data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 35749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young. L... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 35750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greater of the solutions to the equation $x^2 + 15x -54=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 35751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35752,
"solution": "1. **Identify the curves**: The given equations are $x^2 + 4y^2 = 1$ and $4x^2 + y^2 = 4$. These represent an ellipse and another ellipse respectively.\n\n2. **Substitute variables**: Let $a = x^2$ and $b = y^2$. This substitution transforms the equations into:\n ... |
deepscaler | [
{
"role": "user",
"content": "Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. Calculate the degree measure of ∠AMD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67.5"
} | {
"split": "train",
"index": 35753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$ with $\\angle D = 90^\\circ$, side $DE = 9$ cm and side $EF = 15$ cm. Find $\\sin F$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{34}}{34}"
} | {
"split": "train",
"index": 35754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is the graph of an ellipse. (Assume that tick marks are placed every $1$ unit along the axes.)\n[asy]\nsize(8cm);\n\nint x, y;\nfor (y = -4; y <= 2; ++y) {\n draw((-1,y)--(7,y),gray(0.7));\n}\nfor (x = -1; x <= 7; ++x) {\n draw((x,-4)--(x,2),gray(0.7));\n}\n\ndraw(shift... | math | {
"style": "rule",
"ground_truth": "(3+\\sqrt{5},-1)"
} | {
"split": "train",
"index": 35755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Nevada, 580 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the \"Soda\" sector of the graph is $198^\\circ$, to the nearest whole degree. How many of the people surveyed chose \"Soda\"? Express your an... | math | {
"style": "rule",
"ground_truth": "321"
} | {
"split": "train",
"index": 35756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that a parabola has vertex $\\left(\\frac{1}{4},-\\frac{9}{8}\\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 35757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers is $20$. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 35758,
"solution": "\nLet the three numbers be $x$, $y$, and $z$. According to the problem, we have the following conditions:\n1. $x + y + z = 20$\n2. $x = 4(y + z)$\n3. $y = 7z$\n\nWe can use these equations to find the values of $x$, $y$, and $z$.\n\n#### Step 1: Substitute $y$ in t... |
deepscaler | [
{
"role": "user",
"content": "To prepare for her half-marathon, Mary ran a total of $2847_9$ miles. How many miles (in base ten) did Mary run? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2149"
} | {
"split": "train",
"index": 35759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them. How many integers are in Cameron's list? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 35760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"split": "train",
"index": 35761,
"solution": "We notice that the function has reflectional symmetry across both $x=2$ and $x=7$. We also use the fact that $x=0$ is a root. This shows that $x=4$ and $x=14$ are also roots. We then apply the reflection across the other axis to form $x=\\pm 10$ as roots. Continuin... |
deepscaler | [
{
"role": "user",
"content": "How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 35762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A\\left(\\begin{matrix} \\cos \\alpha , & \\sin \\alpha \\end{matrix}\\right)$ and $B\\left(\\begin{matrix} \\cos \\beta , & \\sin \\beta \\end{matrix}\\right)$, where $\\alpha$ and $\\beta$ are acute angles, and the distance between $A$ and $B$ is $\\frac{\\sqrt{... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{25}"
} | {
"split": "train",
"index": 35763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \"fifty percent mirror\" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two \"fifty percent mirrors\" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 35764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answe... | math | {
"style": "rule",
"ground_truth": "0.08"
} | {
"split": "train",
"index": 35765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers defined by $a_{0}=21, a_{1}=35$, and $a_{n+2}=4 a_{n+1}-4 a_{n}+n^{2}$ for $n \\geq 2$. Compute the remainder obtained when $a_{2006}$ is divided by 100. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 35766,
"solution": "No pattern is evident in the first few terms, so we look for a formula for $a_{n}$. If we write $a_{n}=A n^{2}+B n+C+b_{n}$ and put $b_{n+2}=4 b_{n+1}-4 b_{n}$. Rewriting the original recurrence, we find $$\\begin{aligned} A n^{2}+(4 A+B) n+(4 A+2 B+C)+b_{n+2} & \\... |
deepscaler | [
{
"role": "user",
