data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "553"
} | {
"split": "train",
"index": 2200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular $n$-gon is inscribed in a circle with radius $R$, and its area is equal to $3 R^{2}$. Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\text{4:10 P.M.}"
} | {
"split": "train",
"index": 2203,
"solution": "1. **Calculate the time worked before lunch:** Maria starts working at 7:25 A.M. and takes her lunch break at noon. The time interval from 7:25 A.M. to 12:00 P.M. is calculated as follows:\n \\[\n 12:00 - 7:25 = 4 \\text{ hours and } 35 \\text{ minutes}\n \\]\... |
deepscaler | [
{
"role": "user",
"content": "When the three-digit positive integer $N$ is divided by 10, 11, or 12, the remainder is 7. What is the sum of the digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 2204,
"solution": "When $N$ is divided by 10, 11, or 12, the remainder is 7. This means that $M=N-7$ is divisible by each of 10, 11, and 12. Since $M$ is divisible by each of 10, 11, and 12, then $M$ is divisible by the least common multiple of 10, 11, and 12. Since $10=2 \\times 5, 1... |
deepscaler | [
{
"role": "user",
"content": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 poin... | math | {
"style": "rule",
"ground_truth": "0.3072"
} | {
"split": "train",
"index": 2205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many polynomials of degree exactly 5 with real coefficients send the set \\{1,2,3,4,5,6\\} to a permutation of itself? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "714"
} | {
"split": "train",
"index": 2206,
"solution": "For every permutation $\\sigma$ of \\{1,2,3,4,5,6\\}, Lagrange Interpolation gives a polynomial of degree at most 5 with $p(x)=\\sigma(x)$ for every $x=1,2,3,4,5,6$. Additionally, this polynomial is unique: assume that there exist two polynomials $p, q$ of degree \\... |
deepscaler | [
{
"role": "user",
"content": "Suppose $(a_{1}, a_{2}, a_{3}, a_{4})$ is a 4-term sequence of real numbers satisfying the following two conditions: - $a_{3}=a_{2}+a_{1}$ and $a_{4}=a_{3}+a_{2}$ - there exist real numbers $a, b, c$ such that $a n^{2}+b n+c=\\cos \\left(a_{n}\\right)$ for all $n \\in\\{1,2,3,4... | math | {
"style": "rule",
"ground_truth": "-9+3\\sqrt{13}"
} | {
"split": "train",
"index": 2207,
"solution": "Let $f(n)=\\cos a_{n}$ and $m=1$. The second (\"quadratic interpolation\") condition on $f(m), f(m+1), f(m+2), f(m+3)$ is equivalent to having a vanishing third finite difference $f(m+3)-3 f(m+2)+3 f(m+1)-f(m)=0$. This is equivalent to $f(m+3)-f(m) =3[f(m+2)-f(m+1)]... |
deepscaler | [
{
"role": "user",
"content": "Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given ... | math | {
"style": "rule",
"ground_truth": "489"
} | {
"split": "train",
"index": 2208,
"solution": "Note that 4 points can NEVER form 2 triangles. Therefore, we just need to multiply the probability that the first three segments picked form a triangle by 4. We can pick any segment for the first choice, then only segments that share an endpoint with the first one, ... |
deepscaler | [
{
"role": "user",
"content": "Given points A(1, 2, 1), B(-2, $$\\frac{7}{2}$$, 4), and D(1, 1, 1), if $$\\vec{AP} = 2\\vec{PB}$$, then the value of |$$\\vec{PD}$$| is ______. 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": 2209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves?... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 2210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds t... | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"split": "train",
"index": 2211,
"solution": "We look at the number of cottages after each wave of peasants. Let $A_{n}$ be the number of cottages remaining after $8 n$ minutes. During each 8 minute interval, Trodgor burns a total of $480 / 5=96$ peasants and cottages. Trodgor first burns $A_{n}$ peasants and s... |
deepscaler | [
{
"role": "user",
"content": "Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 1... | math | {
"style": "rule",
"ground_truth": "26597"
} | {
"split": "train",
"index": 2212,
"solution": "The probability that all three cards drawn are face cards is $\\left(\\frac{3}{13}\\right)^{3}=\\frac{27}{2197}$. In that case, the sum is 30 and therefore a multiple of 10 . Otherwise, one of the cards is not a face card, so its point value $p$ is drawn uniformly f... |
deepscaler | [
{
"role": "user",
"content": "Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the larg... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 2213,
"solution": "1. **Understanding the Problem:**\n - We start with a square piece of paper. Let's denote the side length of the square as $s$.\n - The paper is folded in half vertically, meaning the new width of the folded paper is $\\frac{s}{2}$, and the height remains $s$.\n... |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $x^5-x^4-x^3+x^2+x$ is divided by $(x^2-4)(x+1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8x^2+13x+20"
} | {
"split": "train",
"index": 2214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "666"
