data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A workshop produces items of types $A$ and $B$. One item of type $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while an item of type $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling an item of type $A$ is 80 thousand rubles, an... | math | {
"style": "rule",
"ground_truth": "2180"
} | {
"split": "train",
"index": 17049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest real number \\( a \\) such that for any non-negative real numbers \\( x \\), \\( y \\), and \\( z \\) that sum to 1, the following inequality holds:\n$$\na(x^{2} + y^{2} + z^{2}) + xyz \\geq \\frac{a}{3} + \\frac{1}{27}.\n$$ Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 17050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, y \\le 10 \\pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"split": "train",
"index": 17051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has a 1 liter bottle of pure orange juice. He pours half of the contents of the bottle into a vat, fills the bottle with water, and mixes thoroughly. He then repeats this process 9 more times. Afterwards, he pours the remaining contents of the bottle into the vat. What fra... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 17052,
"solution": "All the liquid was poured out eventually. 5 liters of water was poured in, and he started with 1 liter of orange juice, so the fraction is \\(\\frac{5}{1+5}=\\frac{5}{6}\\)."
} |
deepscaler | [
{
"role": "user",
"content": "The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17053,
"solution": "1. **Identify the constraints given by the problem:**\n - $\\text{lcm}(a, b) = 12$\n - $\\text{lcm}(b, c) = 15$\n\n2. **Determine the possible values for $b$:**\n - Since $b$ must be a common factor of both 12 and 15, we consider the greatest common divisor (... |
deepscaler | [
{
"role": "user",
"content": "A solid cube of side length 4 cm is cut into two pieces by a plane that passed through the midpoints of six edges. Find the surface area of each half cube created. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 17054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"split": "train",
"index": 17055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\\ne 0$. What is $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 17056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\(AX\\) in the diagram where \\(AC = 27\\) units, \\(BC = 36\\) units, and \\(BX = 30\\) units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 17057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $\\left(\\frac{1}{ \\sqrt[3]{x}} +2x \\sqrt {x}\\right)^{7}$, the coefficient of $x^{5}$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 17058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Addison's age is three times Brenda's age. Janet is six years older than Brenda. Addison and Janet are twins. How old is Brenda? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 17059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left|\\frac56 +2i\\right|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{6}"
} | {
"split": "train",
"index": 17060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equations\n\\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\\]and\n\\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\\]have a common rational root $r$ which is not an integer, and which is positive. What is $r?$ 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": 17061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sum of the digits of a natural number \\( n \\) is subtracted from \\( n \\), the result is 2016. Find the sum of all such natural numbers \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20245"
} | {
"split": "train",
"index": 17062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{5}{4\\sqrt{7} + 3\\sqrt{13}}$ and express your answer in the form $\\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$ where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy\n\\[0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},\\]\nwhere $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$.\n Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "447"
} | {
"split": "train",
"index": 17064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the prime factorization of a positive integer \\( A \\) can be written as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\( \\alpha, \\beta, \\gamma \\) are natural numbers. If half of \\( A \\) is a perfect square, one-third of \\( A \\) is a... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 17065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahed... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 17066,
"solution": "1. **Understanding the Problem**: We need to find the shortest path on the surface of a regular tetrahedron from the midpoint of one edge to the midpoint of an opposite edge. The tetrahedron has edges of length 1.\n\n2. **Unfolding the Tetrahedron**: To visualize t... |
deepscaler | [
{
"role": "user",
"content": "The second hand on a clock is 8 cm long. How far in centimeters does the tip of the second hand travel during a period of 45 minutes? 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": "720\\pi"
} | {
"split": "train",
"index": 17067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \\... | math | {
"style": "rule",
"ground_truth": "6.6"
} | {
"split": "train",
"index": 17068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the curves $C\\_1$: $\\begin{cases} & x=-4+\\cos t, \\ & y=3+\\sin t \\ \\end{cases}$ (with $t$ as the parameter), and $C\\_2$: $\\begin{cases} & x=6\\cos \\theta, \\ & y=2\\sin \\theta \\ \\end{cases}$ (with $\\theta$ as the parameter).\n\n(1) Convert the equations of $C... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}-1"
} | {
"split": "train",
"index": 17069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_{n}\\}$ is an arithmetic sequence with a non-zero common difference, and $a_{1}+a_{10}=a_{9}$, find $\\frac{{a}_{1}+{a}_{2}+…+{a}_{9}}{{a}_{10}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{8}"
} | {
"split": "train",
"index": 17070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 17071,
