data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$ with $\\angle B = 90^\\circ$, we have $$2\\sin A = 3\\cos A.$$What is $\\sin A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{13}}{13}"
} | {
"split": "train",
"index": 22749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (1,1)$ be a point on the parabola $y = x^2.$ The normal to the parabola at $A$ is drawn, intersecting the parabola again at $B.$ Find $B.$\n\n[asy]\nunitsize(1 cm);\n\npair A, B;\n\nA = (1,1);\nB = (-3/2,9/4);\n\nreal parab (real x) {\n return(x^2);\n}\n\ndraw(graph... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{3}{2}, \\frac{9}{4} \\right)"
} | {
"split": "train",
"index": 22750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x<0$, then $|x-\\sqrt{(x-1)^2}|$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1-2x"
} | {
"split": "train",
"index": 22751,
"solution": "1. **Understanding the expression**: We start by simplifying the expression inside the absolute value:\n \\[\n \\left|x - \\sqrt{(x-1)^2}\\right|\n \\]\n We know that the square root of a square gives the absolute value, i.e., $\\sqrt{a^2} = |a|$. Applying ... |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\theta = 5,$ find\n\\[\\frac{1 - \\cos \\theta}{\\sin \\theta} - \\frac{\\sin \\theta}{1 + \\cos \\theta}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 22752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Niffy's favorite number is a positive integer, and Stebbysaurus is trying to guess what it is. Niffy tells her that when expressed in decimal without any leading zeros, her favorite number satisfies the following: - Adding 1 to the number results in an integer divisible by 210 ... | math | {
"style": "rule",
"ground_truth": "1010309"
} | {
"split": "train",
"index": 22753,
"solution": "Note that Niffy's favorite number must end in 9, since adding 1 makes it divisible by 10. Also, the sum of the digits of Niffy's favorite number must be even (because it is equal to twice the number of digits) and congruent to 2 modulo 3 (because adding 1 gives a m... |
deepscaler | [
{
"role": "user",
"content": "Find the number of complex numbers $z$ satisfying $|z| = 1$ and\n\\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 22754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x^2+y^2_{}$ if $x_{}^{}$ and $y_{}^{}$ are positive integers such that \\begin{align*} xy+x+y&=71, \\\\ x^2y+xy^2&=880. \\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 22755,
"solution": "Let $a=x+y$, $b=xy$ then we get the equations \\begin{align*} a+b&=71\\\\ ab&=880 \\end{align*} After finding the prime factorization of $880=2^4\\cdot5\\cdot11$, it's easy to obtain the solution $(a,b)=(16,55)$. Thus \\[x^2+y^2=(x+y)^2-2xy=a^2-2b=16^2-2\\cdot55=\\... |
deepscaler | [
{
"role": "user",
"content": "Let $M = 123456789101112\\dots4950$ be the $95$-digit number formed by writing integers from $1$ to $50$ in order, one after the other. What is the remainder when $M$ is divided by $45$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 22756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$3^3 + 3^3 + 3^3 =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^4"
} | {
"split": "train",
"index": 22757,
"solution": "1. **Identify the expression and simplify**: The given expression is $3^3 + 3^3 + 3^3$.\n\n2. **Factor out the common term**: Notice that each term in the sum is $3^3$. We can factor out $3^3$ as follows:\n \\[\n 3^3 + 3^3 + 3^3 = 3 \\times 3^3\n \\]\n\n3. **... |
deepscaler | [
{
"role": "user",
"content": "A 1$\\%$ late charge was added to Jenna's bill on the $30^{\\text{th}}$ day past its due date. The resulting total was then increased by 1$\\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\\$400$. Exactly how much is the bill now? Let's t... | math | {
"style": "rule",
"ground_truth": "408.04"
} | {
"split": "train",
"index": 22758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:\n\\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3691"
} | {
"split": "train",
"index": 22759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-n representation of $b$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 22760,
"solution": "1. **Understanding the Problem:**\n Given an equation $x^2 - ax + b = 0$ with integer solutions $n$ and $m$, where $n > 8$. The coefficient $a$ in base-$n$ is represented as $18_n$. We need to find the base-$n$ representation of $b$.\n\n2. **Using Vieta's Formula... |
deepscaler | [
{
"role": "user",
"content": "Calculate: \n(1) $2\\log_{2}{10}+\\log_{2}{0.04}$ \n(2) $(\\log_{4}{3}+\\log_{8}{3})\\cdot(\\log_{3}{5}+\\log_{9}{5})\\cdot(\\log_{5}{2}+\\log_{25}{2})$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"split": "train",
"index": 22761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $$\\dfrac{\\sqrt{338}}{\\sqrt{288}}+\\dfrac{\\sqrt{150}}{\\sqrt{96}}.$$Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"split": "train",
