data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\\,836$ is divisible by $9$ but $19\\,825$ is not.\n\nIf $D\\,767\\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible v... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest possible value of \\(x\\) in the simplified form \\(x=\\frac{a+b\\sqrt{c}}{d}\\) if \\(\\frac{7x}{8}-1=\\frac{4}{x}\\), where \\(a, b, c,\\) and \\(d\\) are integers. What is \\(\\frac{acd}{b}\\)? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-105"
} | {
"split": "train",
"index": 22350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$[x-(y-z)] - [(x-y) - z] = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2z"
} | {
"split": "train",
"index": 22351,
"solution": "We start by simplifying the given expression step by step:\n\n1. **Expand and simplify the inner parentheses:**\n \\[\n [x-(y-z)] - [(x-y) - z] = [x-y+z] - [x-y+z]\n \\]\n Here, we used the property that subtracting a negative is equivalent to adding, i.e.,... |
deepscaler | [
{
"role": "user",
"content": "What is $0.1 \\div 0.004$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the proper divisors of $600$? Additionally, determine if $600$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 22354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, $\\overline{AD}$ is perpendicular to $\\overline{DC}$,\n$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\\overline{DC}$, and $\\overline{BE}$ is parallel to $\\overline{AD}$. Find the area of $\\triangle BEC$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 22355,
"solution": "1. **Identify the Shape and Properties**: Given that $ABCD$ is a trapezoid with $\\overline{AD} \\perpendicular \\overline{DC}$, and $AD = AB = 3$, $DC = 6$. Since $\\overline{BE} \\parallel \\overline{AD}$ and both $AD$ and $BE$ are perpendicular to $DC$, $ABED$ f... |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest possible value of \\(\\sum_{i=1}^{10} \\cos(3x_i)\\) for real numbers $x_1,x_2,\\dots,x_{10}$ satisfying \\(\\sum_{i=1}^{10} \\cos(x_i) = 0\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{480}{49}"
} | {
"split": "train",
"index": 22356,
"solution": "The maximum value is $480/49$. Since $\\cos(3x_i) = 4 \\cos(x_i)^3 - 3 \\cos(x_i)$, it is equivalent to maximize $4 \\sum_{i=1}^{10} y_i^3$ for $y_1,\\dots,y_{10} \\in [-1,1]$ with $\\sum_{i=1}^{10} y_i = 0$; note that this domain is compact, so the maximum value i... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest value of $b$ such that $-b^2+7b-10 \\ge 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 22357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(\\alpha) = \\frac{\\sin(\\pi - \\alpha)\\cos(\\pi + \\alpha)\\sin(-\\alpha + \\frac{3\\pi}{2})}{\\cos(-\\alpha)\\cos(\\alpha + \\frac{\\pi}{2})}$.\n\n$(1)$ Simplify $f(\\alpha)$; \n\n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos(\\alpha - \\frac{3\\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}}{5}"
} | {
"split": "train",
"index": 22358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price.... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=( \\frac {1}{3})^{x}$, the sum of the first $n$ terms of the geometric sequence $\\{a\\_n\\}$ is $f(n)-c$, and the first term of the sequence $\\{b\\_n\\}_{b\\_n > 0}$ is $c$. The sum of the first $n$ terms, $S\\_n$, satisfies $S\\_n-S_{n-1}= \\sqrt {S\... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 22360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be inversely proportional to $y$. If $x = 4$ when $y = 2$, find the value of $x$ when $y = -3$ and when $y = 6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 22361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{x^2}{y - 1} + \\frac{y^2}{x - 1}\\]for real numbers $x > 1$ and $y > 1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 22362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) \\) be a function defined on \\(\\mathbf{R}\\). Given that \\( f(x) + x^{2} \\) is an odd function and \\( f(x) + 2^{x} \\) is an even function, find the value of \\( f(1) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{4}"
} | {
"split": "train",
"index": 22363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different prime numbers are factors of $N$ if\n\n$\\log_2 ( \\log_3 ( \\log_5 (\\log_ 7 N))) = 11?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 22364,
"solution": "1. **Rewriting the given logarithmic equation:** \n Start with the equation:\n \\[\n \\log_2 ( \\log_3 ( \\log_5 (\\log_ 7 N))) = 11\n \\]\n Rewriting this equation in terms of exponents, we start from the innermost logarithm and work our way outwards:\n... |
deepscaler | [
{
"role": "user",
"content": "$S=\\{1,2, \\ldots, 6\\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \\subseteq S$ and $A \\cap B=\\phi$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 22365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.\n\nInitially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 22366,
"solution": "\nLet us consider the problem where Amy and Ben take turns placing stones on a 20x20 grid consisting of sites \\((x, y)\\) where \\(x\\) and \\(y\\) are integers between 1 and 20 inclusive. Amy's condition for placing a red stone is that the distance between any tw... |
deepscaler | [
{
"role": "user",