} | {
"split": "train",
"index": 2215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2216,
"solution": "1. **Identify the Requirement**: Danica wants to arrange her model cars in rows with exactly 6 cars in each row. This means that the total number of cars must be a multiple of 6.\n\n2. **Current Number of Cars**: She currently has 23 model cars.\n\n3. **Finding the ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a} = \\begin{pmatrix} 4 \\\\ -3 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ 6 \\end{pmatrix}.$ Calculate the area of the triangle with vertices $\\mathbf{0},$ $\\mathbf{a},$ and $\\mathbf{b}$. Also, determine the vector $\\mathbf{a} + \\mathbf{b}$. ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lucia needs to save 35 files onto disks, each with a capacity of 1.6 MB. 5 of the files are 0.9 MB each, 10 of the files are 0.8 MB each, and the rest are 0.5 MB each. Calculate the smallest number of disks needed to store all 35 files. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{13}"
} | {
"split": "train",
"index": 2219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 16? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 2220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? Let's thin... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 2222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate 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": "218"
} | {
"split": "train",
"index": 2223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angles $C$ and $D$ are supplementary. If the measure of angle $C$ is $12$ times angle $D$, what is the measure of angle $C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "166.15"
} | {
"split": "train",
"index": 2224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 5 different letters from the word \"equation\" to arrange in a row, including the condition that the letters \"qu\" are together and in the same order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 2225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The image shows a grid consisting of 25 small equilateral triangles. How many rhombuses can be formed from two adjacent small triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 2227,
"solution": "To solve this problem, we need to understand the structure of the solid after the smaller cubes are removed from each corner of the larger cube.\n\n1. **Original Cube Characteristics**:\n - The original cube has a side length of $3$.\n - It has $8$ vertices (cor... |
deepscaler | [
{
"role": "user",
"content": "A right triangular prism \\( ABC-A_{1}B_{1}C_{1} \\) has 9 edges of equal length. Point \\( P \\) is the midpoint of \\( CC_{1} \\). The dihedral angle \\( B-A_{1}P-B_{1} \\) is \\( \\alpha \\). What is \\( \\sin \\alpha \\)? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{4}"
} | {
"split": "train",
"index": 2228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least integer value of $x$ for which $3|x| - 2 > 13$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 2229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\\textit{double}$ is a domino that has the same integer on both of its squares. What ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{11}"
} | {
"split": "train",
"index": 2230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assuming that the birth of a boy or a girl is equally likely, what is the probability that the three children in a family include at least one boy and one girl? 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{3}{4}"
} | {
"split": "train",
"index": 2231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x^5$ in the expansion of $(1+2x-3x^2)^6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-168"
} | {
"split": "train",
"index": 2232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with side lengths \\(AB = BC = 80\\) and \\(AC = 96\\).\n\nCircle \\(Q_1\\) is inscribed in the triangle \\(ABC\\). Circle \\(Q_2\\) is tangent to \\(Q_1\\) and to the sides \\(AB\\) and \\(BC\\). Circle \\(Q_3\\) is tangent to \\(Q_2\\) and also to t... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 2233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest prime that divides $$ 1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2. $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 2234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Most countries in the world use the Celsius temperature scale to forecast the weather, but countries like the United States and the United Kingdom still use the Fahrenheit temperature scale. A student looked up data and found the following information in the table:\n\n| Celsius... | math | {
"style": "rule",
"ground_truth": "-\\frac{160}{9}"
} | {
"split": "train",
"index": 2235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let equilateral triangle $ABC$ have side length $7$. There are three distinct triangles $AD_1E_1$, $AD_1E_2$, and $AD_2E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \\sqrt{21}$. Find $\\sum_{k=1}^3 (CE_k)^2$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "294"
