data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = -\\frac{1}{3}x^3 + x^2 + ax + b$ reaches an extreme value of $4$ at $x=3$, determine the maximum value of $f(x)$ on the interval $[-2,1]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 17949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three points \\( A \\), \\( B \\), and \\( C \\) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \\( \\triangle ABC \\) do not exceed \\( \\sqrt{3} \\). 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": 17950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three tanks contain water. The number of litres in each is shown in the table: Tank A: 3600 L, Tank B: 1600 L, Tank C: 3800 L. Water is moved from each of Tank A and Tank C into Tank B so that each tank contains the same volume of water. How many litres of water are moved from ... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 17951,
"solution": "In total, the three tanks contain $3600 \\mathrm{~L} + 1600 \\mathrm{~L} + 3800 \\mathrm{~L} = 9000 \\mathrm{~L}$. If the water is divided equally between the three tanks, each will contain $\\frac{1}{3} \\cdot 9000 \\mathrm{~L} = 3000 \\mathrm{~L}$. Therefore, $36... |
deepscaler | [
{
"role": "user",
"content": "If \\( 6x + t = 4x - 9 \\), what is the value of \\( x + 4 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 17952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive multiples of $7$ that are less than $1000$ end with the digit $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 17955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $-12 \\leq n \\leq 12$ satisfy $(n-3)(n+5)(n+9)<0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer with $n \\geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2n-2"
} | {
"split": "train",
"index": 17957,
"solution": "The answer is $2n-2$. Write $p(x) = a_nx^n+\\cdots+a_1x+a_0$ and $p(x)^2 = b_{2n}x^{2n}+\\cdots+b_1x+b_0$. Note that $b_0 = a_0^2$ and $b_{2n} = a_n^2$. We claim that not all of the remaining $2n-1$ coefficients $b_1,\\ldots,b_{2n-1}$ can be negative, whence the la... |
deepscaler | [
{
"role": "user",
"content": "Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \\[a_3-b_3=2(a_2-b_2),\\]then what is the sum of the common ratios of the two sequences? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a \\ge b > 1$, what is the largest possible value of $\\log_a (a/b) + \\log_b (b/a)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 17959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the first day, Barry Sotter used his magic wand to make an object's length increase by $\\frac{1}{2}$, meaning that if the length of the object was originally $x,$ then it is now $x + \\frac{1}{2} x.$ On the second day he increased the object's longer length by $\\frac{1}{3... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 17960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 17961,
"solution": "1. Let $p$ represent the number of pennies Leah has, and $n$ represent the number of nickels Leah has. We know from the problem statement that the total number of coins Leah has is 13. Therefore, we can write the equation:\n \\[\n n + p = 13\n \\]\n\n2. The p... |
deepscaler | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, and $x+y=1$, find the minimum value of $\\frac{2{x}^{2}-x+1}{xy}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}+1"
} | {
"split": "train",
"index": 17962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ and its derivative $f''(x)$ on $\\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \\leqslant f(-a)+2a+1$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "-\\dfrac {1}{2}"
} | {
"split": "train",
"index": 17963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The dimensions of a triangle are tripled to form a new triangle. If the area of the new triangle is 54 square feet, how many square feet were in the area of the original triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{8}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 17965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $a$, let $\\lfloor a \\rfloor$ denote the greatest integer less than or equal to $a$. Let $\\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\\lfloor x \\rfloor ^2 + \\lfloor y \\rfloor ^2 = 25$. The region $\\math... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 17966,
"solution": "1. **Understanding the Problem:**\n The problem involves finding the minimum radius $r$ of a disk that completely contains the region $\\mathcal{R}$ defined by $\\lfloor x \\rfloor^2 + \\lfloor y \\rfloor^2 = 25$ in the coordinate plane. Here, $\\lfloor a \\rfloo... |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \\cdot a_5 = -15$, find the value of $a_{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{25}{3}"
} | {
"split": "train",