"index": 22762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x<-2$, then $|1-|1+x||$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2-x"
} | {
"split": "train",
"index": 22763,
"solution": "1. **Understanding the expression**: We need to evaluate $|1-|1+x||$ for $x < -2$.\n\n2. **Analyzing $1+x$**: Since $x < -2$, adding 1 to both sides gives $x + 1 < -1$. This implies that $1 + x$ is negative.\n\n3. **Applying the absolute value property**: For any r... |
deepscaler | [
{
"role": "user",
"content": "The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dots.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{150}"
} | {
"split": "train",
"index": 22764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 22765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Plot points $A, B, C$ at coordinates $(0,0),(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $A B$ and $B C$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise 45 degrees about point $A$. Let $X_{2}$ be th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 22766,
"solution": "It's easy to see $X_{1}=X_{2}$. Simple cutting and pasting shows that $X_{1}$ equals the area of $\\frac{1}{8}$ of a circle with radius $A C=\\sqrt{2}$, so $\\frac{X_{1}+X_{2}}{2}=X_{1}=\\frac{1}{8} \\pi(\\sqrt{2})^{2}=\\frac{\\pi}{4}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 22767,
"solution": "A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number o... |
deepscaler | [
{
"role": "user",
"content": "Among the triangles with natural number side lengths, a perimeter not exceeding 100, and the difference between the longest and shortest sides not greater than 2, there are a total of different triangles that are not congruent to each other. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 22768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of an event occurring in each of 900 independent trials is 0.5. Find a positive number $\\varepsilon$ such that with a probability of 0.77, the absolute deviation of the event frequency from its probability of 0.5 does not exceed $\\varepsilon$. Let's think step... | math | {
"style": "rule",
"ground_truth": "0.02"
} | {
"split": "train",
"index": 22769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\\ldots+a_{7}(x-1)^{7}$$, find the value of $a_{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 22770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many real triples $(a, b, c)$ are there such that the polynomial $p(x)=x^{4}+a x^{3}+b x^{2}+a x+c$ has exactly three distinct roots, which are equal to $\\tan y, \\tan 2 y$, and $\\tan 3 y$ for some real $y$ ? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 22771,
"solution": "Let $p$ have roots $r, r, s, t$. Using Vieta's on the coefficient of the cubic and linear terms, we see that $2 r+s+t=r^{2} s+r^{2} t+2 r s t$. Rearranging gives $2 r(1-s t)=\\left(r^{2}-1\\right)(s+t)$. If $r^{2}-1=0$, then since $r \\neq 0$, we require that $1-s ... |
deepscaler | [
{
"role": "user",
"content": "By definition, a polygon is regular if all its angles and sides are equal. Points \\( A, B, C, D \\) are consecutive vertices of a regular polygon (in that order). It is known that the angle \\( ABD = 135^\\circ \\). How many vertices does this polygon have? Let's think step by... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of the triangle is S, and it is given that 2$\\sqrt {3}$S - $\\overrightarrow {AB}$•$\\overrightarrow {AC}$ = 0, and c = 2.\n(I) Find the measure of angle A.\n(II) If a² + b² - c² ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 4\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 22773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Winnie has 17 red balloons, 33 white balloons, 65 green balloons, and 83 chartreuse balloons. With no regard to color, Winnie gives each of her 8 nearest and dearest friends some balloons. Each friend gets the same number of balloons and she hands out as many balloons as poss... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 22775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $w = \\dfrac{\\sqrt{3} + i}{2}$ and $z = \\dfrac{-1 + i\\sqrt{3}}{2},$ where $i = \\sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \\cdot w^r = z^s.$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "834"
} | {
"split": "train",
"index": 22776,
"solution": "We rewrite $w$ and $z$ in polar form: \\begin{align*} w &= e^{i\\cdot\\frac{\\pi}{6}}, \\\\ z &= e^{i\\cdot\\frac{2\\pi}{3}}. \\end{align*} The equation $i \\cdot w^r = z^s$ becomes \\begin{align*} e^{i\\cdot\\frac{\\pi}{2}} \\cdot \\left(e^{i\\cdot\\frac{\\pi}{6}}... |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$ . If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$ . Let's... | math | {
"style": "rule",
"ground_truth": "1594"
} | {
"split": "train",
"index": 22777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 22778,
