data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\nCalculate the lengths of arcs of curves given by the parametric equations.\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array}{l}\nx=2(t-\\sin t) \\\\\ny=2(1-\\cos t)\n\\end{array}\\right. \\\\\n& 0 \\leq t \\leq \\frac{\\pi}{2}\n\\end{aligned}\n$$ Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "8 - 4\\sqrt{2}"
} | {
"split": "train",
"index": 39449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the number of degrees in the acute angle formed by the hands of a clock at 6:44? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62^\\circ"
} | {
"split": "train",
"index": 39450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $X Y Z$ be an equilateral triangle, and let $K, L, M$ be points on sides $X Y, Y Z, Z X$, respectively, such that $X K / K Y=B, Y L / L Z=1 / C$, and $Z M / M X=1$. Determine the ratio of the area of triangle $K L M$ to the area of triangle $X Y Z$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 39451,
"solution": "First, we note that $$[K L M]=[X Y Z]-[X K M]-[Y L K]-[Z M L]$$ Then, note that $$\\begin{gathered} {[X K M]=\\frac{X K}{X Y} \\cdot \\frac{X M}{X Z} \\cdot[X Y Z]=\\frac{B}{B+1} \\cdot \\frac{1}{2} \\cdot[X Y Z]} \\\\ {[Y L K]=\\frac{Y L}{Y Z} \\cdot \\frac{Y K}{Y... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ meets side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, then what is $RS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side edge of a regular tetrahedron \\( S-ABC \\) is 2, and the base is an equilateral triangle with side length 1. A section passing through \\( AB \\) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{15}}"
} | {
"split": "train",
"index": 39453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number selected uniformly at random between 100 and 200. If $\\lfloor {\\sqrt{x}} \\rfloor = 12$, find the probability that $\\lfloor {\\sqrt{100x}} \\rfloor = 120$. ($\\lfloor {v} \\rfloor$ means the greatest integer less than or equal to $v$.) Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{241}{2500}"
} | {
"split": "train",
"index": 39454,
"solution": "1. **Determine the range for $x$ based on $\\lfloor \\sqrt{x} \\rfloor = 12$:**\n Since $\\lfloor \\sqrt{x} \\rfloor = 12$, it implies that $12 \\leq \\sqrt{x} < 13$. Squaring both sides of the inequality, we get:\n \\[\n 12^2 \\leq x < 13^2 \\implies 144 \\l... |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\sqrt{2}\\sin \\left( 2x+ \\frac{π}{4} \\right)$.\n\n(1) Find the equation of the axis of symmetry of the graph of the function $f(x)$;\n\n(2) Find the interval(s) where $f(x)$ is monotonically increasing;\n\n(3) Find the maximum and minimum values of the function... | math | {
"style": "rule",
"ground_truth": "- \\sqrt{2}"
} | {
"split": "train",
"index": 39455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$$\n\\text{Consider the system of inequalities:}\n\\begin{cases}\nx + 2y \\leq 6 \\\\\n3x + y \\geq 3 \\\\\nx \\leq 4 \\\\\ny \\geq 0\n\\end{cases}\n$$\nDetermine the number of units in the length of the longest side of the polygonal region formed by this system. Express your a... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 39456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha$ is a root of the equation $3x^2 - x - 2 = 0$ and $\\alpha$ is an angle in the third quadrant, find the value of $\\frac{\\sin (-\\alpha + \\frac{3\\pi}{2}) \\cos (\\frac{3\\pi}{2} + \\alpha) \\tan^2 (\\pi - \\alpha)}{\\cos (\\frac{\\pi}{2} + \\alpha) ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 39457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 39458,
"solution": "1. **Identify the median position:** Since there are $100$ customers, the median will be the average of the $50\\text{th}$ and $51\\text{st}$ largest amounts of cans per person.\n\n2. **Minimize the first 49 customers' purchases:** To maximize the median, we should... |
deepscaler | [
{
"role": "user",
"content": "There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. \n$(1)$ How many unique three-digit numbers can be formed without repetition? \n$(2)$ How many unique four-digit even numbers can be formed without repetition? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "2296"
} | {
"split": "train",
"index": 39459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ be unit vectors, and $|\\overrightarrow{a}+\\overrightarrow{b}|=1$. Determine the angle $\\theta$ between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 39460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers greater than 1 such that $a b=100$. The maximum possible value of $a^{(\\log_{10} b)^{2}}$ can be written in the form $10^{x}$ for some real number $x$. Find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{27}"
} | {
