor $1 \\frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\\textdollar 3$ per hour. How much did he earn for the week? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\$15"
} | {
"split": "train",
"index": 2820,
"solution": "1. **Convert all time worked into minutes:**\n - **Monday:** Aaron worked for $1 \\frac{1}{4}$ hours. Converting this to minutes:\n \\[\n 1 \\frac{1}{4} \\text{ hours} = \\frac{5}{4} \\text{ hours} \\times 60 \\text{ minutes/hour} = 75 \\text{ minutes}\n ... |
deepscaler | [
{
"role": "user",
"content": "Suppose point \\(P\\) is inside triangle \\(ABC\\). Let \\(AP, BP\\), and \\(CP\\) intersect sides \\(BC, CA\\), and \\(AB\\) at points \\(D, E\\), and \\(F\\), respectively. Suppose \\(\\angle APB=\\angle BPC=\\angle CPA, PD=\\frac{1}{4}, PE=\\frac{1}{5}\\), and \\(PF=\\frac{1... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{12}"
} | {
"split": "train",
"index": 2821,
"solution": "The key is the following lemma: Lemma: If \\(\\angle X=120^{\\circ}\\) in \\(\\triangle XYZ\\), and the bisector of \\(X\\) intersects \\(YZ\\) at \\(T\\), then \\(\\frac{1}{XY}+\\frac{1}{XZ}=\\frac{1}{XT}\\). Proof of the Lemma. Construct point \\(W\\) on \\(XY\\) ... |
deepscaler | [
{
"role": "user",
"content": "In the first week after his birthday, Bill's uncle gave him some money for his piggy bank. Every week after that Bill put $2 into his piggy bank. At the end of the ninth week after his birthday, Bill had trebled the amount he started with. How much did he have in total at the e... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 2822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 2823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with pen... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 2824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$ , such that this is always possible if: $a)$ The stu... | math | {
"style": "rule",
"ground_truth": "k = 2"
} | {
"split": "train",
"index": 2825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To measure the height of tower AB on the opposite bank of a river, a point C is chosen on the bank such that the base of the tower A is exactly to the west of point C. At this point, the elevation angle to the top of the tower B is measured to be 45°. Then, moving from point C ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), angle \\(C\\) is a right angle, and \\(AC: AB = 4: 5\\). A circle with its center on leg \\(AC\\) is tangent to the hypotenuse \\(AB\\) and intersects leg \\(BC\\) at point \\(P\\), such that \\(BP: PC = 2: 3\\). Find the ratio of the radius of the circle... | math | {
"style": "rule",
"ground_truth": "13/20"
} | {
"split": "train",
"index": 2827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha + \\cos \\alpha = -\\frac{3}{\\sqrt{5}}$, and $|\\sin \\alpha| > |\\cos \\alpha|$, find the value of $\\tan \\frac{\\alpha}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5} + 1}{2}"
} | {
"split": "train",
"index": 2828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p \\) and \\( q \\) be the two distinct solutions to the equation\n\\[ (x-6)(3x+10) = x^2 - 19x + 50. \\]\nWhat is \\( (p + 2)(q + 2) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 2829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 2830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $A = 3009 \\div 3$, $B = A \\div 3$, and $Y = A - B$, then what is the value of $Y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "669"
} | {
"split": "train",
"index": 2831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the product of the digits $X$ and $Y$ such that the number $590\\,X34\\,52Y$ is divisible by $45$. What is the sum of all distinct possible products $XY$, with each product counted only once regardless of the pair ($X$, $Y$) variations that produce it? Let's think ste... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{366}{31 \\times 24}"
} | {
"split": "train",
"index": 2833,
"solution": "1. **Identify the total rainfall and the time period**: The problem states that 366 inches of rain fell in one month (July). \n\n2. **Convert the time period from months to hours**:\n - July has 31 days.\n - Each day has 24 hours.\n - Therefore, July has \\(31... |
deepscaler | [
{
"role": "user",
"content": "The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $α$ is an acute angle, and $\\sin (α - \\frac{π}{4})= \\frac{3}{5}$, then $\\cos 2α$ equals ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{25}"
} | {
"split": "train",
"index": 2835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers \\( n \\), with \\( 2 \\leq n \\leq 80 \\), is \\( \\frac{(n-1)(n)(n+1)}{8} \\) equal to an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 2836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{81}"
} | {
