data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.\n\n(Here $n!$ denotes $1\\times\\cdots\\times n$ for each positive integer $n$.) Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A, Person B, Person C, and Person D share 2013 candies. Person A gets 10 more candies than twice the amount Person B has, 18 more candies than three times the amount Person C has, and 55 less candies than five times the amount Person D has. How many candies does Person A... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 38550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increase... | math | {
"style": "rule",
"ground_truth": "\\frac{26^2}{10}"
} | {
"split": "train",
"index": 38551,
"solution": "1. **Calculate the number of old license plates**: \n Each old license plate consists of one letter followed by four digits. Since there are 26 possible letters and 10 possible digits for each position, the total number of old license plates is calculated as:\n ... |
deepscaler | [
{
"role": "user",
"content": "Suppose $z$ and $w$ are complex numbers such that\n\\[|z| = |w| = z \\overline{w} + \\overline{z} w= 1.\\]Find the largest possible value of the real part of $z + w.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 38552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order. \n$(1)$ If the vectors $\\overrightarrow{m}=(3,\\sin B)$ and $\\overrightarrow{n}=(2,\\sin C)$ are colline... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 38553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum\n$$\n\\cos \\left(\\frac{2\\pi}{18}\\right) + \\cos \\left(\\frac{4\\pi}{18}\\right) + \\cdots + \\cos \\left(\\frac{34\\pi}{18}\\right).\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ 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": 38555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=\\cos \\left(\\omega x+\\varphi \\right)\\left(\\omega\\ \\ \\gt 0,0 \\lt \\varphi\\ \\ \\lt \\pi \\right)$, if $f\\left(x\\right)$ is an odd function and monotonically decreasing on $(-\\frac{π}{3},\\frac{π}{6})$, then the maximum value of... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 38556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $k$ is a nonnegative integer, compute the value of\n\\[\n \\frac{\\binom{1/2}{2015} \\cdot 4^{2015}}{\\binom{4030}{2015}} \\, .\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4030 \\cdot 4029 \\cdot 4028}"
} | {
"split": "train",
"index": 38557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x = \\frac{\\sum\\limits_{n=1}^{30} \\cos n^\\circ}{\\sum\\limits_{n=1}^{30} \\sin n^\\circ}$. What is the smallest integer that does not fall below $100x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 38558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify completely: $$\\sqrt[3]{40^3 + 70^3 + 100^3}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\sqrt[3]{1407}"
} | {
"split": "train",
"index": 38559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side \\( AB \\) of the parallelogram \\( ABCD \\), a point \\( F \\) is chosen, and on the extension of the side \\( BC \\) beyond the vertex \\( B \\), a point \\( H \\) is taken such that \\( \\frac{AB}{BF} = \\frac{BC}{BH} = 5 \\). The point \\( G \\) is chosen such t... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given: $2x^2 - 4xy + 4y^2 + 6x + 9 = 0$, then $x + y =$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"split": "train",
"index": 38561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player... | math | {
"style": "rule",
"ground_truth": "5/11"
} | {
"split": "train",
"index": 38562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $a=(\\cos α, \\sin α)$ and $b=(\\cos β, \\sin β)$, with $|a-b|= \\frac{2 \\sqrt{5}}{5}$, find the value of $\\cos (α-β)$.\n\n(2) Suppose $α∈(0,\\frac{π}{2})$, $β∈(-\\frac{π}{2},0)$, and $\\cos (\\frac{5π}{2}-β) = -\\frac{5}{13}$, find the value of $\\sin α$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"split": "train",
"index": 38563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(x^{2}+1)^{2}(x-1)^{6}$, the coefficient of $x^{5}$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-52"
} | {
"split": "train",