} | {
"split": "train",
"index": 2236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers less than 1000000 which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\\max \\left(0,20-80\\left|1-\\frac{X}{Y}\\right|\\right)$ rounded to the nearest inte... | math | {
"style": "rule",
"ground_truth": "247548"
} | {
"split": "train",
"index": 2237,
"solution": "$\\mathrm{N}=1000000$ $\\mathrm{s}=[0] * \\mathrm{~N}$ ans $=0$ for i in range(1, N): if i <= s[i]: ans $+=1$ for $j$ in range(i + i, N, i): $s[j]+=$ i print(ans)"
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 2238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\[-2x^2 = \\frac{4x + 2}{x + 2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence \\(\\left\\{a_{n}\\right\\}\\), it is known that \\(a_{1}=1\\) and \\(a_{n+1}>a_{n}\\), and that \\(a_{n+1}^{2}+a_{n}^{2}+1=2\\left(a_{n+1}+a_{n}+2 a_{n+1} a_{n}\\right)\\). Find \\(\\lim \\limits_{n \\rightarrow \\infty} \\frac{S_{n}}{n a_{n}}\\). Let's think s... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 2240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 2241,
"solution": "1. **Identify the center and radius of the circle:**\n Given points $(0,0)$ and $(8,6)$ are endpoints of a diameter. The center of the circle, $(h,k)$, is the midpoint of the diameter. Using the midpoint formula:\n \\[\n h = \\frac{0+8}{2} = 4, \\quad k = \\fr... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ with $AB=AC,$ point $D$ lies strictly between $A$ and $C$ on side $\\overline{AC},$ and point $E$ lies strictly between $A$ and $B$ on side $\\overline{AB}$ such that $AE=ED=DB=BC.$ The degree measure of $\\angle ABC$ is $\\tfrac{m}{n},$ where $m$ and $n$ ar... | math | {
"style": "rule",
"ground_truth": "547"
} | {
"split": "train",
"index": 2242,
"solution": "Let $\\angle{BAC}$ be $x$ in degrees. $\\angle{ADE}=x$. By Exterior Angle Theorem on triangle $AED$, $\\angle{BED}=2x$. By Exterior Angle Theorem on triangle $ADB$, $\\angle{BDC}=3x$. This tells us $\\angle{BCA}=\\angle{ABC}=3x$ and $3x+3x+x=180$. Thus $x=\\frac{180... |
deepscaler | [
{
"role": "user",
"content": "In the following equation, each of the letters represents uniquely a different digit in base ten:\n\\[(YE) \\cdot (ME) = TTT\\]\nThe sum $E+M+T+Y$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 2243,
"solution": "1. **Understanding the Equation**: The equation given is \\((YE) \\cdot (ME) = TTT\\), where \\(YE\\) and \\(ME\\) are two-digit numbers and \\(TTT\\) is a three-digit number with all digits being \\(T\\). This can be rewritten as \\(111T\\), since \\(TTT\\) represe... |
deepscaler | [
{
"role": "user",
"content": "There are 3 male students and 2 female students. Calculate the total number of different arrangement methods under the following different requirements: \n(1) All are arranged in a row where \"A\" can only be in the middle or on the sides. There are \\_\\_\\_\\_\\_\\_ ways to... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 2244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 2245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that all four of the numbers \\[3 - 2\\sqrt{2}, \\; -3-2\\sqrt{2}, \\; 1+\\sqrt{7}, \\; 1-\\sqrt{7}\\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ the point of intersection of the lines $x + 2y = 7t + 3$ and $x - y = 2t - 2$ is plotted. All the plotted points lie on a line. Find the slope of this line. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"split": "train",
"index": 2247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 2248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $x^{2}-4y^{2}=4$ with left and right foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects the left branch at points $A$ and $B$. If $|AB|=3$, find the perimeter of $\\triangle AF_{2}B$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number $56439.2071$, the value of the place occupied by the digit 6 is how many times as great as the value of the place occupied by the digit 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10,000"
} | {
"split": "train",
"index": 2250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a_1, a_2, a_3, \\dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \\]*Proposed by Evan Chen* Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 2251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two brothers had tickets to a stadium located 10 km from their home. Initially, they planned to walk to the stadium. However, they changed their plan and decided to use a bicycle. They agreed that one would start on the bicycle and the other would walk simultaneously. After cov... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 2252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers $y$ satisfy $5678y + 123 \\equiv 890 \\pmod{29}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "310"
} | {
"split": "train",
"index": 2253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it sa... | math | {