"split": "train",
"index": 2837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 + i) z,$ and $2 \\overline{z},$ respectively, where $|z| = 1.$ When $P,$ $Q$, and $R$ are not collinear, let $S$ be the fourth vertex of the parallelogram $PQSR.$ What is the maximum distance between... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \\(4: 3\\). Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{23}"
} | {
"split": "train",
"index": 2839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of intersection points of 10 lines, given that only two of them are parallel and exactly three of these lines intersect at one point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 2840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a certain basketball player has a 50% chance of making each shot, we use a random simulation method to estimate the probability that the player makes exactly two out of four shots: First, we generate a random integer between 0 and 9 using a calculator, where 0, 1, 2,... | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"split": "train",
"index": 2841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store decides to discount a certain type of clothing due to the change of season. If each piece of clothing is sold at 50% of the marked price, there will be a loss of 20 yuan per piece. However, if sold at 80% of the marked price, there will be a profit of 40 yuan per piece.... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 2842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers 1 to 2031 are written on a blackboard.\n1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.\n2. Then select another number on the blackboard, find the absolute value of its difference from the previo... | math | {
"style": "rule",
"ground_truth": "2030"
} | {
"split": "train",
"index": 2843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Another professor enters the same room and says, 'Each of you has to write down an integer between 0 and 200. I will then compute $X$, the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 2844,
"solution": "Use the same logic to get 7. Note 6 and 8 do not work."
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\\sin \\frac{A}{2},\\cos \\frac{A}{2})$ and $n=(\\cos \\frac{A}{2},-\\cos \\frac{A}{2})$, and $2m\\cdot n+|m|=\\frac{ \\sqrt{2}}{2}$, find $\\angle A=$____. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi }{12}"
} | {
"split": "train",
"index": 2845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi pain... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 2846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = ax^6 + bx^4 - cx^2 + 3.$ If $f(91) = 1$, find $f(91) + f(-91)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time... | math | {
"style": "rule",
"ground_truth": "8.4"
} | {
"split": "train",
"index": 2848,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Problem\nSolve in integers the equation \\[x^2+xy+y^2 = \\left(\\frac{x+y}{3}+1\\right)^3.\\] \nSolution\nWe first notice that both sides must be integers, so $\\frac{x+y}{3}$ must be an integer.\nWe can therefore perform the substitution $x+y = 3t$ where $t$ is an integer.\nTh... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{2} \\left(3p(p-1) \\pm \\sqrt{4(p(p-1)+1)^3 - 27(p(p-1))^2} \\right), 3p(p-1) - \\frac{1}{2} \\left(3p(p-1) \\pm \\sqrt{4(p(p-1)+1)^3 - 27(p(p-1))^2} \\right) \\right)"
} | {
"split": "train",
"index": 2849,
"solution": "Let $n = \\frac{x+y}{3}$ .\nThus, $x+y = 3n$ .\nWe have \\[x^2+xy+y^2 = \\left(\\frac{x+y}{3}+1\\right)^3 \\implies (x+y)^2 - xy = \\left(\\frac{x+y}{3}+1\\right)^3\\] Substituting $n$ for $\\frac{x+y}{3}$ , we have \\[9n^2 - x(3n-x) = (n+1)^3\\] Treating $x$ as a v... |
deepscaler | [
{
"role": "user",
"content": "Jerry and Neil have a 3-sided die that rolls the numbers 1, 2, and 3, each with probability $\\frac{1}{3}$. Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is 3. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{18}"
} | {
"split": "train",
"index": 2850,
"solution": "If Jerry rolls $k$, then there is a $\\frac{1}{4-k}$ probability that Neil's number is 3, since Neil has an equal chance of rolling any of the $4-k$ integers not less than $k$. Thus, the answer is $$\\frac{1}{3}\\left(1+\\frac{1}{2}+\\frac{1}{3}\\right)=\\frac{11}{1... |
deepscaler | [
{
"role": "user",
"content": "Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of max... | math | {
"style": "rule",
"ground_truth": "618"
} | {
"split": "train",
"index": 2851,
"solution": "It is well known that $\\lim_{n\\rightarrow\\infty} \\frac{F_{n-1}}{F_n} = \\phi - 1 =\\frac{1 + \\sqrt{5}}{2} - 1 \\approx .61803$, so $1000 \\cdot \\frac{F_{n-1}}{F_n}$ approaches $x = \\boxed{618}.$"
} |
deepscaler | [
{
"role": "user",