"content": "A line with slope 3 intersects a line with slope 5 at the point $(10,15)$. What is the distance between the $x$-intercepts of these two lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 22367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cheesecake is shaped like a $3 \\times 3 \\times 3$ cube and is decorated such that only the top and three of the vertical faces are covered with icing. The cake is cut into $27$ smaller cubes, each $1 \\times 1 \\times 1$ inch. Calculate the number of the smaller cubes that ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(n) = n^2 \\cos(n\\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \\cdots + a_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 22369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a box, there are 6 cards labeled with numbers 1, 2, ..., 6. Now, one card is randomly drawn from the box, and its number is denoted as $a$. After adjusting the cards in the box to keep only those with numbers greater than $a$, a second card is drawn from the box. The probabi... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{45}"
} | {
"split": "train",
"index": 22370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle $C$ is defined by the equation $x^2 + y^2 = 1$. After the transformation $\\begin{cases} x' = 2x \\\\ y' = \\sqrt{2}y \\end{cases}$, we obtain the curve $C_1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-a... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{2}}{5}"
} | {
"split": "train",
"index": 22371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \\leq y \\leq 8$. How many such integer points $(x, y)$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pedestrian departed from point \\( A \\) to point \\( B \\). After walking 8 km, a second pedestrian left point \\( A \\) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \\( B \\), and both pedestrians arr... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 22373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right triangle \\(ABC\\) with an acute angle of \\(30^\\circ\\), an altitude \\(CD\\) is drawn from the right angle vertex \\(C\\). Find the distance between the centers of the inscribed circles of triangles \\(ACD\\) and \\(BCD\\), if the shorter leg of triangle \\(ABC\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}-1}{\\sqrt{2}}"
} | {
"split": "train",
"index": 22374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $AC=2$, $D$ is the midpoint of $AB$, $CD=\\frac{1}{2}BC=\\sqrt{7}$, $P$ is a point on $CD$, and $\\overrightarrow{AP}=m\\overrightarrow{AC}+\\frac{1}{3}\\overrightarrow{AB}$. Find $|\\overrightarrow{AP}|$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{13}}{3}"
} | {
"split": "train",
"index": 22375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum point of the function $f(x)=x+2\\cos x$ on the interval $[0, \\pi]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"split": "train",
"index": 22376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a pentagonal prism with seven faces, fifteen edges, and ten vertices. One of its faces will be used as the base for a new pyramid. Calculate the maximum value of the sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid). Let... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 22377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express this sum as a common fraction: $.\\overline{8} + .\\overline{2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{9}"
} | {
"split": "train",
"index": 22378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is . Let's t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 22379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 700, inclusive, contain the digit 5 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "214"
} | {
"split": "train",
"index": 22380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1936"
} | {
"split": "train",
"index": 22381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The midpoints of the sides of a triangle with area $T$ are joined to form a triangle with area $M$. What is the ratio of $M$ to $T$? 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{1}{4}"
} | {
"split": "train",
"index": 22382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{10}"
} | {
"split": "train",
"index": 22383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB=3$, $AC=2$, $\\angle BAC=60^{\\circ}$, $D$ is the midpoint of $BC$, $\\cos \\angle BAD=$ __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{19}}{19}"
} | {
"split": "train",
"index": 22384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, \\sqrt{39}) \\cup (\\sqrt{89}, 13)"
} | {
"split": "train",
"index": 22385,
"solution": "\nTo determine the real values of \\( x \\) such that the triangle with sides 5, 8, and \\( x \\) is obtuse, we need to apply the properties of an obtuse triangle. In an obtuse triangle, the square of the longest side is greater than the sum of the squares of the o... |
deepscaler | [
{
"role": "user",
"content": "Three different numbers are chosen from the set $\\{-4, -3, -1, 3, 5, 8\\}$ and multiplied. Find the largest possible positive product. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 22386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers \\( x \\), \\( \\lfloor x \\rfloor \\) denotes the greatest integer less than or equal to \\( x \\). Find the largest positive integer \\( n \\) such that the following equation holds:\n\\[\n\\lfloor \\log_2 1 \\rfloor + \\lfloor \\log_2 2 \\rfloor + \\lfloor \... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 22387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle where $\\angle$ **B=55** and $\\angle$ **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**. $\\angle$ **FDO** = ? Le... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\sin (\\omega x+\\dfrac{\\pi }{3})$ ($\\omega > 0$), $f(\\dfrac{\\pi }{6})=f(\\dfrac{\\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\\dfrac{\\pi }{6},\\dfrac{\\pi }{3})$, find the value of $\\omega$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{3}"
