data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In rectangle $PQRS$, $PS=6$ and $SR=3$. Point $U$ is on $QR$ with $QU=2$. Point $T$ is on $PS$ with $\\angle TUR=90^{\\circ}$. What is the length of $TR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 36949,
"solution": "Since $PQRS$ is a rectangle, then $QR=PS=6$. Therefore, $UR=QR-QU=6-2=4$. Since $PQRS$ is a rectangle and $TU$ is perpendicular to $QR$, then $TU$ is parallel to and equal to $SR$, so $TU=3$. By the Pythagorean Theorem, since $TR>0$, then $TR=\\sqrt{TU^{2}+UR^{2}}=... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $8x^4 - 18x^3 + 27x^2 - 14x - 30$ is divided by $4x-12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "333"
} | {
"split": "train",
"index": 36950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(0,0)\\,$, $(a,11)\\,$, and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 36951,
"solution": "Consider the points on the complex plane. The point $b+37i$ is then a rotation of $60$ degrees of $a+11i$ about the origin, so:\n\\[(a+11i)\\left(\\mathrm{cis}\\,60^{\\circ}\\right) = (a+11i)\\left(\\frac 12+\\frac{\\sqrt{3}i}2\\right)=b+37i.\\]\n Equating the real... |
deepscaler | [
{
"role": "user",
"content": "It is known that, for all positive integers $k$,\n$1^2+2^2+3^2+\\ldots+k^{2}=\\frac{k(k+1)(2k+1)}6$.\nFind the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $200$.\n Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 36952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of h... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 36953,
"solution": "1. **Identify the dimensions of the square and the pieces**: The square has a side length of $2$. Points $E$ and $F$ are midpoints of opposite sides, so $EF = 2$. Since $AG$ is perpendicular to $BF$, and $E$ and $F$ are midpoints, $AE = EB = 1$ and $BF = 2$.\n\n2. ... |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $b_1, b_2, b_3, \\ldots$ by $b_n = \\sum\\limits_{k=1}^n \\cos{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $b_n < 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "632"
} | {
"split": "train",
"index": 36954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"split": "train",
"index": 36955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of the set of numbers $\\{87,85,80,83,84,x\\}$ is 83.5. What is the median of the set of six numbers? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83.5"
} | {
"split": "train",
"index": 36956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the number of faces, edges, and vertices of a square pyramid that has a square base? Additionally, find the number of diagonals in the square base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$, use Horner's Rule to calculate the value of $v_4$ when $x = -4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 36959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The imaginary part of the complex number $z = -4i + 3$ is $-4i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 36960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all three-digit numbers \\( \\overline{\\mathrm{MGU}} \\) consisting of distinct digits \\( M, \\Gamma, \\) and \\( U \\) for which the equality \\( \\overline{\\mathrm{MGU}} = (M + \\Gamma + U) \\times (M + \\Gamma + U - 2) \\) holds. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 36961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $\\frac{{c\\sin C}}{{\\sin A}} - c = \\frac{{b\\sin B}}{{\\sin A}} - a$ and $b = 2$, find:\n\n$(1)$ The measure of angle $B$;\n\n$(2)$ ... | math | {
"style": "rule",
"ground_truth": "1 + \\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 36962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A prism is constructed so that its vertical edges are parallel to the $z$-axis. Its cross-section is a square of side length 10.\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C, D, E, F, G, H;\n\nA = (1,1,0);\nB = (1,-1,0);\nC =... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 36963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 integers written on the board. The sums of these integers taken in pairs resulted in the following set of 10 numbers: $6, 9, 10, 13, 13, 14, 17, 17, 20, 21$. Determine which numbers are written on the board. Provide their product as the answer. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "4320"
} | {
"split": "train",
"index": 36964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be odd integers greater than $1.$ An $m\\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ thr... | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 36965,