"content": "Let $y=f(x)$ be a quadratic function, and the equation $f(x)=0$ has two equal real roots. Also, $f'(x)=2x+2$.\n1. Find the expression for $y=f(x)$.\n2. Find the area of the shape enclosed by the graph of $y=f(x)$ and the two coordinate axes.\n3. If the line $x=-t$ ($0<t<1$) div... | math | {
"style": "rule",
"ground_truth": "1-\\frac{1}{32}"
} | {
"split": "train",
"index": 2852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\(a, b, c\\) and a positive number \\(\\lambda\\) such that \\(f(x)=x^{3}+a x^{2}+b x+c\\) has three real roots \\(x_{1}, x_{2}, x_{3}\\), satisfying:\n(1) \\(x_{2}-x_{1}=\\lambda\\);\n(2) \\(x_{3}>\\frac{1}{2}\\left(x_{1}+x_{2}\\right)\\).\n\nFind the maximum val... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 2853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 2854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\( AB \\) and \\( CD \\) of the trapezoid \\( ABCD \\) are equal to 65 and 31 respectively, and its lateral sides are mutually perpendicular. Find the dot product of the vectors \\( \\overrightarrow{AC} \\) and \\( \\overrightarrow{BD} \\). Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "-2015"
} | {
"split": "train",
"index": 2855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C:\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right vertex at point $A$, a circle $A$ is created with center at $A$ and radius $b$. Circle $A$ intersects with one of the asymptotes of hyperbola $C$ at points $M$ and $N$. If $\\angle... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 2856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the cubic equation $10x^3 - 25x^2 + 8x - 1 = 0$, whose roots are $p$, $q$, and $s$, all of which are positive and less than 1. Calculate the sum of\n\\[\\frac{1}{1-p} + \\frac{1}{1-q} + \\frac{1}{1-s}.\\] Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 2857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all positive reals $ a$ , $ b$ , and $ c$ , what is the value of positive constant $ k$ satisfies the following inequality?\r $ \\frac{a}{c\\plus{}kb}\\plus{}\\frac{b}{a\\plus{}kc}\\plus{}\\frac{c}{b\\plus{}ka}\\geq\\frac{1}{2007}$ . Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "6020"
} | {
"split": "train",
"index": 2858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mayar and Rosie are 90 metres apart. Starting at the same time, they run towards each other. Mayar runs twice as fast as Rosie. How far has Mayar run when they meet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 2859,
"solution": "Suppose that Rosie runs \\(x\\) metres from the time that they start running until the time that they meet. Since Mayar runs twice as fast as Rosie, then Mayar runs \\(2x\\) metres in this time. When Mayar and Rosie meet, they will have run a total of 90 m, since be... |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ and $T_{n}$ denote the sum of the first $n$ terms of the arithmetic sequences ${ a_{n} }$ and ${ b_{n} }$, respectively. Given that $\\frac{S_{n}}{T_{n}} = \\frac{7n}{n+3}$, find the value of $\\frac{a_{5}}{b_{5}}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{4}"
} | {
"split": "train",
"index": 2860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To solve the problem, we need to find the value of $\\log_{4}{\\frac{1}{8}}$.\nA) $-\\frac{1}{2}$\nB) $-\\frac{3}{2}$\nC) $\\frac{1}{2}$\nD) $\\frac{3}{2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 2861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $3 \\times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c_i$ denote the $i$ th composite integer so that $\\{c_i\\}=4,6,8,9,...$ Compute\n\\[\\prod_{i=1}^{\\infty} \\dfrac{c^{2}_{i}}{c_{i}^{2}-1}\\]\n(Hint: $\\textstyle\\sum^\\infty_{n=1} \\tfrac{1}{n^2}=\\tfrac{\\pi^2}{6}$ ) Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{\\pi^2}"
} | {
"split": "train",
"index": 2863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class participates in a tree-planting event, divided into three groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x) - 2 f \\left( \\frac{1}{x} \\right) = 4^x\\]for all $x \\neq 0.$ Find $f(2).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{20}{3}"
} | {
"split": "train",
"index": 2865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid \\(AEFG\\) (\\(EF \\parallel AG\\)) is positioned inside a square \\(ABCD\\) with a side length of 14, such that points \\(E\\), \\(F\\), and \\(G\\) lie on sides \\(AB\\), \\(BC\\), and \\(CD\\) respectively. The diagonals \\(AF\\) and \\(EG\\) are perpendicular, a... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 2866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 180$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac1{72}"
} | {
"split": "train",