} | {
"split": "train",
"index": 22389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \\leqslant a \\leqslant 4$ and $a \\in R)$ units of the nutrient solution are released, its concentration $y (\\text{g}/\\text{L})$ changes over time $x (\\text{days})$ according to the func... | math | {
"style": "rule",
"ground_truth": "24-16\\sqrt{2}"
} | {
"split": "train",
"index": 22390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a bin contains 10 kg of peanuts, 2 kg of peanuts are removed and 2 kg of raisins are added and thoroughly mixed in, and then 2 kg of this mixture are removed and 2 kg of raisins are added and thoroughly mixed in again, determine the ratio of the mass of peanuts to th... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 22391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The math teacher wants to arrange 6 multiple-choice questions into a large practice exam paper, requiring that the two easy questions, $A$ and $B$, must be adjacent, and the two difficult questions, $E$ and $F$, cannot be adjacent. There are a total of $\\_\\_\\_\\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 22392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real function $f$ has the property that, whenever $a, b, n$ are positive integers such that $a+b=2^{n}$, the equation $f(a)+f(b)=n^{2}$ holds. What is $f(2002)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 22393,
"solution": "We know $f(a)=n^{2}-f\\left(2^{n}-a\\right)$ for any $a$, $n$ with $2^{n}>a$; repeated application gives $$f(2002)=11^{2}-f(46)=11^{2}-\\left(6^{2}-f(18)\\right)=11^{2}-\\left(6^{2}-\\left(5^{2}-f(14)\\right)\\right) =11^{2}-\\left(6^{2}-\\left(5^{2}-\\left(4^{2}-f... |
deepscaler | [
{
"role": "user",
"content": "Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 22394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a number is divided by 3, the result is 50 more than if the number had been divided by 4. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 22395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 22396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, a square is divided into 4 identical rectangles, each of which has a perimeter of 20 centimeters. What is the area of this square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 22397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. ... | math | {
"style": "rule",
"ground_truth": "18 minutes"
} | {
"split": "train",
"index": 22398,
"solution": "1. **Calculate Dave's walking speed**: \n Dave walks at a rate of $90$ steps per minute, with each step being $75$ cm long. Therefore, his walking speed is:\n \\[\n 90 \\text{ steps/min} \\times 75 \\text{ cm/step} = 6750 \\text{ cm/min}\n \\]\n\n2. **Calc... |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime divisor of $25^2+72^2$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"split": "train",
"index": 22399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\\frac{\\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\\angle F_{1}PF_{2}=60^{\\circ}$, then $e=$ ______. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 22400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,\\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005$? Let's... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22401,
"solution": "Suppose that the $n$th term of the sequence $S_k$ is $2005$. Then $1+(n-1)k=2005$ so $k(n-1)=2004=2^2\\cdot 3\\cdot 167$. The ordered pairs $(k,n-1)$ of positive integers that satisfy the last equation are $(1,2004)$,$(2,1002)$, $(3,668)$, $(4,501)$, $(6,334)$, $(1... |
deepscaler | [
{
"role": "user",
"content": "For all complex numbers $z$, let \\[f(z) = \\left\\{\n\\begin{array}{cl}\nz^{2}&\\text{ if }z\\text{ is not real}, \\\\\n-z^2 &\\text{ if }z\\text{ is real}.\n\\end{array}\n\\right.\\]Find $f(f(f(f(1+i))))$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "-256"
} | {
"split": "train",
"index": 22402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy]\nunitsize(3mm); defaultpen(linewidth(0.8pt));\npath p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);\npath p2=(0,1)--(1,1)--(1,0);\... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 22403,
"solution": "1. **Identify the basic unit of tiling**: The tiling pattern consists of a large square divided into 9 smaller squares, each with side length $a$. The large square thus has a side length of $3a$.\n\n2. **Calculate the area of the large square**: The area of the lar... |
deepscaler | [
{
"role": "user",
"content": "Given that a bridge is $800$ meters long, a train passes over it and it takes $1$ minute for the train to completely pass through the bridge. The train is entirely on the bridge for $40$ seconds. Calculate the speed of the train. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22405,
"solution": "We label the players as A, B, C, D, and E. The total number of matches played will be equal to the number of pairs of players that can be formed times the number of matches that each pair plays. The possible pairs of players are AB, AC, AD, AE, BC, BD, BE, CD, CE, ... |
deepscaler | [
{
"role": "user",
"content": "Given 6 digits: \\(0, 1, 2, 3, 4, 5\\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1769580"
} | {
"split": "train",