"index": 17967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \\times b$ table. Isabella fills it up with numbers $1,2, \\ldots, a b$, putting the numbers $1,2, \\ldots, b$ in the first row, $b+1, b+2, \\ldots, 2 b$ in the second row, and so on. Vidur fills i... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 17968,
"solution": "Using the formula $1+2+\\cdots+n=\\frac{n(n+1)}{2}$, we get $$\\begin{aligned} \\frac{a b(a b+1)}{2}-\\frac{a(a+1)}{2} \\cdot \\frac{b(b+1)}{2} & =\\frac{a b(2(a b+1)-(a+1)(b+1))}{4} \\\\ & =\\frac{a b(a b-a-b+1)}{4} \\\\ & =\\frac{a b(a-1)(b-1)}{4} \\\\ & =\\frac{... |
deepscaler | [
{
"role": "user",
"content": "1. Given that the terminal side of angle $\\alpha$ passes through point $P(4, -3)$, find the value of $2\\sin\\alpha + \\cos\\alpha$.\n2. Given that the terminal side of angle $\\alpha$ passes through point $P(4a, -3a)$ ($a \\neq 0$), find the value of $2\\sin\\alpha + \\cos\\a... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 17969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \\le i \\le 2010$ and $f(i) = 1 + \\sqrt{i} + i$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagonal prism has a height of 165 cm. Its two hexagonal faces are regular hexagons with sides of length 30 cm. Its other six faces are rectangles. A fly and an ant start at point \\(X\\) on the bottom face and travel to point \\(Y\\) on the top face. The fly flies directly ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 17971,
"solution": "Throughout this solution, we remove the units (cm) as each length is in these same units. First, we calculate the distance flown by the fly, which we call \\(f\\). Let \\(Z\\) be the point on the base on the prism directly underneath \\(Y\\). Since the hexagonal ba... |
deepscaler | [
{
"role": "user",
"content": "The cafe \"Burattino\" operates 6 days a week with a day off on Mondays. Kolya made two statements: \"from April 1 to April 20, the cafe worked 18 days\" and \"from April 10 to April 30, the cafe also worked 18 days.\" It is known that he made a mistake once. How many days did ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 17972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1, what is the maximum number of roots it can have within the interval \\( (0,1) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 17973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular pentagon $PQRST$, what is the measure of $\\angle PRS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72^{\\circ}"
} | {
"split": "train",
"index": 17974,
"solution": "Join $P$ to $R$. Since $PQRST$ is a regular pentagon, then $\\angle PQR = \\angle QRS = 108^{\\circ}$. Since $PQ = QR$, then $\\triangle PQR$ is isosceles with $\\angle QPR = \\angle QRP$. Since $\\angle PQR = 108^{\\circ}$, then $\\angle PQR + \\angle QPR + \\angl... |
deepscaler | [
{
"role": "user",
"content": "The quadratic $-6x^2+36x+216$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "261"
} | {
"split": "train",
"index": 17975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $\\sqrt{3v-2}$, $\\sqrt{3v+1}$, and $2\\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 17976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a large square of area 100 square units, points $P$, $Q$, $R$, and $S$ are the midpoints of the sides of the square. A line is drawn from each corner of the square to the midpoint of the opposite side, creating a new, smaller, central polygon. What is the area of this centra... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 17977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon has an area of $150\\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76.5\\sqrt{3}"
} | {
"split": "train",
"index": 17978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the following three statements are true:\nI. All freshmen are human.\nII. All graduate students are human.\nIII. Some graduate students are pondering.\n\nConsidering the following four statements:\n(1) All freshmen are graduate students.\n(2) Some humans are pondering.\n(... | math | {
"style": "rule",
"ground_truth": "(2)"
} | {
"split": "train",
"index": 17979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, we have $\\angle A = 90^\\circ$ and $\\sin B = \\frac{3}{5}$. Find $\\cos C$. 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": 17980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{2}-2$, and let $f^{n}$ denote the function $f$ applied $n$ times. Compute the remainder when $f^{24}(18)$ is divided by 89. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 17981,
"solution": "Let $L_{n}$ denote the Lucas numbers given by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$. Note that $L_{n}^{2}-2=L_{2 n}$ when $n$ is even (one can show this by induction, or explicitly using $L_{n}=$ $\\left.\\left(\\frac{1+\\sqrt{5}}{2}\\right)^{n}+\\left(\\... |
deepscaler | [
{
"role": "user",