"index": 38564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $E$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\\_1$ perpendicular to the $x$-axis and $\\sin \\angle MF\\_2F\\_1 = \\frac{1}{3}$. Find the eccentricity of $E$. Let's t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle \\(XYZ\\) with \\(XY = YZ\\) and an angle at the vertex equal to \\(96^{\\circ}\\). Point \\(O\\) is located inside triangle \\(XYZ\\) such that \\(\\angle OZX = 30^{\\circ}\\) and \\(\\angle OXZ = 18^{\\circ}\\). Find the measure of angle \\(\\angle... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 38566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five different products, A, B, C, D, and E, are to be arranged in a row on a shelf. Products A and B must be placed together, while products C and D cannot be placed together. How many different arrangements are possible? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \\perp BC, BE \\perp CA, CF \\perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \\perp EF$ and $HQ \\perp EF$. ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38568,
"solution": "\nIn an acute scalene triangle \\(ABC\\), points \\(D, E, F\\) lie on sides \\(BC, CA, AB\\), respectively, such that \\(AD \\perp BC\\), \\(BE \\perp CA\\), \\(CF \\perp AB\\). Altitudes \\(AD, BE, CF\\) meet at orthocenter \\(H\\). Points \\(P\\) and \\(Q\\) lie ... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^2}{4} + \\frac{y^2}{1} = 1$ with one of its foci at $F = (\\sqrt{3}, 0)$, find a point $P = (p, 0)$ where $p > 0$ such that for any chord $\\overline{AB}$ passing through $F$, the angles $\\angle APF$ and $\\angle BPF$ are equal. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 38569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{16}"
} | {
"split": "train",
"index": 38570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 38571,
"solution": "1. **Define the polynomials**: Let $P(x) = x^2 + Bx + C$ and $Q(x) = x^2 + Ex + F$, where both are monic quadratic polynomials.\n\n2. **Expression for $P(Q(x))$**:\n \\[\n P(Q(x)) = (x^2 + Ex + F)^2 + B(x^2 + Ex + F) + C\n \\]\n Expanding and simplifying, w... |
deepscaler | [
{
"role": "user",
"content": "The points $(1, 3)$ and $(5, -1)$ are adjacent vertices of a square. What is the area of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 38572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are e... | math | {
"style": "rule",
"ground_truth": "3432"
} | {
"split": "train",
"index": 38573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $7.21\\times 10^{11}$ has how many digits in the original number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( S = x^{2} + y^{2} - 2(x + y) \\), where \\( x \\) and \\( y \\) satisfy \\( \\log_{2} x + \\log_{2} y = 1 \\), find the minimum value of \\( S \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 - 4\\sqrt{2}"
} | {
"split": "train",
"index": 38575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest constant $ C$ such that for all real $ x,y$ \r\n\\[ 1\\plus{}(x\\plus{}y)^2 \\leq C \\cdot (1\\plus{}x^2) \\cdot (1\\plus{}y^2)\\]\r\nholds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 38576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\\triangle ACD$. Find $s.$\n Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 38578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\{a_{n}\\}\\left(n\\in N*\\right)$ is an arithmetic sequence with a common difference of $-2$, and $a_{6}$ is the geometric mean of $a_{2}$ and $a_{8}$. Let $S_{n}$ be the sum of the first $n$ terms of $\\{a_{n}\\}$. Find the value of $S_{10}$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 38579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\clubsuit$ and the second card dealt is a $\\heartsuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"split": "train",
"index": 38580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1059"
} | {
"split": "train",
"index": 38581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38582,
"solution": "1. **Determine the number of students on the council**: \n Let the number of students on the council be $x$. The number of ways to select a two-person welcoming committee is given by the combination formula $\\binom{x}{2}$, which represents the number of ways to ... |
deepscaler | [
{
"role": "user",
"content": "Three men, Alpha, Beta, and Gamma, working together, do a job in 6 hours less time than Alpha alone, in 1 hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together, to d... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 38583,
"solution": "1. **Define Variables:**\n Let $A$, $B$, and $C$ be the number of hours needed by Alpha, Beta, and Gamma, respectively, to complete the job alone. Their respective rates of work are $\\frac{1}{A}$, $\\frac{1}{B}$, and $\\frac{1}{C}$ jobs per hour.\n\n2. **Set Up ... |
deepscaler | [
{
"role": "user",
"content": "There exist constants $c_1$ and $c_2$ such that\n\\[c_1 \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} + c_2 \\begin{pmatrix} -2 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -1 \\\\ 4 \\end{pmatrix}.\\]Enter the ordered pair $(c_1,c_2).$ Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{3}{16}, \\frac{11}{16} \\right)"