"index": 22406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pair of fair 8-sided dice are rolled, yielding numbers $a$ and $b$. Determine the probability that both digits $a$ and $b$ as well as the two-digit number $ab$ formed by them are all divisible by 4. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 22407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the country of Taxonia, each person pays as many thousandths of their salary in taxes as the number of tugriks that constitutes their salary. What salary is most advantageous to have?\n\n(Salary is measured in a positive number of tugriks, not necessarily an integer.) Let's ... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 22408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 22409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of a positive number and its square is 156. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0\\le x_3 \\le x_2 \\le x_1\\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\\frac{1}{4},$ find $x_1.$ 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": 22411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( P \\) is the product of \\( 3,659,893,456,789,325,678 \\) and \\( 342,973,489,379,256 \\), find the number of digits of \\( P \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 22412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a number line, point $P$ is at 3 and $V$ is at 33. The number line between 3 and 33 is divided into six equal parts by the points $Q, R, S, T, U$. What is the sum of the lengths of $PS$ and $TV$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22413,
"solution": "The segment of the number line between 3 and 33 has length $33 - 3 = 30$. Since this segment is divided into six equal parts, then each part has length $30 \\div 6 = 5$. The segment $PS$ is made up of 3 of these equal parts, and so has length $3 \\times 5 = 15$. Th... |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the first $20$ rows of Pascal's Triangle (from the $0$th row to the $19$th row)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 22414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\\sqrt{3}$ and $8\\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22415,
"solution": "To solve this problem, we first need to determine the side lengths of the equilateral triangles and the square based on the given areas.\n\n1. **Calculate the side length of the equilateral triangles:**\n - For an equilateral triangle, the area $A$ is given by $A... |
deepscaler | [
{
"role": "user",
"content": "If two numbers will be randomly chosen without replacement from $\\{3, 4, 5, 6\\}$, what is the probability that their product will be a multiple of 9? 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{1}{6}"
} | {
"split": "train",
"index": 22416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My friend Ana likes numbers that are divisible by 8. How many different pairs of last two digits are possible in numbers that Ana likes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 22417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 22418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 21 different pairs of digits (a, b) such that $\\overline{5a68} \\times \\overline{865b}$ is divisible by 824. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 22419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ is it true that $\\sqrt{n} \\le \\sqrt{4n - 6} < \\sqrt{2n + 5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line initially 1 inch long grows according to a new law, where the first term is the initial length and an additional linear term is introduced.\n\\[1+\\frac{1}{3}\\sqrt{3}+2\\cdot\\frac{1}{9}+3\\cdot\\frac{1}{27}\\sqrt{3}+4\\cdot\\frac{1}{81}+\\cdots\\]\nDetermine the limit ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(4+\\sqrt{3})"
} | {
"split": "train",
"index": 22421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^{x}=16$, what is the value of $2^{x+3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 22422,
"solution": "Since $2^{x}=16$, then $2^{x+3}=2^{3} 2^{x}=8(16)=128$."
} |
deepscaler | [
{
"role": "user",
"content": "For dessert, Melinda eats a spherical scoop of ice cream with diameter 2 inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x=n, y=n$, and $z=n$ for every integer $n... | math | {
"style": "rule",
"ground_truth": "7+\\frac{13 \\pi}{3}"
} | {
"split": "train",
"index": 22423,
"solution": "Note that if we consider the division of \\mathbb{R}^{3}$ into unit cubes by the given planes, we only need to compute the sum of the probabilities that the ice cream scoop intersects each cube. There are three types of cubes that can be intersected: - The cube $0 ... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \\log_b 9^6$. Find the value of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 22424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "09:38"
} | {
"split": "train",
"index": 22425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$\n Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 22426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the common ratio of the geometric sequence $\\{a_n\\}$ be $q$, and the sum of the first $n$ terms be $S_n$. If $S_{n+1}$, $S_n$, and $S_{n+2}$ form an arithmetic sequence, find the value of $q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 22427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the residue of $182\\cdot 12 - 15\\cdot 7 + 3\\pmod{14}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a store, when a length or a width is reported as $x$ inches that means it is at least $x - 0.5$ inches and at most $x + 0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rect... | math | {
"style": "rule",
"ground_truth": "3.75"
} | {
"split": "train",
"index": 22429,