"content": "In a store, we paid with a 1000 forint bill. On the receipt, the amount to be paid and the change were composed of the same digits but in a different order. What is the sum of the digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies\n\\[a_1 + a_2 + a_3 + \\dots + a_n = n^2 a_n\\]for all $n \\ge 2.$ If $a_{63} = 1,$ find $a_1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 17983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 4 balls in 3 boxes if the balls are not distinguishable and neither are the boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 17984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polynomial \\(x^2 - kx + 24\\) has only positive integer roots, find the average of all distinct possibilities for \\(k\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\diamondsuit$ and $\\Delta$ are whole numbers and $\\diamondsuit \\times \\Delta =36$. The largest possible value of $\\diamondsuit + \\Delta$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 17986,
"solution": "To find the largest possible value of $\\diamondsuit + \\Delta$ given that $\\diamondsuit \\times \\Delta = 36$, we need to consider all pairs of whole numbers $(\\diamondsuit, \\Delta)$ that multiply to 36. We will then calculate the sum $\\diamondsuit + \\Delta$ ... |
deepscaler | [
{
"role": "user",
"content": "Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 17987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 49 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then he eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 49 minutes? Let's think st... | math | {
"style": "rule",
"ground_truth": "1176"
} | {
"split": "train",
"index": 17988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a building with 10 mailboxes, a distributor places a flyer in 5 of the mailboxes. Later, another distributor also places a flyer in 5 of the mailboxes. What is the probability that at least 8 mailboxes receive a flyer? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 17989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 2).$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\pi}{2}"
} | {
"split": "train",
"index": 17990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A reader mentioned that his friend's house in location $A$, where he was invited for lunch at 1 PM, is located 1 km from his own house in location $B$. At 12 PM, he left $B$ in his wheelchair heading towards location $C$ for a stroll. His friend, intending to join him and help ... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 17991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\\frac13$ at pH $3.55$ , while its charge is $\\frac12$ at pH $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "5.97"
} | {
"split": "train",
"index": 17992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the volume of the solid formed by the set of vectors $\\mathbf{v}$ such that\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} -6 \\\\ 18 \\\\ 12 \\end{pmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi \\cdot 126 \\sqrt{126}"
} | {
"split": "train",
"index": 17993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 17994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $\\frac{8}{1+a^{8}} + \\frac{4}{1+a^{4}} + \\frac{2}{1+a^{2}} + \\frac{1}{1+a} + \\frac{1}{1-a}$ and find its value when $a=2^{-\\frac{1}{16}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 17995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(3)=1$ and $f(2x)=2f(x)$ for all $x$, find $f^{-1}(64)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 17996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangular pyramid \\( S-ABC \\) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \\( AS = BS = \\sqrt{19} \\) and \\( CS = 3 \\). Find the surface area of the circumscribed sphere of the triangular pyramid \\( S-ABC \\). Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{268\\pi}{11}"
} | {
"split": "train",
"index": 17997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with sides 6 inches is shown. If $P$ is a point such that the segment $\\overline{PA}$, $\\overline{PB}$, $\\overline{PC}$ are equal in length, and segment $\\overline{PC}$ is perpendicular to segment $\\overline{FD}$, what is the area, in square inches, of triangle $A... | math | {
"style": "rule",
"ground_truth": "\\dfrac{27}{4}"
} | {
"split": "train",
"index": 17998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"split": "train",
"index": 17999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{\\lfloor \\sqrt{1} \\rfloor \\cdot \\lfloor \\sqrt{2} \\rfloor \\cdot \\lfloor \\sqrt{3} \\rfloor \\cdot \\lfloor \\sqrt{5} \\rfloor \\dotsm \\lfloor \\sqrt{15} \\rfloor}{\\lfloor \\sqrt{2} \\rfloor \\cdot \\lfloor \\sqrt{4} \\rfloor \\cdot \\lfloor \\sqrt{6}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 18000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest prime factor of $15! + 18!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \\in N$ . Then find the value of $\\lim_{p \\to \\infty} \\sum_{n=1}^p a_n$ Let'... | math | {