} | {
"split": "train",
"index": 38584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Topsoil costs $\\$6$ per cubic foot. What is the cost, in dollars, of 5 cubic yards of topsoil? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 38585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two couriers start from two locations $A$ and $B$; the first heading towards $B$, and the second towards $A$. When they meet, the first courier has traveled 12 miles more than the second. If they continue their journeys at their original speeds, the first courier will reach the... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 38586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\\frac{-1+i\\sqrt{3}}{2} \\cdot r$? (Note that $i=\\sqrt{-1}$) Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 38587,
"solution": "1. **Identify the form of the polynomial and the transformation of roots:** \n Let $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers. We are given that if $r$ is a root, then $\\frac{-1+i\\sqrt{3}}{2} \\cdot r$ is also ... |
deepscaler | [
{
"role": "user",
"content": "A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and ev... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 38588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three 12 cm $\\times$12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedr... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"split": "train",
"index": 38589,
"solution": "Note that gluing two of the given polyhedra together along a hexagonal face (rotated $60^\\circ$ from each other) yields a cube, so the volume is $\\frac12 \\cdot 12^3 = 864$, so our answer is $\\boxed{864}$.\n Image: \n Sidenote: What if this simple and beautiful ... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_1=5$, and $a_{n+1}-a_n=3+4(n-1)$. Then, calculate the value of $a_{50}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4856"
} | {
"split": "train",
"index": 38591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 7th oblong number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 38592,
"solution": "The 7th oblong number is the number of dots in a rectangular grid of dots with 7 columns and 8 rows. Thus, the 7th oblong number is $7 \\times 8=56$."
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer with 3 digits $\\overline{ABC}$ is $Lusophon$ if $\\overline{ABC}+\\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110,143,242,341,440,164,263,362,461,560,198,297,396,495,594,693,792,891,990"
} | {
"split": "train",
"index": 38593,
"solution": "To find all three-digit Lusophon numbers \\( \\overline{ABC} \\), we first need to establish the conditions under which a number meets the Lusophon criteria. A number is defined as Lusophon if the sum of the number and its digit reversal is a perfect square. Theref... |
deepscaler | [
{
"role": "user",
"content": "Suppose that for some $a,b,c$ we have $a+b+c = 1$, $ab+ac+bc = abc = -4$. What is $a^3+b^3+c^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from ... | math | {
"style": "rule",
"ground_truth": "$0"
} | {
"split": "train",
"index": 38595,
"solution": "1. **Calculate Jack's total:**\n - Jack first calculates the total price including tax on the original price:\n \\[\n 90.00 \\times 1.06 = 95.40 \\text{ dollars}\n \\]\n - Then, he applies the 20% discount to this total:\n \\[\n 95.40 \\time... |
deepscaler | [
{
"role": "user",
"content": "Let $min|a, b|$ denote the minimum value between $a$ and $b$. When positive numbers $x$ and $y$ vary, let $t = min|2x+y, \\frac{2y}{x^2+2y^2}|$, then the maximum value of $t$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ unique integers $a_k$ ($1\\le k\\le r$) with each $a_k$ either $1$ or $- 1$ such that \\[a_13^{n_1} + a_23^{n_2} + \\cdots + a_r3^{n_r} = 1025.\\]\nFind $n_1 + n_2 + \\cdots + n_r$. Let's think step... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{2}$ and on region $B$ is $\\frac{1}{5}$. What is the probability of the arrow stopping on region $C$? Express your answer as a commo... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 38598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $A B C$, points $M$ and $N$ are the midpoints of $A B$ and $A C$, respectively, and points $P$ and $Q$ trisect $B C$. Given that $A, M, N, P$, and $Q$ lie on a circle and $B C=1$, compute the area of triangle $A B C$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{12}"
} | {
"split": "train",
"index": 38599,
"solution": "Note that $M P \\parallel A Q$, so $A M P Q$ is an isosceles trapezoid. In particular, we have $A M=M B=B P=P Q=\\frac{1}{3}$, so $A B=\\frac{2}{3}$. Thus $A B C$ is isosceles with base 1 and legs $\\frac{2}{3}$, and the height from $A$ to $B C$ is $\\frac{\\sqrt{7... |