"content": "The expression $x^2 - 16x + 60$ can be written in the form $(x - a)(x - b)$, where $a$ and $b$ are both nonnegative integers and $a > b$. What is the value of $3b - a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 35767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\\cos B - b\\cos A = c$, and $C = \\frac{π}{5}$, calculate the measure of $\\angle B$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{10}"
} | {
"split": "train",
"index": 35768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the list $-50, -44, -38, \\ldots, 68, 74$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{21}"
} | {
"split": "train",
"index": 35770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "325/648"
} | {
"split": "train",
"index": 35771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $15336$, what is the greatest common divisor of $f(x)=(3x+4)(7x+1)(13x+6)(2x+9)$ and $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 35772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\[\\frac{x^2 + 3x + 4}{x + 5} = x + 6.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{13}{4}"
} | {
"split": "train",
"index": 35773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 35774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute all values of $b$ for which the following system has a solution $(x,y)$ in real numbers:\n\\begin{align*}\n\\sqrt{xy} &= b^b, \\\\\n\\log_b (x^{\\log_b y}) + \\log_b (y^{\\log_b x}) &= 4b^4.\n\\end{align*} Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\left( 0, \\frac{1}{\\sqrt{2}} \\right]"
} | {
"split": "train",
"index": 35775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of the parameter \\(a\\) for which the quadratic trinomial \\(\\frac{1}{3} x^2 + \\left(a+\\frac{1}{2}\\right) x + \\left(a^2 + a\\right)\\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of suc... | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"split": "train",
"index": 35776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the number of positive integers $n \\leq 10^{6}$ such that $n^{2}+1$ has a prime factor greater than $n$. Submit a positive integer $E$. If the correct answer is $A$, you will receive $\\max \\left(0,\\left\\lfloor 20 \\cdot \\min \\left(\\frac{E}{A}, \\frac{10^{6}-E}{... | math | {
"style": "rule",
"ground_truth": "757575"
} | {
"split": "train",
"index": 35777,
"solution": "Let $N$ denote $10^{6}$. We count by summing over potential prime factors $p$. For any prime $p>2$, we have that $p \\mid n^{2}+1$ for two values of $n$ if $p \\equiv 1(\\bmod 4)$, and zero values otherwise. Pretending these values are equally likely to be any of $... |
deepscaler | [
{
"role": "user",
"content": "A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "139.5"
} | {
"split": "train",
"index": 35778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $a$ for which there exists a non-constant function $f :\\Bbb R \\to \\Bbb R$ satisfying the following two equations for all $x\\in \\Bbb R:$\ni) $f(ax) = a^2f(x)$ and\nii) $f(f(x)) = a f(x).$ Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "0 \\text{ and } 1"
} | {
"split": "train",
"index": 35779,
"solution": "\nTo solve this problem, we need to find all real numbers \\( a \\) for which there exists a non-constant function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfying the two conditions:\n\n1. \\( f(ax) = a^2f(x) \\) for all \\( x \\in \\mathbb{R} \\).\n2. \\( f(f(... |
deepscaler | [
{
"role": "user",
"content": "Consider an equilateral triangular grid $G$ with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points.... | math | {
"style": "rule",
"ground_truth": "52 \\sqrt{3}"
} | {
"split": "train",
"index": 35780,
"solution": "Imagine deforming the triangle lattice such that now it looks like a lattice of 45-45-90 right triangles with legs of length 1. Note that by doing this, the area has multiplied by $\\frac{2}{\\sqrt{3}}$, so we need to readjust out answer on the isosceles triangle l... |
deepscaler | [
{
"role": "user",
"content": "Given that the amount of cultural and tourism vouchers issued is $2.51 million yuan, express this amount in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.51 \\times 10^{6}"
} | {
"split": "train",
"index": 35781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 35782,
"solution": "To solve this problem, we need to label each vertex of a cube with integers from $1$ to $8$ such that the sum of the numbers on the vertices of each face is the same. Additionally, we consider two arrangements the same if one can be obtained from the other by rotat... |
deepscaler | [
{
"role": "user",
"content": "Let $G$ be the set of polynomials of the form $$ P(z)=z^n+c_{n-1}z^{n-1}+\\cdots+c_2z^2+c_1z+50, $$where $ c_1,c_2,\\dots, c_{n-1} $ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "528"
} | {
"split": "train",
"index": 35783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 35784,