"solution": "To find the number of scalene triangles with integral side lengths and a perimeter less than 13, we need to consider the properties of scalene triangles and the triangle inequality theorem. A scalene triangle has all sides of different lengths, and the sum of the l... |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6/5"
} | {
"split": "train",
"index": 22779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(PR, PS, QS, QT,\\) and \\(RT\\) are straight line segments. \\(QT\\) intersects \\(PR\\) and \\(PS\\) at \\(U\\) and \\(V\\), respectively. If \\(PU = PV\\), \\(\\angle UPV = 24^\\circ\\), \\(\\angle PSQ = x^\\circ\\), and \\(\\angle TQS = y^\\circ\\), what i... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 22780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose an amoeba is placed in a container one day, and on that day it splits into three amoebas. Each subsequent day, every surviving amoeba splits into three new amoebas. However, at the end of every second day starting from day two, only half of the amoebas survive. How many... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 22781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the graph of a power function passes through the points $(2,16)$ and $(\\frac{1}{2},m)$, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 22782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box of 100 personalized pencils costs $\\$30$. How many dollars does it cost to buy 2500 pencils? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$750"
} | {
"split": "train",
"index": 22783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 22784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{0} \\) be the largest (real) root of the equation \\( x^{4} - 16x - 12 = 0 \\). Evaluate \\( \\left\\lfloor 10 x_{0} \\right\\rfloor \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 22785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 22786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \\[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\\geq (xyz)^{2}-xyz+1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 22787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{x}\\cos x-x$.\n\n(Ⅰ) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f (0))$;\n\n(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\\left[0, \\frac{\\pi}{2}\\right]$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 22788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b, c$ are integers with $a b c=60$, and that complex number $\\omega \\neq 1$ satisfies $\\omega^{3}=1$, find the minimum possible value of $\\left|a+b \\omega+c \\omega^{2}\\right|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 22789,
"solution": "Since $\\omega^{3}=1$, and $\\omega \\neq 1, \\omega$ is a third root of unity. For any complex number $z,|z|^{2}=z \\cdot \\bar{z}$. Letting $z=a+b \\omega+c \\omega^{2}$, we find that $\\bar{z}=a+c \\omega+b \\omega^{2}$, and $$|z|^{2} =a^{2}+a b \\omega+a c \\om... |
deepscaler | [
{
"role": "user",
"content": "(a) In the diagram, $\\triangle PQR$ is right-angled at $R$. If $PQ = 25$ and $RQ = 24$, determine the perimeter and area of $\\triangle PQR$.\n\n(b) In the diagram, $\\triangle ABC$ is right-angled at $C$ with $AB = c$, $AC = b$, and $BC = a$. Also, $\\triangle ABC$ has a peri... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 22790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in the blanks:\n\n(1) In $\\triangle ABC$, $A=60^{\\circ}$, $b = 1$, and the area of $\\triangle ABC$ is $\\sqrt{3}$. Find the value of $\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}$.\n\n(2) In an arithmetic sequence with a common difference not equal to $0$, ${a_1}+{a_3}=8$, a... | math | {
"style": "rule",
"ground_truth": "765"
} | {
"split": "train",
"index": 22791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 22792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x+2$ and $g(x)=x/3.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute \\[f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of positive integers $a$, $b$, and $c$ equals 2450. What is the minimum possible value of the sum $a + b + c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 22794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive 3-digit numbers are multiples of 25, but not of 60? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 22795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bus with programmers departed from Novosibirsk to Pavlodar. After traveling 70 km, another car with Pavel Viktorovich left Novosibirsk on the same route and caught up with the bus in Karasuk. After that, Pavel traveled another 40 km, while the bus traveled only 20 km in the s... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 22796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lceil-\\sqrt{\\frac{49}{4}}\\right\\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 22797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(α) = \\frac{\\sin(2π-α)\\cos(π+α)\\cos\\left(\\frac{π}{2}+α\\right)\\cos\\left(\\frac{11π}{2}-α\\right)}{2\\sin(3π + α)\\sin(-π - α)\\sin\\left(\\frac{9π}{2} + α\\right)}$. \n(1) Simplify $f(α)$; \n(2) If $α = -\\frac{25}{4}π$, find the value of $f(α)$. Let's think s... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 22798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 22799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nFind the smallest positive integer \\( n > 1 \\) such that the arithmetic mean of \\( 1^2, 2^2, 3^2, \\cdots, n^2 \\) is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 22800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\\angle A D C=90^{\\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right tri... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{13}"