"solution": "First of all, note that the nonnegative integer condition really does not matter, since even if we have a nonnegative power, there is always a power of $m$ we can multiply to get to non-negative powers. Now we see that our problem is just a matter of m-chopping blo... |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle \\( A B C \\) where \\( A B = A C \\) and \\( \\angle A B C = 53^\\circ \\). Point \\( K \\) is such that \\( C \\) is the midpoint of \\( A K \\). Point \\( M \\) is chosen such that:\n\n- \\( B \\) and \\( M \\) are on the same side of the line \\(... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 17072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \\geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{11+6\\sqrt{2}}{8}"
} | {
"split": "train",
"index": 17073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $a^7xy-a^6y-a^5x=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^2)=a^4b^4$ for some integers $m$, $n$, and $p$. Find $mnp$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence ${a_n}$ is $S_n$, if $S_2=3$, $S_4=15$, find the common ratio $q$ and $S_6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 17075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $\\sqrt{1\\frac{7}{9}}=$______, $\\sqrt[3]{(-3)^{3}}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 17076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickd... | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 17077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If angle $A$ lies in the second quadrant and $\\sin A = \\frac{3}{4},$ find $\\cos A.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 17078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $15$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $BC$ extended through $C$ intersects the circle at point $E$. Find the number of degrees in angle $AEC$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 17079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{485}{486}"
} | {
"split": "train",
"index": 17080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a random variable $\\eta = 8 - \\xi$, if $\\xi \\sim B(10, 0.6)$, then calculate $E\\eta$ and $D\\eta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 17081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $1357_{10}$ to base 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20412_5"
} | {
"split": "train",
"index": 17082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A typesetter scattered part of a set - a set of a five-digit number that is a perfect square, written with the digits $1, 2, 5, 5,$ and $6$. Find all such five-digit numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15625"
} | {
"split": "train",
"index": 17083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid with a perimeter of 8 and an area of 2, a circle can be inscribed. Find the distance from the point of intersection of the diagonals of the trapezoid to its shorter base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 - \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 17084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a five-digit number without repeating any digit, the probability that the number is even and the digits $1$, $2$ are adjacent is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 17085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a game called \"Twenty-Four Points\" with the following rules: Choose four natural numbers between $1$ and $13$, and perform addition, subtraction, multiplication, and division operations on these four numbers (each number can be used only once and parentheses can be u... | math | {
"style": "rule",
"ground_truth": "3 \\times \\left(10 - 6 + 4\\right)"
} | {
"split": "train",
"index": 17086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nConsider all possible quadratic polynomials $x^2 + px + q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any polynomial with the described properties, the sum of the hundredth po... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17088,
"solution": "Squaring, we find that $(9 + bi)^2 = 81 + 18bi - b^2$. Cubing and ignoring the real parts of the result, we find that $(81 + 18bi - b^2)(9 + bi) = \\ldots + (9\\cdot 18 + 81)bi - b^3i$.\n Setting these two equal, we get that $18bi = 243bi - b^3i$, so $b(b^2 - 225) ... |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = (2^t - 3, 4^t - 5 \\cdot 2^t - 1)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"split": "train",
"index": 17089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bacteria in a jar triple every 20 seconds. After three minutes, there are 275,562 bacteria in the jar. How many were in the jar at the beginning of the experiment? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcabcabc\\ldots=0.\\overline{abc}$, where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how m... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 17091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "187.5"
} | {
"split": "train",
"index": 17092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of 49 consecutive integers is $7^5$. What is their median? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 17093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(ABCD\\) is a square-based pyramid with base \\(ABCD\\) and apex \\(E\\). Point \\(E\\) is directly above point \\(A\\), with \\(AE = 1024\\) units and \\(AB = 640\\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \\(ABCD\\), at a heigh... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 17094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"split": "train",
"index": 17095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments measuring 6 cm and 7 cm. Calculate the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 17096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive odd integers greater than 1 and less than $150$ are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 17097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A reservoir is equipped with an inlet tap A and outlet taps B and C. It is known that opening taps A and B together for 8 hours or opening taps A, B, and C together for a whole day can fill an empty reservoir, and opening taps B and C together for 8 hours can empty a full reser... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, the measure of $\\angle SIP$ is $48^\\circ$ and $OS=12$ cm. Find the number of centimeters in the length of arc $SP$ and also determine the length of arc $SXP$, where $X$ is a point on the arc $SP$ such that $\\angle SXP = 24^\\circ$. Express your a... | math | {