"solution": "To solve for $\\boxed{\\boxed{11}}$, we first need to determine $\\boxed{11}$, which is the sum of the positive factors of $11$.\n\n1. **Finding $\\boxed{11}$**:\n Since $11$ is a prime number, its only positive factors are $1$ and $11$ itself. Therefore, the sum... |
deepscaler | [
{
"role": "user",
"content": " $n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \\lim_{n \\to \n\\infty}e_n. $$ Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with the sum of its first n terms $S_n$, if $S_2 = 4$, and $a_{n+1} = 2S_n + 1$ for $n \\in N^*$, find the values of $a_1$ and $S_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 35786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The last 5 digits of $99 \\times 10101 \\times 111 \\times 1001001$ are _____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88889"
} | {
"split": "train",
"index": 35787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$ \\frac{d^{100}}{dx^{100}}\\bigg(\\frac{p(x)}{x^3-x}\\bigg)=\\frac{f(x)}{g(x)} $$ for some polynomials $f(x)$ and $g(x)$ then find the smallest... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 35788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the system of equations $\\begin{cases} x - 2y = z - 2u \\\\ 2yz = ux \\end{cases}$, for every set of positive real number solutions $\\{x, y, z, u\\}$ where $z \\geq y$, there exists a positive real number $M$ such that $M \\leq \\frac{z}{y}$. Find the maximum value of $... | math | {
"style": "rule",
"ground_truth": "6 + 4\\sqrt{2}"
} | {
"split": "train",
"index": 35789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $x^2 + y^2$ if $x - y = 25$ and $xy = 36$. Additionally, find what $x+y$ equals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{769}"
} | {
"split": "train",
"index": 35790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the largest positive integer such that $\\frac{2007!}{2007^{n}}$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 35791,
"solution": "Note that $2007=3^{2} \\cdot 223$. Using the fact that the number of times a prime $p$ divides $n!$ is given by $$\\left\\lfloor\\frac{n}{p}\\right\\rfloor+\\left\\lfloor\\frac{n}{p^{2}}\\right\\rfloor+\\left\\lfloor\\frac{n}{p^{3}}\\right\\rfloor+\\cdots$$ it foll... |
deepscaler | [
{
"role": "user",
"content": "What is $100(100-3)-(100 \\cdot 100-3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-297"
} | {
"split": "train",
"index": 35792,
"solution": "We are given the expression \\(100(100-3)-(100\\cdot100-3)\\) and need to simplify it.\n\n1. **Distribute and simplify inside the parentheses:**\n \\[\n 100(100-3) = 100 \\times 97 = 9700\n \\]\n Here, we calculate \\(100 - 3 = 97\\) and then multiply by 10... |
deepscaler | [
{
"role": "user",
"content": "The square \\( STUV \\) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \\( 40 \\text{ cm} \\). What is the area, in \\( \\text{cm}^2 \\), of the square \\( STUV \\)?\n(a) 400 \n(b) 200 \n(c) 160 \n(d) 100 \n(e) 80 \n Let's think ste... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 35793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positiv... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 35794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From point \\( A \\), two rays are drawn intersecting a given circle: one at points \\( B \\) and \\( C \\), and the other at points \\( D \\) and \\( E \\). It is known that \\( AB = 7 \\), \\( BC = 7 \\), and \\( AD = 10 \\). Determine \\( DE \\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"split": "train",
"index": 35796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the roots of the equation \\[x\\sqrt{x} - 6x + 7\\sqrt{x} - 1 = 0,\\]given that all of the roots are real and nonnegative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 ... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 35798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan. \n(1) There are 57 students in the ninth grade class 1, and each ... | math | {
"style": "rule",
"ground_truth": "51.30"
} | {
"split": "train",
"index": 35799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$ and $b$ are positive integers such that $\\gcd(a,b)$ is divisible by exactly $7$ distinct primes and $\\mathop{\\text{lcm}}[a,b]$ is divisible by exactly $28$ distinct primes.\n\nIf $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinc... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 35800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is \\(1\\tfrac{1}{2}\\) divided by \\(\\tfrac{5}{6}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{9}{5}"
} | {
"split": "train",
"index": 35801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate $N=\\prod_{n=1}^{\\infty} n^{n^{-1.25}}$. An estimate of $E>0$ will receive $\\lfloor 22 \\min (N / E, E / N)\\rfloor$ points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9000000"
} | {