"solution": "Let us take some cases. Since $m$ and $n$ are odds, and $200$ is in the top row and $2000$ in the bottom, $m$ has to be $3$, $5$, $7$, or $9$. Also, taking a look at the diagram, the slope of the line connecting those centers has to have an absolute value of $< 1$.... |
deepscaler | [
{
"role": "user",
"content": "There are 22 people standing in a circle, and each of them is either a knight (who always tells the truth) or a liar (who always lies). Each person says: \"The next 10 people clockwise from me are liars.\" How many liars are there among these 22 people? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A subset $S$ of the nonnegative integers is called supported if it contains 0, and $k+8, k+9 \\in S$ for all $k \\in S$. How many supported sets are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1430"
} | {
"split": "train",
"index": 36967,
"solution": "Note that every supported set $S$ contains $0,8,9,16,17,18,24-27,32-36,40-45$, 48-54, and all $n \\geq 55$. Now define $\\bar{S}:=\\mathbb{Z}^{+} \\backslash S$, which is a subset of $\\{1-7,10-15,19-23,28-31,37,38,39,46,47,55\\}$ satisfying the opposite property t... |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic polynomial \\( P(x) \\). It is known that the equations \\( P(x) = x - 2 \\) and \\( P(x) = 1 - x / 2 \\) each have exactly one root. What is the discriminant of \\( P(x) \\)? 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": 36968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.\n\n[asy]\nfill((0,0)--(12,0)--(12,-4)--(4,-4)--(4,-2)--(0,... | math | {
"style": "rule",
"ground_truth": "8.6"
} | {
"split": "train",
"index": 36969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different pattern... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 36970,
"solution": "Let $a_n$ be the number of ways if the first house has mail, and let $b_n$ be the number of ways if the first house does not get mail.\n$a_n=a_{n-2}+a_{n-3}$ because if the first house gets mail, the next house that gets mail must either be the third or fourth hous... |
deepscaler | [
{
"role": "user",
"content": "We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\\le i\\le k}$ with $$1=d_1<d_2<\\dots <d_k=n\\quad (k\\ge 6).$$\nFind all positive integers $n$ such that $$n=d_5^2+d_6^2.$$ Let's think... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 36971,
"solution": "\nLet \\( n \\) be a positive integer with at least six positive divisors. The sequence of divisors of \\( n \\) is \\( (d_i)_{1 \\le i \\le k} \\) where:\n\n\\[\n1 = d_1 < d_2 < \\cdots < d_k = n \\quad (k \\ge 6).\n\\]\n\nWe need to find all \\( n \\) such that:\... |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = $\\sqrt{|x+2|+|6-x|-m}$, whose domain is R,\n\n(I) Find the range of the real number m;\n\n(II) If the maximum value of the real number m is n, and the positive numbers a and b satisfy $\\frac{8}{3a+b}$ + $\\frac{2}{a+2b}$ = n, find the minimum value o... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"split": "train",
"index": 36972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\tan A = 2\\tan B$, $b = \\sqrt{2}$, find the value of $a$ when the area of $\\triangle ABC$ is maximized. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 36973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest common divisor (gcd) of all the numbers of the form\n\n$$\n(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)\n$$\n\nwhere $a, b, c, d$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ACB$ is 40 degrees. If ray $CA$ is rotated 480 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?\n\n[asy]\ndraw((0,0)--dir(40),linewidth(1),Arrow);\ndraw((0,0)--dir(0),linewidth(1)... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 36975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that\n\\[2g(a^2 + b^2) = [g(a)]^2 + [g(b)]^2\\] for all nonnegative integers $a$ and $b.$\n\nLet $n$ be the number of possible values of $g(16),$ and let $s$ be the sum of the possible value... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 36976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $(3b-c)\\cos A - a\\cos C = 0$.\n(1) Find $\\cos A$;\n(2) If $a = 2\\sqrt{3}$ and the area of $\\triangle ABC$ is $S_{\\triangle ABC} = 3\\sqrt{2}$, determine the sh... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 36977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a square of side length $1$ , and let $P$ be a variable point on $\\overline{CD}$ . Denote by $Q$ the intersection point of the angle bisector of $\\angle APB$ with $\\overline{AB}$ . The set of possible locations for $Q$ as $P$ varies along $\\o... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 36978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\\frac{m}{n}$, where $m$ and $n$ are relatively prime po... | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 36979,