"style": "rule",
"ground_truth": "3.2\\pi"
} | {
"split": "train",
"index": 17099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\cos α,\\sin α)$ and $\\overrightarrow{b}=(-2,2)$.\n(1) If $\\overrightarrow{a}\\cdot \\overrightarrow{b}= \\frac {14}{5}$, find the value of $(\\sin α+\\cos α)^{2}$;\n(2) If $\\overrightarrow{a}$ is parallel to $\\overrightarrow{b}$, find t... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 17100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\\boxed{32768}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"split": "train",
"index": 17101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call an integer \"extraordinary\" if it has exactly one even divisor other than 2. How many extraordinary numbers exist in the interval $[1 ; 75]$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 17102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 17103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cubic function $f(x)=\\frac{a}{3}x^{3}+bx^{2}+cx+d$ ($a < b$) is monotonically increasing on $\\mathbb{R}$, then the minimum value of $\\frac{a+2b+3c}{b-a}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8+6 \\sqrt{2}"
} | {
"split": "train",
"index": 17104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ are there subject to the constraint that $1 \\leq n \\leq 2020$ and $n^n$ is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1032"
} | {
"split": "train",
"index": 17105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of A not losing is $\\dfrac{1}{3} + \\dfrac{1}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"split": "train",
"index": 17107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $20n$ members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If $n$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching ban... | math | {
"style": "rule",
"ground_truth": "940"
} | {
"split": "train",
"index": 17108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rosie can make two pies out of nine apples. How many pies can she make out of twenty-seven apples? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 17110,
"solution": "1. **Identify the relationship between the angles**: In a right triangle, the sum of the angles is $180^\\circ$. Given one angle is $90^\\circ$, the sum of the other two angles, $a^\\circ$ and $b^\\circ$, must be $90^\\circ$. Thus, we have:\n \\[\n a^\\circ + b... |
deepscaler | [
{
"role": "user",
"content": "A ball is dropped from 10 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 17111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha \\cos \\alpha = \\frac{1}{8}$, and $\\alpha$ is an angle in the third quadrant. Find $\\frac{1 - \\cos^2 \\alpha}{\\cos(\\frac{3\\pi}{2} - \\alpha) + \\cos \\alpha} + \\frac{\\sin(\\alpha - \\frac{7\\pi}{2}) + \\sin(2017\\pi - \\alpha)}{\\tan^2 \\alpha... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 17112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 8$. Two of the roots of $g(x)$ are $r + 5$ and $r + 11$, and\n\\[f(x) - g(x) = 2r\\] for all real numbers $x$. Find $r$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "20.25"
} | {
"split": "train",
"index": 17113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number ot 6-tuples $(x_1, x_2,...,x_6)$ , where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 17114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all four-digit numbers that are 9 times larger than their reversed counterparts. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9801"
} | {
"split": "train",
"index": 17115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three p... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 17116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\{a_n\\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2011} + a_{2012} > 0$, and $a_{2011} \\cdot a_{2012} < 0$, determine the natural number $n$ that maximizes the sum of the first $n$ terms $S_n$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 17117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "0.9232"
} | {
"split": "train",
"index": 17118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $N$ be the number of minimal... | math | {
"style": "rule",
"ground_truth": "760"
} | {
"split": "train",
"index": 17119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 17120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens ... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 17121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different divisors does the number 86,400,000 have (including 1 and the number 86,400,000 itself)? Find the sum of all these divisors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "319823280"
} | {
"split": "train",
"index": 17122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{17}{10}=19448$, $\\binom{17}{11}=12376$ and $\\binom{19}{12}=50388$, find $\\binom{18}{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18564"
} | {
"split": "train",
"index": 17123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two transformations are applied to the complex number $-3 - 8i$:\n\nA $45^\\circ$ rotation around the origin in the counter-clockwise direction.\nA dilation, centered at the origin, with scale factor $\\sqrt{2}.$\n\nWhat is the resulting complex number? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "5 - 11i"
} | {
"split": "train",
"index": 17124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ in the set $\\{4,9,14,19, \\ldots, 2014\\}$ have the property that the sum of the decimal digits of $n$ is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 17125,
"solution": "We know that 2014 does not qualify the property. So, we'll consider $\\{4,9,14, \\ldots, 2009\\}$ instead. Now, we partition this set into 2 sets: $\\{4,14,24, \\ldots, 2004\\}$ and $\\{9,19,29, \\ldots, 2009\\}$. For each so the first and second set are basically ... |