"split": "train",
"index": 39461,
"solution": "Let $p=\\log_{10} a, q=\\log_{10} b$. Since $a, b>1, p$ and $q$ are positive. The condition $a b=100$ translates to $p+q=2$. We wish to maximize $$x=\\log_{10} a^{(\\log_{10} b)^{2}}=(\\log_{10} a)(\\log_{10} b)^{2}=p q^{2}$$ By AM-GM, $$\\frac{27}{4} p q^{2} \\leq... |
deepscaler | [
{
"role": "user",
"content": "If the ratio of the surface areas of three spheres is 1:4:9, then the ratio of their volumes is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 : 8 : 27"
} | {
"split": "train",
"index": 39462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a unit square. A circle with radius $\\frac{32}{49}$ passes through point $D$ and is tangent to side $A B$ at point $E$. Then $D E=\\frac{m}{n}$, where $m, n$ are positive integers and $\\operatorname{gcd}(m, n)=1$. Find $100 m+n$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "807"
} | {
"split": "train",
"index": 39463,
"solution": "Let $O$ be the center of the circle and let $F$ be the intersection of lines $O E$ and $C D$. Also let $r=32 / 49$ and $x=D F$. Then we know $$x^{2}+(1-r)^{2}=D F^{2}+O F^{2}=D O^{2}=r^{2}$$ which implies that $x^{2}+1-2 r=0$, or $1+x^{2}=2 r$. Now, $$D E=\\sqrt{D ... |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\\frac {xy}{x+2y-2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1+ \\sqrt {2}}{2}"
} | {
"split": "train",
"index": 39464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$x_{n+1}= \\left ( 1+\\frac2n \\right )x_n+\\frac4n$ , for every positive integer $n$ . If $x_1=-1$ , what is $x_{2000}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000998"
} | {
"split": "train",
"index": 39465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Athletes A, B, and C, along with 4 volunteers, are to be arranged in a line. If A and B are next to each other and C is not at either end, determine the number of different ways to arrange them. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 39466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram and let $\\overrightarrow{AA^\\prime}$, $\\overrightarrow{BB^\\prime}$, $\\overrightarrow{CC^\\prime}$, and $\\overrightarrow{DD^\\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\\prime} = 10$, $BB^{\\... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39467,
"solution": "1. **Assume the Configuration**: Let $ABCD$ be a unit square with coordinates $A(0,0,0)$, $B(0,1,0)$, $C(1,1,0)$, and $D(1,0,0)$. Assume that the rays $\\overrightarrow{AA'}$, $\\overrightarrow{BB'}$, $\\overrightarrow{CC'}$, and $\\overrightarrow{DD'}$ extend in t... |
deepscaler | [
{
"role": "user",
"content": "An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after workin... | math | {
"style": "rule",
"ground_truth": "$2.60"
} | {
"split": "train",
"index": 39468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 39469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Mr. Thompson can choose between two routes to commute to his office: Route X, which is 8 miles long with an average speed of 35 miles per hour, and Route Y, which is 7 miles long with an average speed of 45 miles per hour excluding a 1-mile stretch with a reduced speed of... | math | {
"style": "rule",
"ground_truth": "1.71"
} | {
"split": "train",
"index": 39470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 39471,
"solution": "Let the smaller number be $x$ and the larger number be $y$. We are given two conditions:\n1. Three times the larger number is four times the smaller number.\n2. The difference between the numbers is 8.\n\nWe can translate these conditions into equations:\n1. \\(3y ... |
deepscaler | [
{
"role": "user",
"content": "If for any $x \\in D$, the inequality $f_1(x) \\leq f(x) \\leq f_2(x)$ holds, then the function $f(x)$ is called a \"compromise function\" of the functions $f_1(x)$ to $f_2(x)$ over the interval $D$. It is known that the function $f(x) = (k-1)x - 1$, $g(x) = 0$, $h(x) = (x+1)\\... | math | {
"style": "rule",
"ground_truth": "\\{2\\}"
} | {
"split": "train",
"index": 39472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange 2002 students numbered from 1 to 2002 in a row from left to right. Counting from left to right from 1 to 11, the student who is counted as 11 remains in place, while the others leave the row. Then, the remaining students count from left to right from 1 to 11 again, and ... | math | {
"style": "rule",
"ground_truth": "1331"
} | {
"split": "train",
"index": 39473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "968"
} | {
"split": "train",
"index": 39474,
"solution": "Any subset of the ten points with three or more members can be made into exactly one such polygon. Thus, we need to count the number of such subsets. There are $2^{10} = 1024$ total subsets of a ten-member set, but of these ${10 \\choose 0} = 1$ have 0 members, ${1... |