"solution": "Use complementary probability and Principle of Inclusion-Exclusion. If we consider the delegates from each country to be indistinguishable and number the chairs, we have \\[\\frac{9!}{(3!)^3} = \\frac{9\\cdot8\\cdot7\\cdot6\\cdot5\\cdot4}{6\\cdot6} = 6\\cdot8\\cdot... |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for ... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 36980,
"solution": "1. **Observation of $f(n)$**: We start by observing the function $f(n)$, which counts the number of ways to write $n$ as a sum of powers of 2, considering the order of terms. For example, $f(4) = 6$ because we can write $4$ as:\n - $4$\n - $2 + 2$\n - $2 + 1 ... |
deepscaler | [
{
"role": "user",
"content": "In convex quadrilateral $ABCD, \\angle A \\cong \\angle C, AB = CD = 180,$ and $AD \\neq BC.$ The perimeter of $ABCD$ is $640$. Find $\\lfloor 1000 \\cos A \\rfloor.$ (The notation $\\lfloor x \\rfloor$ means the greatest integer that is less than or equal to $x.$) Let's think ... | math | {
"style": "rule",
"ground_truth": "777"
} | {
"split": "train",
"index": 36981,
"solution": "[asy] real x = 1.60; /* arbitrary */ pointpen = black; pathpen = black+linewidth(0.7); size(180); real BD = x*x + 1.80*1.80 - 2 * 1.80 * x * 7 / 9; pair A=(0,0),B=(1.8,0),D=IP(CR(A,x),CR(B,BD)),C=OP(CR(D,1.8),CR(B,2.80 - x)); D(MP(\"A\",A)--MP(\"B\",B)--MP(\"C\",C)... |
deepscaler | [
{
"role": "user",
"content": "A line initially 1 inch long grows according to the following law, where the first term is the initial length.\n\\[1+\\frac{1}{4}\\sqrt{2}+\\frac{1}{4}+\\frac{1}{16}\\sqrt{2}+\\frac{1}{16}+\\frac{1}{64}\\sqrt{2}+\\frac{1}{64}+\\cdots\\]\nIf the growth process continues forever,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(4+\\sqrt{2})"
} | {
"split": "train",
"index": 36982,
"solution": "1. **Identify the pattern and separate the terms**:\n The given series is:\n \\[\n 1 + \\frac{1}{4}\\sqrt{2} + \\frac{1}{4} + \\frac{1}{16}\\sqrt{2} + \\frac{1}{16} + \\frac{1}{64}\\sqrt{2} + \\frac{1}{64} + \\cdots\n \\]\n We can observe that this series... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABC$ is right-angled at $A,$ with $AB=45$ and $AC=108.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $AD$ and the ratio of the areas of triangles $ABD$ and $ADC$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "5:12"
} | {
"split": "train",
"index": 36983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are more than 20 and fewer than 30 children in Miss Tree's class. They are all standing in a circle. Anna notices that there are six times as many children between her and Zara going round the circle clockwise, as there are going round anti-clockwise. How many children ar... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 36984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $M$ and $N$ are the common points of circle $A$: $x^{2}+y^{2}-2x=0$ and circle $B$: $x^{2}+y^{2}+2x-4y=0$, find the area of $\\triangle BMN$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 36985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The triangle $\\triangle ABC$ is an isosceles triangle where $AB = 4\\sqrt{2}$ and $\\angle B$ is a right angle. If $I$ is the incenter of $\\triangle ABC,$ then what is $BI$?\n\nExpress your answer in the form $a + b\\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is ... | math | {
"style": "rule",
"ground_truth": "8 - 4\\sqrt{2}"
} | {
"split": "train",
"index": 36986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A soccer ball kicked vertically upward reaches, after $t$ seconds, a height of $s$ meters where $s = 180t - 20t^2$. Find the maximum height reached by the ball. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 36987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)64.83-5\\frac{18}{19}+35.17-44\\frac{1}{19}$;<br/>$(2)(+2.5)+(-3\\frac{1}{3})-(-1)$;<br/>$(3)\\frac{(0.125+\\frac{3}{5})×\\frac{33}{87}}{12.1×\\frac{1}{11}$;<br/>$(4)41\\frac{1}{3}×\\frac{3}{4}+52\\frac{1}{2}÷1\\frac{1}{4}+63\\frac{3}{5}×\\frac{5}{6}$;<br/>$(... | math | {