"style": "rule",
"ground_truth": "2^{mn} - 1"
} | {
"split": "train",
"index": 2254,
"solution": "We claim that any configuration of $0$ 's produces a distinct garden. To verify this claim, we show that, for any cell that is nonzero, the value of that cell is its distance away from the nearest zero, where distance means the shortest chain of adjacent cells conn... |
deepscaler | [
{
"role": "user",
"content": "A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage diff... | math | {
"style": "rule",
"ground_truth": "2.5\\%"
} | {
"split": "train",
"index": 2255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Many calculators have a reciprocal key $\\boxed{\\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\\boxed{00004}$ and the $\\boxed{\\frac{1}{x}}$ key is depressed, then the display becomes $\\boxed{000.25}$. If $\\b... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2256,
"solution": "1. **Define the function**: Let $f(x) = \\frac{1}{x}$ represent the operation of the reciprocal key on the calculator.\n\n2. **Apply the function**: We need to determine how many times we must apply $f(x)$ to return to the original number. Start by applying $f(x)$ t... |
deepscaler | [
{
"role": "user",
"content": "If $x,y>0$, $\\log_y(x)+\\log_x(y)=\\frac{10}{3}$ and $xy=144$, then $\\frac{x+y}{2}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13\\sqrt{3}"
} | {
"split": "train",
"index": 2257,
"solution": "1. **Rewrite the logarithmic equation**: Given $\\log_y(x) + \\log_x(y) = \\frac{10}{3}$, we can use the change of base formula to rewrite this as:\n \\[\n \\frac{\\log x}{\\log y} + \\frac{\\log y}{\\log x} = \\frac{10}{3}\n \\]\n Let $u = \\frac{\\log x}{\... |
deepscaler | [
{
"role": "user",
"content": "Ben \"One Hunna Dolla\" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$. Let... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2258,
"solution": "Let $[K I R]=[R I T]=a$ and $[K E R]=[T E R]=b$. We will relate all areas to $a$ and $b$. First, $$ [R A I N]=[R A I]+[I N R]=\\frac{1}{2} a+\\frac{1}{2} a=a $$ Next, we break up $[M A K E]=[M A D]+[A K D]+[D E M]$. We have $$ \\begin{aligned} & {[M A D]=\\frac{A D ... |
deepscaler | [
{
"role": "user",
"content": "Allen and Ben are painting a fence. The ratio of the amount of work Allen does to the amount of work Ben does is $3:5$. If the fence requires a total of $240$ square feet to be painted, how many square feet does Ben paint? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 2259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct prime factors does the sum of the positive divisors of $400$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ and $g(x)$ be nonzero polynomials such that\n\\[f(g(x)) = f(x) g(x).\\]If $g(2) = 37,$ find $g(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 33x - 33"
} | {
"split": "train",
"index": 2261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \\ldots, a_{k} ; b_{1}, b_{2}, \\ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\\cdots+a_{k}(b_{1}+b_{2}+\\cdots+b_{k})=7$$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2262,
"solution": "Let $k, a_{1}, \\ldots, a_{k}, b_{1}, \\ldots, b_{k}$ be a solution. Then $b_{1}, b_{1}+b_{2}, \\ldots, b_{1}+\\cdots+b_{k}$ is just some increasing sequence of positive integers. Considering the $a_{i}$ as multiplicities, the $a_{i}$ 's and $b_{i}$ 's uniquely dete... |
deepscaler | [
{
"role": "user",
"content": "In the expansion of \\((x + y + z)^8\\), determine the sum of the coefficients of all terms of the form \\(x^2 y^a z^b\\) (\\(a, b \\in \\mathbf{N}\\)). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1792"
} | {
"split": "train",
"index": 2263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rhombus $ABCD$, $\\angle BAD=60^{\\circ}$, $\\overrightarrow{DE}=\\overrightarrow{EC}$, $AB=2$, calculate $\\overrightarrow{AE}\\cdot\\overrightarrow{DB}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The radius of the inscribed circle is 6 cm. What is the number of centimeters in the length of $\\overline{AB}$? Express your answer in simplest radical form. [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\ndraw((sqrt(3),0)--origin--(0,1)--cycle... | math | {
"style": "rule",
"ground_truth": "AB=12+12\\sqrt{3}"
} | {
"split": "train",
"index": 2265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord of length √3 divides a circle of radius 1 into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of a rectangle that can be drawn in R? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 2266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Horses $X, Y$ and $Z$ are entered in a three-horse race in which ties are not possible. The odds against $X$ winning are $3:1$ and the odds against $Y$ winning are $2:3$, what are the odds against $Z$ winning? (By \"odds against $H$ winning are $p:q$\" we mean the probability o... | math | {
"style": "rule",
"ground_truth": "17:3"
} | {
"split": "train",
"index": 2267,