deepscaler | [
{
"role": "user",
"content": "One angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and 9. Compute the length of the third side of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To a natural number \\( N \\), the largest divisor of \\( N \\) that is less than \\( N \\) was added, resulting in a power of ten. Find all such \\( N \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 38601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"PERCEPTION\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "907,200"
} | {
"split": "train",
"index": 38602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 8:15? Express your answer as a decimal to the nearest tenth.\n[asy]\nsize(200);\ndraw(Circle((0,0),5),linewidth(1.2));\npair[] mins;\nfor(int i = 0; i < 60; ++i){\n\nmins[i] = 4... | math | {
"style": "rule",
"ground_truth": "157.5"
} | {
"split": "train",
"index": 38603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 38604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\(\\left(a^a - a(a-2)^a\\right)^a\\) when \\( a = 4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1358954496"
} | {
"split": "train",
"index": 38605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a @ b = \\frac{a \\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 38606,
"solution": "1. **Define the operation**: Given the operation $a @ b = \\frac{a \\times b}{a + b}$, where $a$ and $b$ are positive integers.\n\n2. **Substitute the values**: Substitute $a = 5$ and $b = 10$ into the operation:\n \\[\n 5 @ 10 = \\frac{5 \\times 10}{5 + 10}\n ... |
deepscaler | [
{
"role": "user",
"content": "Let points $A = (0,0)$, $B = (2,4)$, $C = (6,6)$, and $D = (8,0)$. Quadrilateral $ABCD$ is cut into two pieces by a line passing through $A$ and intersecting $\\overline{CD}$ such that the area above the line is twice the area below the line. This line intersects $\\overline{CD... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 38607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram below shows an isosceles triangle with base $21$ and height $28$ . Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square.\n\n[asy]\nsize(170);\ndefaultpen(linewidth(0.8));\ndraw((0,0)--(1,1));\np... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 38608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38609,
"solution": "1. **Convert Naomi's travel time to hours**: Naomi travels $6$ miles in $10$ minutes. To convert minutes to hours, we divide by $60$:\n \\[\n 10 \\text{ minutes} = \\frac{10}{60} \\text{ hours} = \\frac{1}{6} \\text{ hours}\n \\]\n\n2. **Calculate Naomi's ave... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\sin (C-A)=1$, $\\sin B= \\frac{1}{3}$.\n(I) Find the value of $\\sin A$;\n(II) Given $b= \\sqrt{6}$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 38610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 38611,
"solution": "Let's denote the consecutive positive integers as $a, b, c, d, e$ such that $a < b < c < d < e$. Given that $b = a+1$, $c = a+2$, $d = a+3$, and $e = a+4$, we can express the conditions in the problem using these variables.\n\n1. **Sum of $b$, $c$, and $d$ is a per... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( |AB| = 13 \\), \\( |BC| = 14 \\), \\( |CA| = 15 \\), an internal point \\( P \\) satisfies\n\\[ \n\\overrightarrow{BP} \\cdot \\overrightarrow{CA} = 18 \\text{ and } \\overrightarrow{CP} \\cdot \\overrightarrow{BA} = 32.\n\\]\nWhat is \\(... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 38612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers with $m\\le 2000$ and $k=3-\\frac{m}{n}$. Find the smallest positive value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{667}"
} | {
"split": "train",
"index": 38613,
"solution": "\nGiven the problem with positive integers \\( m \\) and \\( n \\) such that \\( m \\leq 2000 \\), and \\( k = 3 - \\frac{m}{n} \\). We are tasked to find the smallest positive value of \\( k \\).\n\nFirstly, to ensure \\( k \\) is positive, we need:\n\\[\nk = 3 - ... |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110.8333"
} | {
"split": "train",
"index": 38614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A subset $S$ of the set $\\{1,2, \\ldots, 10\\}$ is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of $S$. (By convention, the product of the elements of the empty set is... | math | {
"style": "rule",
"ground_truth": "\\frac{375}{8}"
} | {
"split": "train",
"index": 38615,
"solution": "For primes $p=2,3,5,7$, let the random variable $X_{p}$ denote the number of factors of $p$ in the product of the elements of $S$, plus 1 . Then we wish to find $\\mathbb{E}\\left(X_{2} X_{3} X_{5} X_{7}\\right)$. If there were only prime powers between 1 and 10, t... |