"style": "rule",
"ground_truth": "4044\\frac{2022}{2023}"
} | {
"split": "train",
"index": 36988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yu... | math | {
"style": "rule",
"ground_truth": "2640"
} | {
"split": "train",
"index": 36989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number written on the barrel itself? An example of a correct arrangement is having 123 in th... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 36990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two sets $A$ and $B$ that satisfy the conditions $A\\cap B \\neq \\emptyset$ and $A\\cup B = \\{1, 2, 3, 4, 5\\}$. When $A \\neq B$, the pair $(A, B)$ and $(B, A)$ are considered as two different pairs of sets. The total number of such set pairs $(A, B)$ that meet the con... | math | {
"style": "rule",
"ground_truth": "211"
} | {
"split": "train",
"index": 36991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, $a=2$, $\\angle A=\\frac{π}{6}$, $b=2\\sqrt{3}$, find the measure of angle $\\angle C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 36992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ from left to right. For example, $a_3 = 123$ and $a_{11} = 1234567891011$. Compute the remainder when $a_{44}$ is divided by $45$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"split": "train",
"index": 36994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\geq 3$ be an integer. Rowan and Colin play a game on an $n \\times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [... | math | {
"style": "rule",
"ground_truth": "2(n! + 1)"
} | {
"split": "train",
"index": 36995,
"solution": "\nTo determine the number of orderly colorings on an \\( n \\times n \\) grid where each square is either red or blue, we must first understand the conditions of the game as described.\n\nAn orderly coloring must satisfy two main conditions:\n1. No matter how the r... |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?\n\nEnter your answer in the form $\\frac{a \\sqrt{b} - c}{d},$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2} - 3}{2}"
} | {
"split": "train",
"index": 36996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 8\\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 36997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80\\%"
} | {
"split": "train",
"index": 36998,
"solution": "1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.\n\n2. **Determine the number of hits that were not singles**: \n - Home runs: 1\n - Triples: 1\n - Doubles: 5\n - Total non-single hits = 1 (home run) + 1 (triple) + 5 ... |
deepscaler | [
{
"role": "user",
"content": "Consider the arithmetic sequence defined by the set $\\{2, 5, 8, 11, 14, 17, 20\\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the focus of the parabola $y = -3x^2 - 6x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( -1, \\frac{35}{12} \\right)"
} | {
"split": "train",
"index": 37000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to the real number $x$, such as $\\lfloor 3.2 \\rfloor = 3$, $\\lfloor -4.5 \\rfloor = -5$. The area of the shape formed by points $(x, y)$ on the plane that satisfy $\\lfloor x \\rfloor^2 + \\lfloor y \\r... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\dfrac{111}{9999} \\cdot 33.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{37}{101}"
} | {
"split": "train",
"index": 37002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( q(x) = 2x - 5 \\) and \\( r(q(x)) = x^3 + 2x^2 - x - 4 \\). Find \\( r(3) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 37003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_na_{n+1}a_{n+2}a_{n+3}=24$, and $a_1=1$, $a_2=2$, $a_3=3$, find the sum $a_1+a_2+a_3+\\ldots+a_{2013}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5031"
} | {
"split": "train",
"index": 37004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 37005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the lines $ax+2y+6=0$ and $x+\\left(a-1\\right)y+a^{2}-1=0$ are parallel to each other, find the value(s) of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 37006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a permutation $\\sigma$ of $\\{1,2, \\ldots, 2013\\}$, let $f(\\sigma)$ to be the number of fixed points of $\\sigma$ - that is, the number of $k \\in\\{1,2, \\ldots, 2013\\}$ such that $\\sigma(k)=k$. If $S$ is the set of all possible permutations $\\sigma$, compute $$\\... | math | {
"style": "rule",
"ground_truth": "15(2013!)"