} | {
"split": "train",
"index": 22801,
"solution": "We see that $\\angle G A F=\\angle G B F=45^{\\circ}$, hence quadrilateral $G F B A$ is cyclic. Consequently $\\angle A G F+\\angle F B A=180^{\\circ}$. So $\\cos \\angle A G F=-\\cos \\angle F B A$. One can check directly that $\\cos \\angle C B A=\\frac{5}{13}$ (... |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ are there such that $0 \\le n \\le 720$ and $n^2 \\equiv 1$ (mod $720$ )? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of diameter $1$ is removed from a $2\\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?\n[asy]\nfill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);\ndraw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));\nfill(circle((1,5/4),1/2),wh... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 22803,
"solution": "1. **Calculate the area of the rectangle**: \n The rectangle has dimensions $2 \\times 3$. Therefore, its area is:\n \\[\n \\text{Area}_{\\text{rectangle}} = 2 \\times 3 = 6.\n \\]\n\n2. **Determine the radius and area of the circle**:\n The diameter of t... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a \\) and \\( b \\) are positive integers, and \\( b - a = 2013 \\). If the equation \\( x^{2} - a x + b = 0 \\) has a positive integer solution, what is the smallest value of \\( a \\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 22804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen int... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 22805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(x^x)^{(x^x)}$ at $x = 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 22806,
"solution": "To solve the expression $(x^x)^{(x^x)}$ at $x = 2$, we follow these steps:\n\n1. **Calculate $x^x$ for $x=2$:**\n \\[\n x^x = 2^2 = 4\n \\]\n\n2. **Substitute $4$ back into the expression $(x^x)^{(x^x)}$:**\n \\[\n (x^x)^{(x^x)} = 4^4\n \\]\n\n3. **Calc... |
deepscaler | [
{
"role": "user",
"content": "Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle that is tangent to the side \\(DC\\) of a regular pentagon \\(ABCDE\\) at point \\(D\\) and tangent to the side \\(AB\\) at point \\(A\\), what is the degree measure of the minor arc \\(AD\\)? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 22808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 22809,
"solution": "To solve this problem, we need to convert the base three number $12112211122211112222_3$ to its base nine representation and find the first digit of that representation.\n\n#### Step 1: Convert from base 3 to base 10\nThe number in base 3 is $12112211122211112222_3... |
deepscaler | [
{
"role": "user",
"content": "Concentric circles $\\Omega_1$ and $\\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\\Omega_1$ and $\\Omega_2$ , respectively. Denote by $\\ell$ the ... | math | {
"style": "rule",
"ground_truth": "10004"
} | {
"split": "train",
"index": 22810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is $\\boxed{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 22811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define\n\\[c_k = k + \\cfrac{1}{2k + \\cfrac{1}{2k + \\cfrac{1}{2k + \\dotsb}}}.\\]Calculate $\\sum_{k = 1}^{11} c_k^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "517"
} | {
"split": "train",
"index": 22812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by: $ \\begin{cases} x=1+t\\cos \\alpha \\\\ y=2+t\\sin \\alpha \\end{cases} (t$ is the parameter, $0\\leqslant \\alpha < \\pi)$, and the polar coordinate system is established with $O$ a... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 22813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \\, \\mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bu... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 22814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \\plus{} 4$, $ p|q^2 \\plus{} 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 2), (5, 29), (29, 5)"
} | {
"split": "train",
"index": 22815,
"solution": "\nWe are given the problem to find all prime numbers \\( p \\) and \\( q \\) less than 2005 such that \\( q \\mid p^2 + 4 \\) and \\( p \\mid q^2 + 4 \\). Let's go through the solution step-by-step.\n\n1. **Understand the Divisibility Conditions:**\n\n We need to... |
deepscaler | [
{
"role": "user",