"split": "train",
"index": 35802,
"solution": "We approximate $\\ln N=\\sum_{n=1}^{\\infty} \\frac{\\ln n}{n^{5 / 4}}$ with an integral as $\\int_{1}^{\\infty} \\frac{\\ln x}{x^{5 / 4}} d x =\\left.\\left(-4 x^{-1 / 4} \\ln x-16 x^{-1 / 4}\\right)\\right|_{1} ^{\\infty} =16$. Therefore $e^{16}$ is a good approx... |
deepscaler | [
{
"role": "user",
"content": "Hooligan Vasya loves running on the metro escalator, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds wil... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 35803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\\geq1$, we have\n$$a_{n+1}=\\frac{n-1}{n+1}a_n-\\frac{n-2}{n^2+n}a_{n-1}.$$\nCalculate the value of $\\frac{a_1}{a_2}-\\frac{a_2}{a_3}+\\frac{a_3}{a_4}-\\frac{a_4}{a_5}+\\ldots+\\frac{a_{2013}}{... | math | {
"style": "rule",
"ground_truth": "3021"
} | {
"split": "train",
"index": 35804,
"solution": "\nWe start by examining the sequence \\((a_n)\\) given by the recurrence relations \\(a_0 = 1\\) and \\(a_1 = 2015\\), with the following recursive formula for \\(n \\geq 1\\):\n\n\\[\na_{n+1} = \\frac{n-1}{n+1}a_n - \\frac{n-2}{n^2+n}a_{n-1}.\n\\]\n\nThe goal is t... |
deepscaler | [
{
"role": "user",
"content": "The decimal number corresponding to the binary number $111011001001_2$ is to be found. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3785"
} | {
"split": "train",
"index": 35805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\vec{a}| = |\\vec{b}| = 2$, and $(\\vec{a} + 2\\vec{b}) \\cdot (\\vec{a} - \\vec{b}) = -2$, find the angle between $\\vec{a}$ and $\\vec{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 35806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.\n\\begin{tabular}{ccc} X & X & X \\\\ X & X & X \\end{tabular}\nIf the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjac... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"split": "train",
"index": 35807,
"solution": "1. **Total Arrangements**: First, we calculate the total number of ways to arrange 6 kids (including Abby and Bridget) in 6 seats. This is simply the number of permutations of 6 items, which is $6!$.\n\n2. **Adjacent Arrangements**: We need to count the number of w... |
deepscaler | [
{
"role": "user",
"content": "Let \\( g(x) = 3x^4 + 2x^3 - x^2 - 4x + s \\). Find the value of \\( s \\) such that \\( g(-1) = 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 35808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the minimum of $\\sqrt{(x-1)^{2}+y^{2}}+\\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \\in \\mathbb{R}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 35809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right $\\Delta ABC$, $\\angle CAB$ is a right angle. Point $M$ is the midpoint of $\\overline{BC}$. What is the number of centimeters in the length of median $\\overline{AM}$? Express your answer as a decimal to the nearest tenth. [asy] pair A,B,C,M;\nA = (0,0); B = (4,0); C... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 35810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 4-digit numbers greater than 1000 are there that use the four digits of 2012? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 35811,
"solution": "To solve this problem, we need to determine how many distinct 4-digit numbers can be formed using the digits of 2012, ensuring that each number is greater than 1000. We will use casework based on the leading digit of the number.\n\n#### Case 1: Leading digit is 1\n... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \\frac{\\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.\n\n(Ⅰ) Find the equation of the ellipse $C$.\n\n(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 35812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f\\left(x\\right)=ax^{2}-1$ and $g\\left(x\\right)=\\ln \\left(ax\\right)$ have an \"$S$ point\", then find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{e}"
} | {
"split": "train",
"index": 35813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 15... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 35814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the planes $x - 3y + 3z = 8$ and $2x - 6y + 6z = 2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7 \\sqrt{19}}{19}"
} | {
"split": "train",
"index": 35815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\\text{th}}$ term of the sequence is $\\log{b^n}$. What is $n$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 35816,
"solution": "\n1. **Define the terms in terms of logarithms**:\n Let $A = \\log(a)$ and $B = \\log(b)$. The given terms of the sequence can be rewritten using logarithmic properties:\n \\[\n \\begin{align*}\n \\log(a^3b^7) &= 3A + 7B, \\\\\n \\log(a^5b^{12}) &= 5A + 1... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 - 3x^2 - 9x + 1$, find the intervals of monotonicity and the extrema of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"split": "train",