deepscaler | [
{
"role": "user",
"content": "Assume that $y_1, y_2, ... , y_8$ are real numbers such that\n\\[\n\\begin{aligned}\ny_1 + 4y_2 + 9y_3 + 16y_4 + 25y_5 + 36y_6 + 49y_7 + 64y_8 &= 3, \\\\\n4y_1 + 9y_2 + 16y_3 + 25y_4 + 36y_5 + 49y_6 + 64y_7 + 81y_8 &= 15, \\\\\n9y_1 + 16y_2 + 25y_3 + 36y_4 + 49y_5 + 64y_6 + 81y... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"split": "train",
"index": 38616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagon inscribed in a circle has three consecutive sides, each of length 4, and three consecutive sides, each of length 7. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 4, and the other with three sides each of leng... | math | {
"style": "rule",
"ground_truth": "1017"
} | {
"split": "train",
"index": 38617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers are there between $(11.1)^3$ and $(11.2)^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 38618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of m... | math | {
"style": "rule",
"ground_truth": "1236"
} | {
"split": "train",
"index": 38619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is: Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "s\\sqrt{2}"
} | {
"split": "train",
"index": 38620,
"solution": "1. **Calculate the area of the square**: \n The area of a square with side length $s$ is given by:\n \\[\n \\text{Area of square} = s^2\n \\]\n\n2. **Determine the length of the diagonal of the square**: \n The diagonal of a square divides it into two 4... |
deepscaler | [
{
"role": "user",
"content": "Find the largest odd natural number that cannot be expressed as the sum of three distinct composite numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum\n \\[\\sum_{(a,b,c)\\in S}\\frac{1}{2^a3^b5^c}\\]\n can be expressed as $\\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .\n... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 38623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[1 \\cdot 1987 + 2 \\cdot 1986 + 3 \\cdot 1985 + \\dots + 1986 \\cdot 2 + 1987 \\cdot 1 = 1987 \\cdot 994 \\cdot x,\\]compute the integer $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "663"
} | {
"split": "train",
"index": 38624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to ... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 38625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio $AC:CB$ is $3:4$, in $\\triangle ABC$. The external angle bisector of $\\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 38626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6561}"
} | {
"split": "train",
"index": 38627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing a... | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"split": "train",
"index": 38628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After the Guts round ends, HMMT organizers will collect all answers submitted to all 66 questions (including this one) during the individual rounds and the guts round. Estimate $N$, the smallest positive integer that no one will have submitted at any point during the tournament... | math | {
"style": "rule",
"ground_truth": "139"
} | {
"split": "train",
"index": 38629,
"solution": "The correct answer was 139. Remark: Until the end of the Guts round, no team had submitted 71 as the answer to any question. One team, however, submitted 71 as their answer to this question, increasing the answer up to 139."
} |
deepscaler | [
{
"role": "user",
"content": "If the product of the first 2011 positive integers $1 \\times 2 \\times \\ldots \\times 2011$ is divisible by $2010^k$, then the maximum value of the positive integer $k$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a parallelogram with $A B=480, A D=200$, and $B D=625$. The angle bisector of $\\angle B A D$ meets side $C D$ at point $E$. Find $C E$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 38631,
"solution": "First, it is known that $\\angle B A D+\\angle C D A=180^{\\circ}$. Further, $\\angle D A E=\\frac{\\angle B A D}{2}$. Thus, as the angles in triangle $A D E$ sum to $180^{\\circ}$, this means $\\angle D E A=\\frac{\\angle B A D}{2}=\\angle D A E$. Therefore, $D A ... |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(\\frac{{2x}^{2}+2x}{{x}^{2}-1}-\\frac{{x}^{2}-x}{{x}^{2}-2x+1})÷\\frac{x}{x+1}$, where $x=|\\sqrt{3}-2|+(\\frac{1}{2})^{-1}-(π-3.14)^0-\\sqrt[3]{27}+1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{3}}{3} + 1"
} | {
"split": "train",
"index": 38632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_n\\}$, the common ratio $q = -2$, and $a_3a_7 = 4a_4$, find the arithmetic mean of $a_8$ and $a_{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-56"