} | {
"split": "train",
"index": 37007,
"solution": "First, note that $$\\sum_{\\sigma \\in S} f(\\sigma)^{4}=\\sum_{\\sigma \\in S} \\sum_{1 \\leq a_{1}, a_{2}, a_{3}, a_{4} \\leq 2013} g\\left(\\sigma, a_{1}, a_{2}, a_{3}, a_{4}\\right)$$ where $g\\left(\\sigma, a_{1}, a_{2}, a_{3}, a_{4}\\right)=1$ if all $a_{i}$ ... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $\\angle BAC=60^\\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 37008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:(0,\\infty)\\rightarrow (0,\\infty)$ such that for any $x,y\\in (0,\\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \\left(f(f(x^2)) + f(f(y^2))\\right).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\frac{1}{x}"
} | {
"split": "train",
"index": 37009,
"solution": "\nLet's find all functions \\( f: (0, \\infty) \\rightarrow (0, \\infty) \\) that satisfy the functional equation:\n\n\\[\nxf(x^2)f(f(y)) + f(yf(x)) = f(xy) \\left( f(f(x^2)) + f(f(y^2)) \\right).\n\\]\n\nTo solve this problem, consider the possibility \\( f(x) = \... |
deepscaler | [
{
"role": "user",
"content": "8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obt... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 37010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) for which \\( (n+1)(n+2)(n+3)(n+4) \\) is divisible by 1000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 37011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's pe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 37012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel... | math | {
"style": "rule",
"ground_truth": "359999"
} | {
"split": "train",
"index": 37013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a regular polygon has a total of nine diagonals, how many sides does it have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 37015,
"solution": "1. **Identify the factors of 128**: We know that the product of the ages of Kiana and her twin brothers is 128. Since 128 can be expressed as $128 = 2^7$, the ages must be factors of 128.\n\n2. **List possible ages for the twins**: The twins are older than Kiana an... |
deepscaler | [
{
"role": "user",
"content": "Determine the least real number $M$ such that the inequality \\[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \\leq M(a^{2}+b^{2}+c^{2})^{2}\\] holds for all real numbers $a$, $b$ and $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "M=\\frac 9{16\\sqrt 2}"
} | {
"split": "train",
"index": 37016,
"solution": "\nTo find the least real number \\( M \\) such that the inequality \n\n\\[\n|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \\leq M(a^{2}+b^{2}+c^{2})^{2}\n\\]\n\nholds for all real numbers \\( a, b, \\) and \\( c \\), we proceed as follows:\n\n### Step 1: Expres... |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\log_8 2 = 0.2525\\) in base 8 (to 4 decimal places), find \\(\\log_8 4\\) in base 8 (to 4 decimal places). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5050"
} | {
"split": "train",
"index": 37017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_{n}\\}$ be an integer sequence such that for any $n \\in \\mathbf{N}^{*}$, the condition \\((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\\) holds. Additionally, \\(2008 \\mid a_{2007}\\). Find the smallest positive integer \\(n \\geqslant 2\\) such that \\(2008 \\mid a_{n}\\... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 37018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three of the four endpoints of the axes of an ellipse are, in some order, \\[(-2, 4), \\; (3, -2), \\; (8, 4).\\]Find the distance between the foci of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"split": "train",
"index": 37019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be given, $n \\geq 4$, and suppose that $P_1, P_2, \\dots, P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i$. What is the probability that at least one of the vertex angles of this pol... | math | {
"style": "rule",
"ground_truth": "n(n-2) 2^{-n+1}"
} | {
"split": "train",
"index": 37020,