deepscaler | [
{
"role": "user",
"content": "Elisenda has a piece of paper in the shape of a triangle with vertices $A, B$, and $C$ such that $A B=42$. She chooses a point $D$ on segment $A C$, and she folds the paper along line $B D$ so that $A$ lands at a point $E$ on segment $B C$. Then, she folds the paper along line ... | math | {
"style": "rule",
"ground_truth": "168+48 \\sqrt{7}"
} | {
"split": "train",
"index": 39475,
"solution": "Let $F$ be the midpoint of segment $D C$. Evidently $\\angle A D B=60^{\\circ}=\\angle B D E=\\angle E D C$. Moreover, we have $B D=D F=F C, A D=D E$, and $A B=B E$. Hence angle bisector on $B D C$ gives us that $B E=42, E C=84$, and hence angle bisector on $A B C$... |
deepscaler | [
{
"role": "user",
"content": "Which number appears most frequently in the second position when listing the winning numbers of a lottery draw in ascending order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 39476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\\overline{BC}$ and $\\overline{CD},$ respectively, so that $\\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\\overline{AE}.$ The length of a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39477,
"solution": "Why not solve in terms of the side $x$ only (single-variable beauty)? By similar triangles we obtain that $BE=\\frac{x}{1-x}$, therefore $CE=\\frac{1-2x}{1-x}$. Then $AE=\\sqrt{2}*\\frac{1-2x}{1-x}$. Using Pythagorean Theorem on $\\triangle{ABE}$ yields $\\frac{x^2... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $2^{3}-2+3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39478,
"solution": "Evaluating, $2^{3}-2+3=2 \times 2 \times 2-2+3=8-2+3=9$."
} |
deepscaler | [
{
"role": "user",
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the lo... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 39479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) + \\left( \\frac{y}{x} + \\frac{z}{y} + \\frac{x}{z} \\right) = 8.\\]Find the minimum value of\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) \\... | math | {
"style": "rule",
"ground_truth": "22 \\sqrt{11} - 57"
} | {
"split": "train",
"index": 39480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the graph of \\( y=g(x) \\) includes the points \\( (1,4), (2,6), \\) and \\( (3,2) \\).\n\nBased only on this information, there are two points that must be on the graph of \\( y=g(g(x)) \\). If we call these points \\( (a,b) \\) and \\( (c,d) \\), what is \\( ab + cd ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain tour group checked the weather conditions on the day of the outing. A weather forecasting software predicted that the probability of rain during the time periods $12:00$ to $13:00$ and $13:00$ to $14:00$ on the day of the outing are $0.5$ and $0.4$ respectively. Then,... | math | {
"style": "rule",
"ground_truth": "0.7"
} | {
"split": "train",
"index": 39482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Hua plays a certain game where each round can be played several times freely. Each score in a round is one of the numbers $8$, $a$ (a natural number), or $0$. The total score for a round is the sum of all individual scores in that round. Xiao Hua has achieved the following... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 39483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the reciprocal of $\\frac{3}{4} + \\frac{4}{5}$?\nA) $\\frac{31}{20}$\nB) $\\frac{20}{31}$\nC) $\\frac{19}{20}$\nD) $\\frac{20}{19}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{31}"
} | {
"split": "train",
"index": 39484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $y = -4.9t^2 + 23.8t$ describes the height (in meters) of a projectile launched from the ground at 23.8 meters per second. In how many seconds will the projectile first reach 28 meters in height? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that\n\\[f(x) f(-x) = f(x^3).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-32"
} | {
"split": "train",
"index": 39487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 39488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a city with $n$ citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following:\n\n[list]\n[*][i]Virus positive[/i]: there is at least one currentl... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"split": "train",
"index": 39489,
"solution": "\nTo determine the smallest number of tests required to ascertain if the sceptervirus is currently present or has been present in the city, let's analyze the given conditions for the test results:\n\n1. **Virus positive**: Indicates there is at least one currently ... |
deepscaler | [
{
"role": "user",