"content": "Marguerite drove 100 miles in 2.4 hours. If Sam drove for 3 hours at the same average rate as Marguerite, how many miles did he drive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125\\textnormal{ miles}"
} | {
"split": "train",
"index": 22816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 22817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\\le x\\le 5$ and $1\\le y\\le 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2158"
} | {
"split": "train",
"index": 22818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $\\cos A= \\frac {3}{4}$, $\\cos C= \\frac {1}{8}$,\n(I) find the ratio $a:b:c$;\n(II) if $| \\overrightarrow{AC}+ \\overrightarrow{BC}|= \\sqrt {46}$, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac {15 \\sqrt {7}}{4}"
} | {
"split": "train",
"index": 22819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 22820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 2-dimensional Cartesian coordinate system, there are 16 lattice points \\((i, j)\\) where \\(0 \\leq i \\leq 3\\) and \\(0 \\leq j \\leq 3\\). If \\(n\\) points are selected from these 16 points, determine the minimum value of \\(n\\) such that there always exist four poin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"split": "train",
"index": 22822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and if the card is black, she simultaneously repla... | math | {
"style": "rule",
"ground_truth": "42648"
} | {
"split": "train",
"index": 22823,
"solution": "The order of the deck does not matter as black cards and white cards commute, therefore we can assume that the cards are alternating black and white, and only worry about the arrangement of the numbers. After each pair of black and white cards, each number is repla... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . What is the maximum value of $A\\cdot M\\cdot C+A\\cdot M+M\\cdot C+C\\cdot A$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 22824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a_1 \\equal{} 11^{11}, \\, a_2 \\equal{} 12^{12}, \\, a_3 \\equal{} 13^{13}$, and $ a_n \\equal{} |a_{n \\minus{} 1} \\minus{} a_{n \\minus{} 2}| \\plus{} |a_{n \\minus{} 2} \\minus{} a_{n \\minus{} 3}|, n \\geq 4.$ Determine $ a_{14^{14}}$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 22825,
"solution": "\nTo determine \\( a_{14^{14}} \\), we need to evaluate the recursive relationship given by \\( a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}| \\) starting from the initial terms \\( a_1 = 11^{11} \\), \\( a_2 = 12^{12} \\), and \\( a_3 = 13^{13} \\).\n\n### Step-... |
deepscaler | [
{
"role": "user",
"content": "Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\\frac{π}{2}$ always holds. Deter... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 22826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the decimal representation of the even number \\( M \\), only the digits \\( 0, 2, 4, 5, 7, \\) and \\( 9 \\) participate, and digits may repeat. It is known that the sum of the digits of the number \\( 2M \\) is 31, and the sum of the digits of the number \\( M / 2 \\) is 2... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 22827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the book \"Nine Chapters on the Mathematical Art,\" a tetrahedron with all four faces being right-angled triangles is called a \"biēnào.\" Given that tetrahedron $ABCD$ is a \"biēnào,\" $AB\\bot $ plane $BCD$, $BC\\bot CD$, and $AB=\\frac{1}{2}BC=\\frac{1}{3}CD$. If the volu... | math | {
"style": "rule",
"ground_truth": "14\\pi"
} | {
"split": "train",
"index": 22828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\[\n\\frac{\\sin ^{2}\\left(135^{\\circ}-\\alpha\\right)-\\sin ^{2}\\left(210^{\\circ}-\\alpha\\right)-\\sin 195^{\\circ} \\cos \\left(165^{\\circ}-2 \\alpha\\right)}{\\cos ^{2}\\left(225^{\\circ}+\\alpha\\right)-\\cos ^{2}\\left(210^{\\circ}-\\alpha\\right)+\\sin 15^{\\circ} ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 22829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of positive integers that are less than or equal to $5000$ and whose base-$3$ representation has more $1$'s than any other digit. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "379"
} | {
"split": "train",
"index": 22830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are constants, and $a \\neq 0$, $f(x) = ax^2 + bx$, $f(2) = 0$.\n\n$(1)$ If the equation $f(x) - x = 0$ has a unique real root, find the expression for the function $f(x)$;\n\n$(2)$ When $a = 1$, find the maximum and minimum values of the function $f(x)$ ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 22831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $0 \\leq k < n$ be integers and $A=\\{a \\: : \\: a \\equiv k \\pmod n \\}.$ Find the smallest value of $n$ for which the expression \n\\[ \\frac{a^m+3^m}{a^2-3a+1} \\]\ndoes not take any integer values for $(a,m) \\in A \\times \\mathbb{Z^+}.$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\\overline{AB}$ and $\\overline{CD}$ are the only parallel sides. The sum of the absolute values of all po... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 22833,