"solution": "The angle at a vertex $P$ is acute if and only if all of the other points lie on an open semicircle. We first deduce from this that if there are any two acute angles at all, they must occur consecutively. Suppose the contrary; label the vertices $Q_1, \\dots, Q_n$ ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ be a diameter of circle $\\omega$. Extend $\\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\\overline{AB} = 18$, and let $m... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 37021,
"solution": "Proceed as follows for Solution 1.\n Once you approach the function $k=(2x-27)/x^2$, find the maximum value by setting $dk/dx=0$.\n Simplifying $k$ to take the derivative, we have $2/x-27/x^2$, so $dk/dx=-2/x^2+54/x^3$. Setting $dk/dx=0$, we have $2/x^2=54/x^3$.\n ... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ where $a_n > 0$, $a_1=1$, $a_{n+2}= \\frac {1}{a_n+1}$, and $a_{100}=a_{96}$, find the value of $a_{2014}+a_3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 37022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To rebuild homes after an earthquake for disaster relief, in order to repair a road damaged during the earthquake, if Team A alone takes 3 months to complete the work, costing $12,000 per month; if Team B alone takes 6 months to complete the work, costing $5,000 per month. How ... | math | {
"style": "rule",
"ground_truth": "34,000"
} | {
"split": "train",
"index": 37023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = 2\\sqrt{3}\\sin x \\cos x + 2\\cos^2x - 1$,\n\n(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;\n\n(2) In $\\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 37024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a pile of apples, the ratio of large apples to small apples is $9:1$. Now, a fruit sorting machine is used for screening, with a probability of $5\\%$ that a large apple is sorted as a small apple and a probability of $2\\%$ that a small apple is sorted as a large apple. Cal... | math | {
"style": "rule",
"ground_truth": "\\frac{855}{857}"
} | {
"split": "train",
"index": 37025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a line $l$ intersects the hyperbola $x^2 - \\frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 37026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $A$ is tangent to circle $B$ at one point, and the center of circle $A$ lies on the circumference of circle $B$. The area of circle $A$ is $16\\pi$ square units. Find the area of circle $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64\\pi"
} | {
"split": "train",
"index": 37027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mike and Alain play a game in which each player is equally likely to win. The first player to win three games becomes the champion, and no further games are played. If Mike has won the first game, what is the probability that Mike becomes the champion? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"split": "train",
"index": 37028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 37029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2$, and point $Q( \\frac{a^{2}}{ \\sqrt{a^{2}-b^{2}}},0)$ on the line $l$: $x=2$.\n(1) Find the standard equation of the ellipse $C$;\n(2) Let $O$ be the coordinate origin, ... | math | {
"style": "rule",
"ground_truth": "\\frac{ \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 37030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a machine that when a positive integer $N$ is entered, the machine's processing rule is:\n\n- If $N$ is odd, output $4N + 2$.\n- If $N$ is even, output $N / 2$.\n\nUsing the above rule, if starting with an input of $N = 9$, after following the machine's process for six... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 37031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We measured the angle of elevation of an antenna tower standing on flat terrain from distances of $100 \\text{ m}$, $200 \\text{ m}$, and $300 \\text{ m}$ from its base. The sum of the three angles is $90^\\circ$. How tall is the tower? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 37032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two linear functions $f(x)$ and $g(x)$ such that the graphs of $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"split": "train",
"index": 37033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the rightmost three digits of $3^{1987}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "187"
} | {