"content": "A natural number's proper divisors are defined as positive integer divisors other than 1 and the number itself. A natural number greater than 1 is called \"good\" if it is equal to the product of all its distinct proper divisors. What is the sum of the first 10 \"good\" natural... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 39490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The secant \\( ABC \\) intercepts an arc \\( BC \\), which contains \\( 112^\\circ \\); the tangent \\( AD \\) at point \\( D \\) divides this arc in the ratio \\( 7:9 \\). Find \\(\\angle BAD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31.5"
} | {
"split": "train",
"index": 39491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n Let's... | math | {
"style": "rule",
"ground_truth": "601"
} | {
"split": "train",
"index": 39492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[\\sqrt{1 + \\sqrt{2 + \\sqrt{x}}} = \\sqrt[3]{1 + \\sqrt{x}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 39493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the first 10 odd positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 39494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1, a_2, \\dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \\le i \\le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 39495,
"solution": "To solve this problem, we need to understand how the lists can be constructed under the given constraints. The key constraint is that for each $a_i$ where $i \\geq 2$, either $a_i + 1$ or $a_i - 1$ (or both) must appear before $a_i$ in the list. This constraint gui... |
deepscaler | [
{
"role": "user",
"content": "In the acute triangle \\(KLM\\), \\(V\\) is the intersection of its heights, and \\(X\\) is the foot of the height onto side \\(KL\\). The angle bisector of angle \\(XVL\\) is parallel to side \\(LM\\), and the angle \\(MKL\\) measures \\(70^\\circ\\).\n\nWhat are the measures ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 39496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the internal angles $A$, $B$, $C$ of $\\triangle ABC$ be opposite to the sides $a$, $b$, $c$ respectively, and $c\\cos B= \\sqrt {3}b\\sin C$. \n$(1)$ If $a^{2}\\sin C=4 \\sqrt {3}\\sin A$, find the area of $\\triangle ABC$; \n$(2)$ If $a=2 \\sqrt {3}$, $b= \\sqrt {7}$, a... | math | {
"style": "rule",
"ground_truth": "\\sqrt {13}"
} | {
"split": "train",
"index": 39497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39498,
"solution": "1. **Define Variables:**\n Let $x$ be the number of points scored by the Cougars, and $y$ be the number of points scored by the Panthers.\n\n2. **Set Up Equations:**\n From the problem, we have two equations based on the total points scored and the margin by wh... |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a,b,c$ and prime $p$ satisfying that\n\\[ 2^a p^b=(p+2)^c+1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 1, 3)"
} | {
"split": "train",
"index": 39499,
"solution": "\nWe need to find all positive integers \\(a, b, c\\) and a prime \\(p\\) that satisfy the equation:\n\\[\n2^a p^b = (p+2)^c + 1.\n\\]\n\nFirst, we note that \\(p\\) cannot be 2 because the left-hand side would be even, while the right-hand side would be odd.\n\n##... |
deepscaler | [
{
"role": "user",
"content": "Find the total number of solutions to the equation $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$ where $a, b, c$ are positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1755"
} | {
"split": "train",
"index": 39500,
"solution": "We write this as $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$. Since $a, b, c$ are positive integers, $a-b<a+b+c$. So, we have three possibilities: $a-b=1$ and $a+b+c=2012$, $a-b=2$ and $a+b+c=1006$, and $a-b=4$ and $a+b+c=503$. The first solution gives $a=b+1$ and $c=2... |
deepscaler | [
{
"role": "user",
"content": "When $n$ is a positive integer, $n! = n \\times (n - 1) \\times \\ldots \\times 2 \\times 1$ is defined as the factorial of $n$ (for example, $10! = 10 \\times 9 \\times \\ldots \\times 2 \\times 1 = 3,628,800$). Then, how many zeros are there at the end of $2010!$? Let's think... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 39501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos x\\bullet \\sin (x+\\frac{\\pi }{3})-\\sqrt{3}\\cos^{2}x+\\frac{\\sqrt{3}}{4}$, where $x\\in R$.\n\n(1) Find the minimum positive period and symmetry center of $f(x)$.\n\n(2) If the graph of the function $y=g(x)$ is obtained by shifting the graph ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 39502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 700, inclusive, contain the digit 0 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "123"
} | {
"split": "train",