"style": "rule",
"ground_truth": "\\pi/2"
} | {
"split": "train",
"index": 18002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \\times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six p... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 18003,
"solution": "Call the sushi pieces $A, B, C$ in the top row and $D, E, F$ in the bottom row of the grid. Note that Wendy must first eat either $A, C, D$, or $F$. Due to the symmetry of the grid, all of these choices are equivalent. Without loss of generality, suppose Wendy eats... |
deepscaler | [
{
"role": "user",
"content": "In a plane, a right angle is given. A circle with a center located outside of this angle is tangent to the bisector of the right angle. It intersects one side of the right angle at points \\(A\\) and \\(B\\) and the extension of the other side at points \\(C\\) and \\(D\\). Giv... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 18004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x+\\sin y=2008$ and $x+2008 \\cos y=2007$, where $0 \\leq y \\leq \\pi / 2$, find the value of $x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2007+\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 18005,
"solution": "Subtracting the two equations gives $\\sin y-2008 \\cos y=1$. But since $0 \\leq y \\leq \\pi / 2$, the maximum of $\\sin y$ is 1 and the minimum of $\\cos y$ is 0 , so we must have $\\sin y=1$, so $y=\\pi / 2$ and $x+y=2007+\\frac{\\pi}{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "Two circles, one with radius 4 and the other with radius 5, are externally tangent to each other and are circumscribed by a third circle. Calculate the area of the shaded region formed between these three circles. Express your answer in terms of $\\pi$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"split": "train",
"index": 18006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the first and third terms of the geometric sequence $\\{a_n\\}$, given that $a_1$ and $a_4$ are the roots of the equation $x^2-2x-3=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 18007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $B(1, 1)$, $I(2, 4)$ and $G(5, 1)$ are plotted in the standard rectangular coordinate system to form triangle $BIG$. Triangle $BIG$ is translated five units to the left and two units upward to triangle $B'I'G'$, in such a way that $B'$ is the image of $B$, $I'$ is th... | math | {
"style": "rule",
"ground_truth": "(-2,3)"
} | {
"split": "train",
"index": 18008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the ... | math | {
"style": "rule",
"ground_truth": "1116"
} | {
"split": "train",
"index": 18009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x+2)$ is an odd function and it satisfies $f(6-x)=f(x)$, and $f(3)=2$, determine the value of $f(2008)+f(2009)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 18010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=4\\cos (2x+\\frac{\\pi}{4})$, determine the direction and magnitude of horizontal shift required to obtain the graph of the function $y=4\\cos 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 18011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, take turns to draw candies from a bag. A starts by taking 1 candy, then B takes 2 candies, A takes 4 candies next, then B takes 8 candies, and so on. This continues. When the number of candies remaining in the bag is less than the number they are supposed t... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 18012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a circle is doubled when its radius $r$ is increased by $n$. Then $r$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n(\\sqrt{2} + 1)"
} | {
"split": "train",
"index": 18013,
"solution": "1. **Identify the relationship between the areas and radii**: Given that the area of a circle is doubled when its radius $r$ is increased by $n$, we can express this mathematically using the formula for the area of a circle, $A = \\pi r^2$. The new radius becomes $... |
deepscaler | [
{
"role": "user",
"content": "Let $\\eta(m)$ be the product of all positive integers that divide $m$ , including $1$ and $m$ . If $\\eta(\\eta(\\eta(10))) = 10^n$ , compute $n$ .\n\n*Proposed by Kevin Sun* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 18014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In terms of $k$, for $k>0$, how likely is it that after $k$ minutes Sherry is at the vertex opposite the vertex where she started? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}+\\frac{1}{3(-2)^{k}}"
} | {
"split": "train",
"index": 18015,
"solution": "The probability that she ends up on the original vertex is equal to the probability that she ends up on the top vertex, and both are equal to $\\frac{1-p(n)}{2}$ for $n \\geq 1$. From the last problem, $$\\begin{aligned} p(n+1) & =1-\\frac{p(n)}{2} \\\\ p(n+1)-\\fr... |
deepscaler | [
{
"role": "user",