"index": 2867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 2868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of\n\\[\\frac{(x^2 + 3x + 1)(y^2 + 3y + 1)(z^2 + 3z + 1)}{xyz}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 2869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 2870,
"solution": "To solve the equation $\\frac{64^{x-1}}{4^{x-1}} = 256^{2x}$, we start by expressing all terms with base 2:\n\n1. **Expressing each term with base 2:**\n - $64 = 2^6$, so $64^{x-1} = (2^6)^{x-1} = 2^{6(x-1)}$\n - $4 = 2^2$, so $4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)}... |
deepscaler | [
{
"role": "user",
"content": "Transport Teams A and B need to deliver a batch of relief supplies to an earthquake-stricken area. Team A can transport 64.4 tons per day, which is 75% more than Team B can transport per day. If both teams transport the supplies simultaneously, when Team A has transported half ... | math | {
"style": "rule",
"ground_truth": "644"
} | {
"split": "train",
"index": 2871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tens digit of the smallest positive integer that is divisible by each of 20, 16, and 2016? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 2872,
"solution": "We note that $20=2^{2} \\cdot 5$ and $16=2^{4}$ and $2016=16 \\cdot 126=2^{5} \\cdot 3^{2} \\cdot 7$. For an integer to be divisible by each of $2^{2} \\cdot 5$, $2^{4}$, and $2^{5} \\cdot 3^{2} \\cdot 7$, it must include at least 5 factors of 2, at least 2 factors ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $(12345679^2 \\times 81 - 1) \\div 11111111 \\div 10 \\times 9 - 8$ in billions. (Answer in billions) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x) = 2\\sin x\\cos x + 2\\cos\\left(x + \\frac{\\pi}{4}\\right)\\cos\\left(x - \\frac{\\pi}{4}\\right)$,\n1. Find the interval(s) where $f(x)$ is monotonically decreasing.\n2. If $\\alpha \\in (0, \\pi)$ and $f\\left(\\frac{\\alpha}{2}\\right) = \\frac{\\sqr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 2874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The radius of the circumcircle of the acute-angled triangle \\(ABC\\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \\(A\\), \\(C\\), and the orthocenter of triangle \\(ABC\\). Find \\(AC\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 2875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(\\sqrt{3}\\cos x,-\\cos x)$ and $\\overrightarrow{n}=(\\cos (x-\\frac{π}{2}),\\cos x)$, satisfying the function $f\\left(x\\right)=\\overrightarrow{m}\\cdot \\overrightarrow{n}+\\frac{1}{2}$. \n$(1)$ Find the interval on which $f\\left(x\\ri... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{3}-5}{26}"
} | {
"split": "train",
"index": 2876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sum of the binomial coefficients of the odd terms in the expansion of ${(x-\\frac{2}{x})}^{n}$ is $16$, then the coefficient of $x^{3}$ in the expansion is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 2877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( g(x) \\) is a rational function such that \\( 4g\\left(\\frac{1}{x}\\right) + \\frac{3g(x)}{x} = x^3 \\) for \\( x \\neq 0 \\). Find \\( g(-3) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{6565}{189}"
} | {
"split": "train",
"index": 2878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the three internal angles are $A$, $B$, and $C$. Find the value of $A$ for which $\\cos A + 2\\cos\\frac{B+C}{2}$ attains its maximum value, and determine this maximum value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 2879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $\nz^{10} + z^9 + z^6+z^5+z^4+z+1\n$ divides $z^k-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 2880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. What is the perimeter of $ABCD$? [asy]\n/* AMC8 1999 #14 Problem */\npen p = linetype(\"4 4\");\ndraw((0,0)--(4,3)--(12,3)--(16,0)--cycle);\ndraw((4,0)--(4,3), p);\ndraw((3.5,0)--(3.5, .5)--(4.0,0.5));\n\nlabel(scale(0.75)... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 2881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of all integers between 50 and 450 that end in 1 or 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19920"
} | {
"split": "train",
"index": 2882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.\n Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 2883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integers that can be expressed as a sum of three distinct numbers chosen from the set $\\{4,7,10,13, \\ldots,46\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 2884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that a randomly drawn positive factor of $60$ is less than $7$? 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": 2885,