"solution": "1. **Understanding the odds against $X$ and $Y$:**\n - The odds against $X$ winning are given as $3:1$. This means the probability of $X$ winning is $\\frac{1}{3+1} = \\frac{1}{4}$.\n - The odds against $Y$ winning are given as $2:3$. This means the probability ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{\\cos \\alpha + \\sin \\alpha}{\\cos \\alpha - \\sin \\alpha} = 2$, find the value of $\\frac{1 + \\sin 4\\alpha - \\cos 4\\alpha}{1 + \\sin 4\\alpha + \\cos 4\\alpha}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 2268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harry has 3 sisters and 5 brothers. His sister Harriet has $\\text{S}$ sisters and $\\text{B}$ brothers. What is the product of $\\text{S}$ and $\\text{B}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2269,
"solution": "1. **Identify the family structure**: Harry has 3 sisters and 5 brothers. This means in total, there are 4 girls (including Harriet) and 6 boys (including Harry) in the family.\n\n2. **Determine Harriet's siblings**: Since Harriet is one of the sisters, she has 3 si... |
deepscaler | [
{
"role": "user",
"content": "Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area... | math | {
"style": "rule",
"ground_truth": "\\frac{500}{3}"
} | {
"split": "train",
"index": 2270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fourth term of a geometric sequence is 512, and the 9th term is 8. Determine the positive, real value for the 6th term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 2271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many boys are on the chess team? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Does there exist a point \\( M \\) on the parabola \\( y^{2} = 2px \\) such that the ratio of the distance from point \\( M \\) to the vertex and the distance from point \\( M \\) to the focus is maximized? If such a point \\( M \\) exists, find its coordinates and the maximum ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{3}}"
} | {
"split": "train",
"index": 2273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $y = -16t^2 + 80t$ describes the height (in feet) of a projectile launched from the ground at 80 feet per second. At what $t$ will the projectile reach 36 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth. Let's think ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 2275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jacob flipped a fair coin five times. In the first three flips, the coin came up heads exactly twice. In the last three flips, the coin also came up heads exactly twice. What is the probability that the third flip was heads? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 2276,
"solution": "How many sequences of five flips satisfy the conditions, and have the third flip be heads? We have __H_-, so exactly one of the first two flips is heads, and exactly one of the last two flips is heads. This gives $2 \\times 2=4$ possibilities. How many sequences of ... |
deepscaler | [
{
"role": "user",
"content": "Two distinct positive integers $a$ and $b$ are factors of 48. If $a\\cdot b$ is not a factor of 48, what is the smallest possible value of $a\\cdot b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a blackboard, the number 123456789 is written. Select two adjacent digits from this number, and if neither of them is 0, subtract 1 from each and swap their positions. For example: \\( 123456789 \\rightarrow 123436789 \\rightarrow \\cdots \\). After performing this operation... | math | {
"style": "rule",
"ground_truth": "101010101"
} | {
"split": "train",
"index": 2278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What integer should 999,999,999 be multiplied by to get a number consisting of only ones? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111111111"
} | {
"split": "train",
"index": 2279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real solutions to $x^{4}+(2-x)^{4}=34$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 \\pm \\sqrt{2}"
} | {
"split": "train",
"index": 2280,
"solution": "Let $y=2-x$, so $x+y=2$ and $x^{4}+y^{4}=34$. We know $$(x+y)^{4}=x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}=x^{4}+y^{4}+2 x y(2 x^{2}+2 y^{2}+3 x y) .$$ Moreover, $x^{2}+y^{2}=(x+y)^{2}-2 x y$, so the preceding equation becomes $2^{4}=34+2 x y(2. 2^{2}-x y)$, or... |
deepscaler | [
{
"role": "user",
"content": "Find an integer $n$, where $100 \\leq n \\leq 1997$, such that \n\\[ \\frac{2^n+2}{n} \\] \nis also an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "946"
} | {
"split": "train",
"index": 2281,
"solution": "To find an integer \\( n \\) such that \\( 100 \\leq n \\leq 1997 \\) and \n\\[\n\\frac{2^n + 2}{n}\n\\] \nis an integer, we need to ensure that \\( n \\mid (2^n + 2) \\). This means that the expression can be rewritten using divisibility:\n\n\\[\n2^n + 2 \\equiv 0 ... |
deepscaler | [
{
"role": "user",
"content": "Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 2282,