"content": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 18016,
"solution": "1. **Identify the common difference**: \n Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:\n \\[\n d = (x... |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ that satisfies the equation $25^{-2} = \\frac{5^{48/x}}{5^{26/x} \\cdot 25^{17/x}}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18017,
"solution": "1. **Simplify the given equation**: Start by simplifying the right-hand side of the equation:\n \\[\n \\frac{5^{48/x}}{5^{26/x} \\cdot 25^{17/x}}.\n \\]\n Since $25 = 5^2$, we can rewrite $25^{17/x}$ as $(5^2)^{17/x} = 5^{34/x}$. Thus, the expression simpli... |
deepscaler | [
{
"role": "user",
"content": "Find $n$ if $n\\cdot n! + n! = 720$, where $n! = n\\cdot (n-1)\\cdot (n-2)\\cdots 2\\cdot 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p \\) and \\( q \\) be positive integers such that \\( \\frac{5}{8}<\\frac{p}{q}<\\frac{7}{8} \\). What is the smallest value of \\( p \\) such that \\( p+q=2005 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "772"
} | {
"split": "train",
"index": 18019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 18020,
"solution": "1. **Identify the Total Number of Outcomes**: \n When Jack removes one ball, there are 10 choices for him. After Jack's turn, there are 9 balls left for Jill. Thus, the total number of ways in which Jack and Jill can each remove one ball is $10 \\times 9 = 90$.\n... |
deepscaler | [
{
"role": "user",
"content": "An entire floor is tiled with blue and white tiles. The floor has a repeated tiling pattern that forms every $8 \\times 8$ square. Each of the four corners of this square features an asymmetrical arrangement of tiles where the bottom left $4 \\times 4$ segment within each $8 \\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 18021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $101 \\times 101$ using a similar mental math technique. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10201"
} | {
"split": "train",
"index": 18022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the row of Pascal's triangle that starts with 1 and then 12, what is the fourth number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 18023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\log_{10} 60 + \\log_{10} 80 - \\log_{10} 15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.505"
} | {
"split": "train",
"index": 18024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $\\sqrt{x-3}+y^{2}-4y+4=0$, find the value of the algebraic expression $\\frac{{x}^{2}-{y}^{2}}{xy}•\\frac{1}{{x}^{2}-2xy+{y}^{2}}÷\\frac{x}{{x}^{2}y-x{y}^{2}}-1$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(1,2)$ and $\\overrightarrow{b}=(-3,2)$, for what value of $k$ does\n(1) $k \\overrightarrow{a}+ \\overrightarrow{b}$ and $\\overrightarrow{a}-3 \\overrightarrow{b}$ are perpendicular?\n(2) $k \\overrightarrow{a}+ \\overrightarrow{b}$ and $\\overright... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 18026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Enlarge each edge of a graph by four times its original size. This is equivalent to enlarging the graph by a scale of \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"split": "train",
"index": 18027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 18$, $\\angle A = 60^\\circ$, and $\\overline{AB} \\parallel \\overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\\overline{AB}$ is a side of maximum length. The length of another side is ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 18028,
"solution": "1. **Identify the properties of the quadrilateral**: Given that $ABCD$ is a convex quadrilateral with $AB = 18$, $\\angle A = 60^\\circ$, and $\\overline{AB} \\parallel \\overline{CD}$. The lengths of the sides form an arithmetic progression, and $AB$ is the longes... |
deepscaler | [
{
"role": "user",
"content": "Anton colors a cell in a \\(4 \\times 50\\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 18029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of all real numbers that are doubled when added to their reciprocals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 18030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "c = a"
} | {
"split": "train",
"index": 18031,