"split": "train",
"index": 37034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain store sells a batch of thermal shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to increase sales and profits, the store has taken appropriate price reduction measures. After investigation, it was found that within a certain ran... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 37035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87429"
} | {
"split": "train",
"index": 37036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "3.75"
} | {
"split": "train",
"index": 37037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two lines are perpendicular. One line has a direction vector of $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix}.$ The other line has a direction vector of $\\begin{pmatrix} a \\\\ 2 \\end{pmatrix}.$ Find $a.$ Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}"
} | {
"split": "train",
"index": 37038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrix\n\\[\\begin{pmatrix} a & 3 \\\\ -8 & d \\end{pmatrix}\\]is its own inverse, for some real numbers $a$ and $d.$ Find the number of possible pairs $(a,d).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive number $x$ has the property that $x\\%$ of $x$ is $4$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 37040,
"solution": "\nTo solve the problem, we need to understand the statement \"$x\\%$ of $x$ is $4$\". The symbol \"$x\\%$\" represents $x$ percent, which mathematically is expressed as $0.01x$. Therefore, \"$x\\%$ of $x$\" translates to $0.01x \\cdot x$.\n\n1. **Express the proble... |
deepscaler | [
{
"role": "user",
"content": "Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 37041,
"solution": "Let $b$ be the finish number of Betty's peanuts. Then \\[3b = 444-(5 + 9 + 25) = 405 = 3 \\cdot 135 \\implies b = 135, b+ 9 = 144.\\]\n Let $k > 1$ be the common ratio. Then \\[\\frac{144}{k} + 144 + k \\cdot 144 = 444 \\implies \\frac{144}{k} + k \\cdot 144 = 300\... |
deepscaler | [
{
"role": "user",
"content": "The positive integer equal to the expression \n \\[ \\sum_{i=0}^{9} \\left(i+(-9)^i\\right)8^{9-i} \\binom{9}{i}\\] \n is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.\n\n*Team #7* Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "835"
} | {
"split": "train",
"index": 37042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=x^{2}-2$, find $\\lim_{{Δx→0}}\\frac{{f(3)-f({3-2Δx})}}{{Δx}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two plane vectors $\\boldsymbol{\\alpha}$ and $\\boldsymbol{\\beta}$ that satisfy\n\\[ |\\boldsymbol{\\alpha} + 2\\boldsymbol{\\beta}| = 3 \\]\n\\[ |2\\boldsymbol{\\alpha} + 3\\boldsymbol{\\beta}| = 4, \\]\nfind the minimum value of $\\boldsymbol{\\alpha} \\cdot \\boldsym... | math | {
"style": "rule",
"ground_truth": "-170"
} | {
"split": "train",
"index": 37044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a \\otimes b = \\dfrac{a + b}{a - b}$, then $(6\\otimes 4)\\otimes 3 =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 37045,
"solution": "1. **Calculate $6 \\otimes 4$:**\n \\[\n 6 \\otimes 4 = \\frac{6 + 4}{6 - 4} = \\frac{10}{2} = 5\n \\]\n\n2. **Calculate $5 \\otimes 3$:**\n \\[\n 5 \\otimes 3 = \\frac{5 + 3}{5 - 3} = \\frac{8}{2} = 4\n \\]\n\n3. **Combine the results:**\n Since $(6 ... |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x) = \\sin \\omega x \\cos \\omega x + \\sqrt{3} \\sin^2 \\omega x - \\frac{\\sqrt{3}}{2}$ ($\\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common diffe... | math | {
"style": "rule",
"ground_truth": "\\frac{11\\pi}{3}"
} | {
"split": "train",
"index": 37046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $2\\sin 2C\\cdot\\cos C-\\sin 3C= \\sqrt {3}(1-\\cos C)$.\n(1) Find the measure of angle $C$;\n(2) If $AB=2$, and $\\sin C+\\sin (B-A)=2\\sin 2A$, 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": "\\dfrac {2 \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 37047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider numbers of the form \\(10n + 1\\), where \\(n\\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \\(10k + 1\\), where \\(k\\) is a positive integer... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 37048,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.