deepscaler | [
{
"role": "user",
"content": "Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers with safe vegetables, a rural cooperative invests 2 million yuan each year to build two pollution-f... | math | {
"style": "rule",
"ground_truth": "282"
} | {
"split": "train",
"index": 17126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$a$ and $b$ are integers such that $a+\\sqrt{b}=\\sqrt{15+\\sqrt{216}}$. Compute $a / b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 17127,
"solution": "Squaring both sides gives $a^{2}+b+2 a \\sqrt{b}=15+\\sqrt{216}$; separating rational from irrational parts, we get $a^{2}+b=15,4 a^{2} b=216$, so $a^{2}$ and $b$ equal 6 and $9 . a$ is an integer, so $a^{2}=9, b=6 \\Rightarrow a / b=3 / 6=1 / 2$. (We cannot have $... |
deepscaler | [
{
"role": "user",
"content": "A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 17128,
"solution": "1. **Understanding the Problem:**\n - We have a list of $2018$ positive integers.\n - The mode (most frequently occurring number) appears exactly $10$ times.\n - We need to find the least number of distinct values in the list.\n\n2. **Setting Up the Equation:... |
deepscaler | [
{
"role": "user",
"content": "The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 17129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All of the roots of $x^3+ax^2+bx+c$ are positive integers greater than $2$ , and the coeffi\u000ecients satisfy $a+b+c+1=-2009$ . Find $a$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-58"
} | {
"split": "train",
"index": 17130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "6000"
} | {
"split": "train",
"index": 17131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,2009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$. Let's think... | math | {
"style": "rule",
"ground_truth": "803"
} | {
"split": "train",
"index": 17132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the base $8$ representation of a perfect square is $ab3c$, where $a\\ne 0$, then $c$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 17133,
"solution": "1. **Understanding the problem**: We are given a number in base $8$ represented as $ab3c$, which is a perfect square. We need to determine the value of $c$.\n\n2. **Expressing the number in decimal**: The number $ab3c$ in base $8$ can be expressed in decimal as:\n ... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse \\( C: \\frac{x^{2}}{2}+y^{2}=1 \\) with left and right foci \\( F_{1}, F_{2} \\) respectively, let \\( P \\) be a point on the ellipse \\( C \\) in the first quadrant. The extended lines \\( P F_{1}, P F_{2} \\) intersect the ellipse \\( C \\) at points \\( Q_... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 17134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the asymptote equation of the hyperbola $y^{2}+\\frac{x^2}{m}=1$ is $y=\\pm \\frac{\\sqrt{3}}{3}x$, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 17135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[\\arcsin (\\sin x) = \\frac{x}{2}.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2 \\pi}{3}, 0, \\frac{2 \\pi}{3}"
} | {
"split": "train",
"index": 17136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 17137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f:\\mathbb{Z} \\to \\mathbb{Z}$ satisfies\n\\begin{align*}\nf(x+4)-f(x) &= 8x+20, \\\\\nf(x^2-1) &= (f(x)-x)^2+x^2-2\n\\end{align*}for all integers $x.$ Enter the ordered pair $(f(0),f(1)).$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "(-1,1)"
} | {
"split": "train",
"index": 17138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $y=\\frac{x^3+11x^2+38x+35}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $y=f\\left(x\\right)$ have domain $D$, and all points on its graph be above the line $y=t$. If the function $f\\left(x\\right)=\\left(x-t\\right)e^{x}$ has the domain $R$ and is a \"$\\left(-\\infty ,+\\infty \\right)-t$ function\", determine the largest intege... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 17140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of each exterior angle of a regular polygon is $30$ degrees. What is the sum of the measures of the interior angles, in degrees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 17141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A conical flask has a base radius of $15$ cm and a height of $30$ cm. The flask is filled with a liquid to a height of $10$ cm from the base, and a sphere is inscribed such that it just touches the liquid surface. What is the radius of the sphere? Express your answer in cm. Let... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 17143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of the sequence $\\{a_n\\}$ be $S_n$, where $S_n=n^2+n$, and the general term of the sequence $\\{b_n\\}$ is given by $b_n=x^{n-1}$.\n\n(1) Find the general term formula for the sequence $\\{a_n\\}$;\n\n(2) Let $c_n=a_nb_n$, and the sum of the... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{4}"
} | {
"split": "train",
"index": 17144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\\sum_{N = 1}^{1024} \\lfloor \\log_2 N \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"split": "train",
"index": 17145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\nax+3 & \\text{ if }x>0, \\\\\nab & \\text{ if }x=0, \\\\\nbx+c & \\text{ if }x<0.\n\\end{array}\n\\right.\\]If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then what is $a+b+c$? Let's think ste... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy\n\\[\\frac{a}{b} + \\frac{b}{a} = \\frac{4}{5}.\\]Find\n\\[\\frac{k_1}{k_2} + \\frac{k_2}{k_1}.\\] Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"split": "train",
"index": 17147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A glass is filled to the brim with salty water. Fresh ice with mass \\( m = 502 \\) floats on the surface. What volume \\( \\Delta V \\) of water will spill out of the glass by the time the ice melts? Neglect surface tension. The density of fresh ice is \\( \\rho_{n} = 0.92 \\,... | math | {
"style": "rule",
"ground_truth": "2.63"
} | {
"split": "train",
"index": 17148,
"solution": ""
} |
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