"solution": "Since square $P Q R S$ has an area of 900, then its side length is $\\sqrt{900}=30$. Thus, $P Q=P S=30$. Since $M$ and $N$ are the midpoints of $P Q$ and $P S$, respectively, then $P N=P M=\\frac{1}{2}(30)=15$. Since $P Q R S$ is a square, then the angle at $P$ is $... |
deepscaler | [
{
"role": "user",
"content": "If $C=\\frac{5}{9}(F-32)$, what is $F$ when $C=20$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 2283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\n\t Let $PROBLEMZ$ be a regular octagon inscribed in a circle of unit radius. Diagonals $MR$ , $OZ$ meet at $I$ . Compute $LI$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of apples to bananas in a box is $3: 2$. What total number of apples and bananas in the box cannot be equal to? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2285,
"solution": "Since the ratio of apples to bananas is $3: 2$, then we can let the numbers of apples and bananas equal $3n$ and $2n$, respectively, for some positive integer $n$. Therefore, the total number of apples and bananas is $3n + 2n = 5n$, which is a multiple of 5. Of the ... |
deepscaler | [
{
"role": "user",
"content": "A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid towar... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 2286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Translate the graph of the function $f(x)=\\sin(2x+\\varphi)$ ($|\\varphi| < \\frac{\\pi}{2}$) to the left by $\\frac{\\pi}{6}$ units. If the resulting graph is symmetric about the origin, determine the minimum value of the function $f(x)$ on the interval $\\left[0, \\frac{\\pi... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 2287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\(ABCD\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\), \\(AB = 73\\) and \\(BC = 88\\). The angle bisector of \\(\\angle ABC\\) intersects segment \\(AD\\) at point \\(E\\) and ray \\(CD\\) at point \\(F\\). Find the length of segment \\(EF\\).\n\n1. 9\n... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an integer \\( n \\geq 2 \\), let \\( f(n) \\) be the second largest positive divisor of \\( n \\). For example, \\( f(12)=6 \\) and \\( f(13)=1 \\). Determine the largest positive integer \\( n \\) such that \\( f(n)=35 \\). Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 2289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x) = 12x + 5$. Find the sum of all $x$ that satisfy the equation $g^{-1}(x) = g((3x)^{-1})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 2290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac19"
} | {
"split": "train",
"index": 2291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Westeros Empire, there were 1000 cities and 2017 roads (each road connects some two cities). From each city, it was possible to travel to any other city. One day, an evil wizard cursed $N$ roads, making them impassable. As a result, 7 kingdoms formed, such that within ea... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 2292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "2 * (2018!)^2"
} | {
"split": "train",
"index": 2293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A kite is inscribed in a circle with center $O$ and radius $60$ . The diagonals of the kite meet at a point $P$ , and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\\sqrt{b}$ , where $a$ and $b$ are positive integers and $b... | math | {
"style": "rule",
"ground_truth": "239"
} | {
"split": "train",
"index": 2294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $|\\sqrt{3}-2|+(\\pi -\\sqrt{10})^{0}-\\sqrt{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3-3\\sqrt{3}"
} | {
"split": "train",
"index": 2295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one bu... | math | {
"style": "rule",
"ground_truth": "770"
} | {
"split": "train",
"index": 2296,
"solution": "Currently there are ${10 \\choose 5}$ possible combinations. With any integer $x$ from $1$ to $9$, the number of ways to choose a set of $x$ buttons is $\\sum^{9}_{k=1}{10 \\choose k}$. Now we can use the identity $\\sum^{n}_{k=0}{n \\choose k}=2^{n}$. So the number... |
deepscaler | [
{
"role": "user",
"content": "In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 2297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos^2x+\\cos^2\\left(x-\\frac{\\pi}{3}\\right)-1$, where $x\\in \\mathbb{R}$,\n\n$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;\n\n$(2)$ The function $f(x)$ is translated to the right by $\\frac{\\pi}{3}$ u... | math | {
"style": "rule",
"ground_truth": "- \\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 2298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$, \nare on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ \nhave equal areas, then that area is Let's... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2299,
"solution": "1. **Identify Key Relationships**: Given that $AD = 2$, $DB = 3$, and $[ABE] = [DBEF]$, where $[X]$ denotes the area of figure $X$.\n\n2. **Area of $\\triangle ABC$**: The area of $\\triangle ABC$ is given as $10$.\n\n3. **Ratio of Segments on $AB$**: Since $AD = 2$... |
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