"solution": "1. **Start by normalizing the quadratic equation**: Given the quadratic equation $ax^2 + bx + c = 0$, we can divide each term by $a$ (assuming $a \\neq 0$) to simplify the equation:\n \\[\n x^2 + \\frac{b}{a}x + \\frac{c}{a} = 0\n \\]\n\n2. **Identify the roo... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a point on the circle $x^2 + y^2 + 4x - 4y + 4 = 0$, and let $B$ be a point on the parabola $y^2 = 8x$. Find the smallest possible distance $AB$. 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": 18032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the roots of $x^3 + 2x^2 + 5x - 8 = 0$ are $p$, $q$, and $r$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find $w$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jar A has exactly four red buttons and eight blue buttons. Carla then removes the same number of red buttons as blue buttons from Jar A and places them in an empty Jar B. Jar A now has $\\frac{2}{3}$ of its original number of buttons. If Carla were now to randomly choose a b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 18034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{-1}^{0}(x+2)^{3} \\cdot \\ln ^{2}(x+2) \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\ln^{2} 2 - 2 \\ln 2 + \\frac{15}{32}"
} | {
"split": "train",
"index": 18035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\\left(-1\\right)^{n+1}\\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\\{\\left(-1\\right)^{n}a_{n}\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{41}"
} | {
"split": "train",
"index": 18036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{3}"
} | {
"split": "train",
"index": 18037,
"solution": "Suppose that $AB$ is the length that is more than $1$ . Let spheres with radius $1$ around $A$ and $B$ be $S_A$ and $S_B$ . $C$ and $D$ must be in the intersection of these spheres, and they must be on the circle created by the intersection to maximize the distance... |
deepscaler | [
{
"role": "user",
"content": "A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 18038,
"solution": "To solve this problem, we need to determine the probability of correctly guessing the match between each celebrity and their corresponding baby picture.\n\n1. **Total Possible Matches**:\n There are three celebrities and each has one corresponding baby picture. T... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 120^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 18039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest number of solid $2 \\times 2 \\times 1$ blocks that can fit in a $3 \\times 2 \\times 3$ box? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18040,
"solution": "1. **Calculate the volume of the larger box and the smaller blocks:**\n - The volume of the larger box is calculated by multiplying its dimensions:\n \\[\n V_{\\text{box}} = 3 \\text{ in} \\times 2 \\text{ in} \\times 3 \\text{ in} = 18 \\text{ in}^3\n ... |
deepscaler | [
{
"role": "user",
"content": "George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{27}"
} | {
"split": "train",
"index": 18041,
"solution": "Notice that Travis can only have the hot potato at the end if he has two potatoes before his turn. A little bit of casework shows that this can only happen when Case 1: George gives Travis his potato, while Jeff gives Brian his potato, which in then goes to Travis.... |
deepscaler | [
{
"role": "user",
"content": "A certain store in Hefei plans to sell a newly launched stationery item, with a purchase price of 20 yuan per item. During the trial marketing phase, it was found that when the selling price is 25 yuan per item, the daily sales volume is 150 items; for every 1 yuan increase in ... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 18042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at const... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 18043,
"solution": "Using a similar approach to Solution 1, let the speed of the east bound train be $a$ and the speed of the west bound train be $b$.\n So $a-20=b+20$ and $a+20=10(b-20)$.\n From the first equation, $a=b+40$. Substituting into the second equation, \\[b+60=10b-200\\] \... |
deepscaler | [
{
"role": "user",
"content": "There is a $6 \\times 6$ square in which all cells are white. In one move, you are allowed to change the color of both cells in any domino (rectangle consisting of two cells) to the opposite color. What is the minimum number of moves required to obtain a square with a checkerbo... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In every cell of a \\(5 \\times 5\\) square, there is a number. The numbers in any given row (any row) and any given column (any column) form an arithmetic progression. The numbers in the corners of the square are \\(1, 25, 17, 81\\). What number is in the center cell? Do not f... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 18045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu, 45]=180$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "292"
} | {
"split": "train",
"index": 18046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The oranges are placed in a pyramid-like stack with the base being a rectangle that is 5 oranges wide and 7 oranges long. Each orange above the first layer rests in a pocket formed by four oranges in the layer below. Determine the total number of oranges in the stack. Let's thi... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 18047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 18048,
"solution": ""
} |
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