"solution": "1. **Understanding the Problem:**\n The problem states that when a dimension is reported as $x$ inches, the actual dimension is between $x - 0.5$ inches and $x + 0.5$ inches. For a rectangle reported to have dimensions $2$ inches by $3$ inches, the actual dimensi... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=3$, $BC=4$, and $\\angle B=60^{\\circ}$. Find the length of $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {13}"
} | {
"split": "train",
"index": 22430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains four balls of the same shape and size, numbered $1$, $2$, $3$, $4$ respectively.\n$(1)$ Randomly draw two balls from the bag, and calculate the probability that the sum of the numbers on the drawn balls is even.\n$(2)$ First, randomly draw a ball from the bag, an... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 22431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a quadrilateral piece of paper, there are a total of 10 points, and if the vertices of the quadrilateral are included, there are a total of 14 points. It is known that any three of these points are not collinear. According to the following rules, cut this piece of paper into... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 22432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\), not exceeding 333300, such that \\( k^{2} - 2k \\) is exactly divisible by 303. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4400"
} | {
"split": "train",
"index": 22433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two v... | math | {
"style": "rule",
"ground_truth": "199776"
} | {
"split": "train",
"index": 22434,
"solution": "Using the given rules, the words that are 1 letter long are A, B, C, D, E. Using the given rules, the words that are 2 letters long are AB, AC, AD, BA, BC, BD, BE, CA, CB, CD, CE, DA, DB, DC, DE, EB, EC, ED. Let $v_{n}$ be the number of words that are $n$ letters l... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 10 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 22435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $$\\left\\lfloor\\frac{2005^{3}}{2003 \\cdot 2004}-\\frac{2003^{3}}{2004 \\cdot 2005}\\right\\rfloor$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 22436,
"solution": "Let $x=2004$. Then the expression inside the floor brackets is $$\\frac{(x+1)^{3}}{(x-1) x}-\\frac{(x-1)^{3}}{x(x+1)}=\\frac{(x+1)^{4}-(x-1)^{4}}{(x-1) x(x+1)}=\\frac{8 x^{3}+8 x}{x^{3}-x}=8+\\frac{16 x}{x^{3}-x}$$ Since $x$ is certainly large enough that $0<16 x /... |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 22437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope of the tangent line to the curve $y=\\frac{1}{3}{x^3}-\\frac{2}{x}$ at $x=1$ is $\\alpha$. Find $\\frac{{sin\\alpha cos2\\alpha}}{{sin\\alpha+cos\\alpha}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"split": "train",
"index": 22438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 22439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The pentagon \\( A B C D E \\) is inscribed around a circle.\n\nThe angles \\( \\angle A B C \\), \\( \\angle B A E \\), and \\( \\angle C D E \\) each measure \\( 104^\\circ \\). Find \\( \\angle A D B \\). Provide the answer in degrees (only the number, without units). Let's ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 22440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\\%$ of his bill and Joe tipped $20\\%$ of his bill. What was the difference, in dollars between their bills? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22441,
"solution": "1. **Define the variables:**\n Let $m$ represent Mike's bill and $j$ represent Joe's bill.\n\n2. **Set up the equations based on the given percentages:**\n - Mike tipped $10\\%$ of his bill, which is $2$ dollars. Therefore, we have the equation:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ whose first $n$ terms have a sum of $S_n$, and the point $(n, \\frac{S_n}{n})$ lies on the line $y = \\frac{1}{2}x + \\frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \\in \\mathbb{N}^*$), and $b_3 = 11$, with the sum... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 22442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have 6 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one ... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 22443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not tak... | math | {
"style": "rule",
"ground_truth": "10999"
} | {
"split": "train",
"index": 22444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every second, the computer displays a number equal to the sum of the digits of the previous number multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the screen on the 2020th second? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "310"
} | {
"split": "train",
"index": 22445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 4 shoes from 5 pairs of different-sized shoes, find the number of possibilities where at least 2 of the 4 shoes can be paired together. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 22446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that α is an angle in the third quadrant, and sin(π - α) = -3/5, find the value of tan 2α. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"split": "train",
"index": 22447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the distinct points of intersection of the plane curves given by $x^{2}=x+y+4$ and $y^{2}=y-15 x+36$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 22448,
"solution": "Substituting $y=x^{2}-x-4$ into the second equation yields $$\\begin{aligned} 0 & =\\left(x^{2}-x-4\\right)^{2}-\\left(x^{2}-x-4\\right)+15 x-36 \\\\ & =x^{4}-2 x^{3}-7 x^{2}+8 x+16-x^{2}+x+4+15 x-36 \\\\ & =x^{4}-2 x^{3}-8 x^{2}+24 x-16 \\\\ & =(x-2)\\left(x^{3}-8... |
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