"solution": "\nTo solve this problem, we need to determine the total number of positive factors of $60$ and how many of these factors are less than $7$. We then calculate the probability by dividing the number of favorable outcomes (factors less than $7$) by the total number of ... |
deepscaler | [
{
"role": "user",
"content": "**\nConsider a set of numbers $\\{1, 2, 3, 4, 5, 6, 7\\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.\n\n** Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{21}"
} | {
"split": "train",
"index": 2886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$ for every positive integer $n$ , where $a_0 = a_1 = 1$ . Calculate the sum $$ \\frac{a_0}{a_1} + \\frac{a_1}{a_2} + ... + \\frac{a_{2008}}... | math | {
"style": "rule",
"ground_truth": "2009 * 1005"
} | {
"split": "train",
"index": 2887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\\{1,2,3, 4, 5,6\\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 2888,
"solution": "To determine how many quadratic equations of the form $x^2 + bx + c = 0$ have real roots with $b$ and $c$ chosen from the set $\\{1, 2, 3, 4, 5, 6\\}$, we need to ensure that the discriminant of the quadratic equation is non-negative. The discriminant $\\Delta$ for ... |
deepscaler | [
{
"role": "user",
"content": "Given that b is an even number between 1 and 11 (inclusive) and c is any natural number, determine the number of quadratic equations x^{2} + bx + c = 0 that have two distinct real roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 2889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\triangle ABC$ and a point $P$ on one of its sides, call line $\\ell$ the $\\textit{splitting line}$ of $\\triangle ABC$ through $P$ if $\\ell$ passes through $P$ and divides $\\triangle ABC$ into two polygons of equal perimeter. Let $\\triangle ABC$ be a triangle where... | math | {
"style": "rule",
"ground_truth": "459"
} | {
"split": "train",
"index": 2890,
"solution": "We wish to solve the Diophantine equation $a^2+ab+b^2=3^2 \\cdot 73^2$. It can be shown that $3|a$ and $3|b$, so we make the substitution $a=3x$ and $b=3y$ to obtain $x^2+xy+y^2=73^2$ as our new equation to solve for.\n Notice that $r^2+r+1=(r-\\omega)(r-{\\omega}^2... |
deepscaler | [
{
"role": "user",
"content": "What fraction of the pizza is left for Wally if Jovin takes $\\frac{1}{3}$ of the pizza, Anna takes $\\frac{1}{6}$ of the pizza, and Olivia takes $\\frac{1}{4}$ of the pizza? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 2891,
"solution": "Since Jovin, Anna and Olivia take $\\frac{1}{3}, \\frac{1}{6}$ and $\\frac{1}{4}$ of the pizza, respectively, then the fraction of the pizza with which Wally is left is \n $$ 1-\\frac{1}{3}-\\frac{1}{6}-\\frac{1}{4}=\\frac{12}{12}-\\frac{4}{12}-\\frac{2}{12}-\\frac{... |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $\... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 2892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "a) In Mexico City, there are regulations that restrict private cars from driving on city streets on two specified days of the week. A family needs to have at least ten cars available every day. What is the minimum number of cars the family needs if they can choose the restricte... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n,$ let $\\langle n \\rangle$ denote the closest integer to $\\sqrt{n}.$ Evaluate\n\\[\\sum_{n = 1}^\\infty \\frac{2^{\\langle n \\rangle} + 2^{-\\langle n \\rangle}}{2^n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rthea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical pr... | math | {
"style": "rule",
"ground_truth": "28812"
} | {
"split": "train",
"index": 2895,
"solution": "There are $4 \\cdot 3=12$ ways to choose the first base pairs, and regardless of which base pair it is, there are 3 possibilities for the next base on one strand and 3 possibilities for the next base on the other strand. Among these possibilities, exactly 2 of them ... |
deepscaler | [
{
"role": "user",
"content": "Let $x=-2016$. What is the value of $| ||x|-x|-|x| | -x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 2896,
"solution": "Given the problem, we need to evaluate the expression $\\bigg|$ $||x|-x|-|x|$ $\\bigg|$ $-x$ for $x = -2016$.\n\n1. **Calculate $|x|$**:\n Since $x = -2016$, we have $|x| = |-2016| = 2016$.\n\n2. **Evaluate $||x|-x|$**:\n \\[ ||x|-x| = ||2016 - (-2016)| = |2016 ... |
deepscaler | [
{
"role": "user",
"content": "The time right now is exactly midnight. What time will it be in 1234 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\!:\\!34 \\text{ p.m.}"
} | {
"split": "train",
"index": 2897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 2898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 2899,
"solution": ""
} |
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