"solution": "A very natural solution: . Shift $A$ to the origin. Suppose point $B$ was $(x, kx)$. Note $k$ is the slope we're looking for. Note that point $C$ must be of the form: $(x \\pm 1, kx \\pm 7)$ or $(x \\pm 7, kx \\pm 1)$ or $(x \\pm 5, kx \\pm 5)$. Note that we want t... |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be angles such that\n\\begin{align*}\n\\sin a &= \\cot b, \\\\\n\\sin b &= \\cot c, \\\\\n\\sin c &= \\cot a.\n\\end{align*}\nFind the largest possible value of $\\cos a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3 - \\sqrt{5}}{2}}"
} | {
"split": "train",
"index": 22834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 22835,
"solution": "We are given a problem where a frog jumps randomly in one of the four cardinal directions from a point $(1,2)$ within a square bounded by $(0,0), (0,4), (4,4),$ and $(4,0)$. We need to find the probability that the frog first reaches a vertical side of the square.\... |
deepscaler | [
{
"role": "user",
"content": "The line \\(y = M\\) intersects the graph of the function \\(y = x^{3} - 84x\\) at points with abscissas \\(a\\), \\(b\\), and \\(c\\) (\\(a < b < c\\)). It is given that the distance between \\(a\\) and \\(b\\) is half the distance between \\(b\\) and \\(c\\). Find \\(M\\). Le... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 22836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 22837,
"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": 22838,
"solution": "\nWe are given that $x\\%$ of $x$ is $4$. We need to interpret and solve this equation mathematically.\n\n1. **Understanding the percentage operation**: The expression \"$x\\%$ of $x$\" translates to $x\\% \\times x$. Since $x\\%$ means $x$ percent, which is $\\fra... |
deepscaler | [
{
"role": "user",
"content": "Using the trapezoidal rule with an accuracy of 0.01, calculate $\\int_{2}^{3} \\frac{d x}{x-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.6956"
} | {
"split": "train",
"index": 22839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $3$-digit positive integers have digits whose product equals $30$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( AB = AC \\), \\( AD \\) and \\( BE \\) are the angle bisectors of \\( \\angle A \\) and \\( \\angle B \\) respectively, and \\( BE = 2 AD \\). What is the measure of \\( \\angle BAC \\)? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 22841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Allen and Brian are playing a game in which they roll a 6-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to 7 and the latter is at most 3. What is the probability that Allen wins? Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 22842,
"solution": "Note that at any point in the game after the first roll, the probability that Allen wins depends only on the most recent roll, and not on any rolls before that one. So we may define $p$ as the probability that Allen wins at any point in the game, given that the las... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of line $l$ as $ \\begin{cases} x=m+ \\frac { \\sqrt {2}}{2}t \\\\ y= \\frac { \\sqrt {2}}{2}t \\end{cases} (t\\text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\\heartsuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"split": "train",
"index": 22844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the mass of the body $\\Omega$ with density $\\mu = 20z$, bounded by the surfaces\n\n$$\nz = \\sqrt{1 - x^{2} - y^{2}}, \\quad z = \\sqrt{\\frac{x^{2} + y^{2}}{4}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 22845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We color some cells in $10000 \\times 10000$ square, such that every $10 \\times 10$ square and every $1 \\times 100$ line have at least one coloring cell. What minimum number of cells we should color ? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 22846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using three rectangular pieces of paper (A, C, D) and one square piece of paper (B), an area of 480 square centimeters can be assembled into a large rectangle. It is known that the areas of B, C, and D are all 3 times the area of A. Find the total perimeter of the four pieces o... | math | {
"style": "rule",
"ground_truth": "184"
} | {
"split": "train",
"index": 22847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ratio of the area of $\\triangle BCY$ to the area of $\\triangle ABY$ in the diagram if $CY$ bisects $\\angle BCA$. [asy]\nimport markers;\n\nreal t=34/(34+28);\npair A=(-17.18,0);\npair B=(13.82,0);\npair C=(0,30);\npair Y=t*B+(1-t)*A;\n\ndraw(C--A--B--C--Y);\n\nlabel... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{14}"
} | {
"split": "train",
"index": 22848,
"solution": ""
} |
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