} | {
"split": "train",
"index": 38633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk sp... | math | {
"style": "rule",
"ground_truth": "18 \\pi"
} | {
"split": "train",
"index": 38634,
"solution": "Suppose the center of the largest teacup is at the origin in the complex plane, and let $z=\\frac{2}{3} e^{\\pi i t / 6}$. The center of the second disk is at $5 e^{\\pi i t / 6}$ at time $t$; that is, \\frac{15}{2} z$. Then the center of the third disk relative to... |
deepscaler | [
{
"role": "user",
"content": "Given the set of digits {1, 2, 3, 4, 5}, find the number of three-digit numbers that can be formed with the digits 2 and 3, where 2 is positioned before 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the graph of $y=f(x)$, which consists of five line segments as described below:\n- From $(-5, -4)$ to $(-3, 0)$\n- From $(-3, 0)$ to $(-1, -1)$\n- From $(-1, -1)$ to $(1, 3)$\n- From $(1, 3)$ to $(3, 2)$\n- From $(3, 2)$ to $(5, 6)$\n\nWhat is the sum of the $x$-coordi... | math | {
"style": "rule",
"ground_truth": "4.35"
} | {
"split": "train",
"index": 38636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many values of $x,-19<x<98$, satisfy $$\\cos ^{2} x+2 \\sin ^{2} x=1 ?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 38637,
"solution": "For any $x, \\sin ^{2} x+\\cos ^{2} x=1$. Subtracting this from the given equation gives \\(\\sin ^{2} x=0\\), or \\(\\sin x=0\\). Thus $x$ must be a multiple of \\(\\pi\\), so \\(-19<k \\pi<98\\) for some integer $k$, or approximately \\(-6.1<k<31.2\\). There are ... |
deepscaler | [
{
"role": "user",
"content": "Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3430"
} | {
"split": "train",
"index": 38638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of consecutive integers whose sum is $45?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 38639,
"solution": "1. **Understanding the Problem:**\n We need to find the greatest number of consecutive integers that sum up to $45$. These integers can be positive, negative, or zero.\n\n2. **Exploring Small Cases:**\n - If we consider only positive integers starting from $1$,... |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( g(x) \\) is a rational function such that \\( 4g\\left(\\dfrac{1}{x}\\right) + \\dfrac{3g(x)}{x} = 2x^2 \\) 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{98}{13}"
} | {
"split": "train",
"index": 38640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x) = \\tan(2x - \\frac{\\pi}{6})$, then the smallest positive period of $f(x)$ is \\_\\_\\_\\_\\_\\_; $f\\left(\\frac{\\pi}{8}\\right)=$ \\_\\_\\_\\_\\_\\_. 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": 38641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\\div 14=\\frac{12}{7}$.\nWhat is $f(768)-f(384)?$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{192}"
} | {
"split": "train",
"index": 38642,
"solution": "1. **Define the function $f(n)$**: \n Given $f(n) = \\frac{\\sigma(n)}{n}$, where $\\sigma(n)$ is the sum of all positive divisors of $n$.\n\n2. **Prime factorization and properties of $\\sigma(n)$**:\n If $n = \\prod_{i=1}^{k} p_i^{e_i}$ is the prime factoriza... |
deepscaler | [
{
"role": "user",
"content": "Below is pictured a regular seven-pointed star. Find the measure of angle \\(a\\) in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{7}"
} | {
"split": "train",
"index": 38643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 38644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "second (1-2)"
} | {
"split": "train",
"index": 38645,
"solution": "To determine during which hour the average speed of the airplane was the largest, we need to analyze the slope of the graph of distance versus time. The average speed for any given hour is calculated as the change in distance divided by the change in time (which is... |
deepscaler | [
{
"role": "user",
"content": "Multiply $555.55$ by $\\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-148.15"
} | {
"split": "train",
"index": 38646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, and $d$ are distinct positive integers, and $abcd = 441$, calculate the value of $a+b+c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 38647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\((x, y)\\) is randomly selected from the rectangular region with vertices at \\((0, 0), (3000, 0), (3000, 4000),\\) and \\((0, 4000)\\). What is the probability that \\(x > 4y\\)? Express your answer as a common fraction. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"split": "train",
"index": 38648,
"solution": ""
} |
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