"index": 35817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x) = m\\ln{x} + nx$ whose tangent at point $(1, f(1))$ is parallel to the line $x + y - 2 = 0$, and $f(1) = -2$, where $m, n \\in \\mathbb{R}$,\n(Ⅰ) Find the values of $m$ and $n$, and determine the intervals of monotonicity for the function $f(x)$;\n(Ⅱ) Let... | math | {
"style": "rule",
"ground_truth": "\\frac{e(e - 2)}{e - 1}"
} | {
"split": "train",
"index": 35818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(n\\\\) be a positive integer, and \\\\(f(n) = 1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{n}\\\\). It is calculated that \\\\(f(2) = \\frac{3}{2}\\\\), \\\\(f(4) > 2\\\\), \\\\(f(8) > \\frac{5}{2}\\\\), and \\\\(f(16) > 3\\\\). Observing the results above, ac... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 35819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the foci of a hyperbola, a line passing through $F\\_2$ perpendicular to the real axis intersects the hyperbola at points $A$ and $B$. If $BF\\_1$ intersects the $y$-axis at point $C$, and $AC$ is perpendicular to $BF\\_1$, determine the eccen... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 35820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the volume of the solid formed by the set of vectors $\\mathbf{v}$ such that:\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} 12 \\\\ -34 \\\\ 6 \\end{pmatrix}\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi (334)^{3/2}"
} | {
"split": "train",
"index": 35821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a singing contest, a Rooster, a Crow, and a Cuckoo were contestants. Each jury member voted for one of the three contestants. The Woodpecker tallied that there were 59 judges, and that the sum of votes for the Rooster and the Crow was 15, the sum of votes for the Crow and th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have $ 23^2 = 529 $ ordered pairs $ (x, y) $ with $ x $ and $ y $ positive integers from 1 to 23, inclusive. How many of them have the property that $ x^2 + y^2 + x + y $ is a multiple of 6? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 35823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "463"
} | {
"split": "train",
"index": 35824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5} x^2"
} | {
"split": "train",
"index": 35825,
"solution": "1. **Define the dimensions of the rectangle**: Let the width of the rectangle be $w$. Since the rectangle's length is twice its width, the length is $2w$.\n\n2. **Apply the Pythagorean Theorem**: The diagonal of a rectangle forms a right triangle with the rectangle... |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of\n\\[2xy \\sqrt{6} + 8yz.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{22}"
} | {
"split": "train",
"index": 35826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\sqrt{64 \\times \\sqrt{49}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{7}"
} | {
"split": "train",
"index": 35827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last three digits of $7^{103}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 35828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the focus $F$ of the parabola $x=4y^{2}$ intersects the parabola at points $M$ and $N$, and $|MF|= \\frac{1}{8}$, find the value of $|MN|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 35829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 35830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n \\in \\{1, 3\\}"
} | {
"split": "train",
"index": 35831,
"solution": "\nWe are tasked with finding all positive integers \\( n \\) such that the decimal representation of \\( n^2 \\) consists only of odd digits. To approach this problem, we need to analyze the properties of squares of integers and the constraints that arise due to ha... |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence \n$1,-2,3,-4,5,-6,\\ldots,$\nwhose $n$th term is $(-1)^{n+1}\\cdot n$. What is the average of the first $200$ terms of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-0.5"
} | {
"split": "train",
"index": 35832,
"solution": "1. **Identify the sequence**: The given sequence is $1, -2, 3, -4, 5, -6, \\ldots$, where the $n$th term is given by $a_n = (-1)^{n+1} \\cdot n$.\n\n2. **Pairing terms for summation**: We observe that the sequence alternates in sign. Pairing the terms, we have:\n ... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3} - 3}{13} \\times 100\\%"
} | {
"split": "train",
"index": 35833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the probability of Team A winning a single game is $\\frac{2}{3}$, calculate the probability that Team A will win in a \"best of three\" format, where the first team to win two games wins the match and ends the competition. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{27}"
} | {
"split": "train",
"index": 35834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a permutation $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ of $\\{1,2,3,4,5\\}$. We say the tuple $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ is flawless if for all $1 \\leq i<j<k \\leq 5$, the sequence $\\left(a_{i}, a_{j}, a_{k}\\right)$ is not an arithm... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35835,
"solution": "We do casework on the position of 3. - If $a_{1}=3$, then the condition is that 4 must appear after 5 and 2 must appear after 1. It is easy to check there are six ways to do this. - If $a_{2}=3$, then there are no solutions; since there must be an index $i \\geq 3$... |
deepscaler | [
{
"role": "user",
"content": "Square \\( ABCD \\) has a side length of 12 inches. A segment \\( AE \\) is drawn where \\( E \\) is on side \\( DC \\) and \\( DE \\) is 5 inches long. The perpendicular bisector of \\( AE \\) intersects \\( AE, AD, \\) and \\( BC \\) at points \\( M, P, \\) and \\( Q \\) resp... | math | {
"style": "rule",
"ground_truth": "5:19"
} | {
"split": "train",
"index": 35836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The U.S. produces about 5.5 million tons of apples each year. Of the total, $20\\%$ is mixed with other products, with $50\\%$ of the remainder used for apple juice and the other $50\\%$ sold fresh. How many million tons of apples are used for apple juice? Express your answer a... | math | {
"style": "rule",
"ground_truth": "2.2"
} | {
"split": "train",
"index": 35837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point moving in the positive direction of the $O x$ axis has the abscissa $x(t)=5(t+1)^{2}+\\frac{a}{(t+1)^{5}}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \\geqslant 24$ for all $t \\geqslant 0$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^7}"
} | {
"split": "train",
"index": 35838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the lengths of the three sides of $\\triangle ABC$ form an arithmetic sequence with a common difference of 2, and the sine of the largest angle is $\\frac{\\sqrt{3}}{2}$, the sine of the smallest angle of this triangle is \\_\\_\\_\\_\\_\\_. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{14}"
} | {
"split": "train",
"index": 35839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a>0$ and $b>0,$ a new operation $\\nabla$ is defined as follows: $$a \\nabla b = \\dfrac{a + b}{1 + ab}.$$For example, $$3 \\nabla 6 = \\frac{3 + 6}{1 + 3 \\times 6} = \\frac{9}{19}.$$Calculate $2 \\nabla 5.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{11}"
} | {
"split": "train",
"index": 35840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be \"Selecting a part from the $100$ parts at random, and the part is qualified\", and event $B$ be \... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{8}"
} | {
"split": "train",
"index": 35841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30} \\cdot 5^6 \\cdot 3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35843,
"solution": "1. Let the three numbers be $x$, $y$, and $z$. Arrange them in ascending order, so $x \\leq y \\leq z$. Given that the median of the three numbers is $5$, we have $y = 5$.\n\n2. The mean of the three numbers is given as $10$ more than the least of the numbers and $... |
deepscaler | [
{
"role": "user",
"content": "Line $\\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\\sqrt n$ ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following numbers is less than $\\frac{1}{20}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{25}"
} | {
"split": "train",
"index": 35845,
"solution": "If $0<a<20$, then $\\frac{1}{a}>\\frac{1}{20}$. Therefore, $\\frac{1}{15}>\\frac{1}{20}$ and $\\frac{1}{10}>\\frac{1}{20}$. Also, $\\frac{1}{20}=0.05$ which is less than both 0.5 and 0.055. Lastly, $\\frac{1}{20}>\\frac{1}{25}$ since $0<20<25$. Therefore, $\\frac{1... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $AB = 5$ , $AC = 8$ , and $BC = 7$ . Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$ . Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\\overline{ID}$ and $\\overl... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA numerical sequence is defined by the conditions: \\( a_{1} = 1 \\), \\( a_{n+1} = a_{n} + \\left\\lfloor \\sqrt{a_{n}} \\right\\rfloor \\).\n\nHow many perfect squares are there among the first terms of this sequence that do not exceed \\( 1{,}000{,}000 \\)? Let's think ste... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, where $|\\overrightarrow{a}|= \\sqrt {2}$, $|\\overrightarrow{b}|=2$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\perp \\overrightarrow{a}$, calculate the angle between vector $\\overrightarrow{a}$ and $\\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 35848,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.