"index": 39503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the final stage of a professional bowling tournament, the competition between the top five players is conducted as follows: the fifth and fourth place players compete first, the loser gets the 5th place prize; the winner competes with the third place player, the loser gets t... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\\overrightarrow{a}+y\\overrightarrow{b}|=1$ and $xy\\geqslant 0$, the inequality $|x+2y|\\leqslant \\frac{8}{\\sqrt{15}}$ holds. Find t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 39505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the set of all three-digit numbers composed of the digits 0, 1, 2, 3, 4, 5, without any repeating digits, how many such numbers have a digit-sum of 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 39507,
"solution": "1. **Identify the type of function:** The function given for elevation $s$ in terms of time $t$ is $s = 160t - 16t^2$. This is a quadratic function in the standard form $s = at^2 + bt + c$ where $a = -16$, $b = 160$, and $c = 0$.\n\n2. **Determine the vertex:** The... |
deepscaler | [
{
"role": "user",
"content": "Find the area of a trapezoid with diagonals of 7 cm and 8 cm, and bases of 3 cm and 6 cm. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12 \\sqrt{5}"
} | {
"split": "train",
"index": 39508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 4, \\frac{5 \\pi}{3}, \\frac{\\pi}{2} \\right)$ in spherical coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, -2 \\sqrt{3}, 0)"
} | {
"split": "train",
"index": 39509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle touches the longer leg of a right triangle, passes through the vertex of the opposite acute angle, and has its center on the hypotenuse of the triangle. What is the radius of the circle if the lengths of the legs are 5 and 12? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{65}{18}"
} | {
"split": "train",
"index": 39510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with angles $\\angle A = 60^{\\circ}, \\angle B = 75^{\\circ}, \\angle C = 45^{\\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto ... | math | {
"style": "rule",
"ground_truth": "1132"
} | {
"split": "train",
"index": 39511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle \\( A B C \\) (with right angle at \\( C \\)), the medians \\( A M \\) and \\( B N \\) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 39512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the three sides $a, b, c$ form an arithmetic sequence, and $\\angle A = 3 \\angle C$. Find $\\cos \\angle C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{33}}{8}"
} | {
"split": "train",
"index": 39513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tim continues the prank into the next week after a successful first week. This time, he starts on Monday with two people willing to do the prank, on Tuesday there are three options, on Wednesday everyone from Monday and Tuesday refuses but there are six new people, on Thursday ... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 39514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 5,$ find $\\begin{vmatrix} 2a & 2b \\\\ 2c & 2d \\end{vmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\\sqrt{2}$, and passing through the point $A(\\frac{3}{2}, -\\frac{1}{2})$.\n(1) Find the equation of the ellipse;\n(2) Find the coordinates of a point $P$ on the ellipse ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 39517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $\\frac{1}{3^{2}+1}+\\frac{1}{4^{2}+2}+\\frac{1}{5^{2}+3}+\\cdots$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13/36"
} | {
"split": "train",
"index": 39518,
"solution": "Each term takes the form $$\\frac{1}{n^{2}+(n-2)}=\\frac{1}{(n+2) \\cdot(n-1)}$$ Using the method of partial fractions, we can write (for some constants $A, B$ ) $$\\begin{gathered} \\frac{1}{(n+2) \\cdot(n-1)}=\\frac{A}{(n+2)}+\\frac{B}{(n-1)} \\\\ \\Rightarrow 1=... |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers satisfying the conditions\n$\\quad\\bullet\\ \\gcd(m+n,210)=1,$\n$\\quad\\bullet\\ m^m$ is a multiple of $n^n,$ and\n$\\quad\\bullet\\ m$ is not a multiple of $n.$\n Find the least possible value of $m+n.$ Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "407"
} | {
"split": "train",
"index": 39519,
"solution": "Taking inspiration from $4^4 \\mid 10^{10}$ we are inspired to take $n$ to be $p^2$, the lowest prime not dividing $210$, or $11 \\implies n = 121$. Now, there are $242$ factors of $11$, so $11^{242} \\mid m^m$, and then $m = 11k$ for $k \\geq 22$. Now, $\\gcd(m+n,... |
deepscaler | [
{
"role": "user",
"content": "In the third season of \"China Poetry Conference\", there were many highlights under the theme of \"Life has its own poetry\". In each of the ten competitions, there was a specially designed opening poem recited in unison by a hundred people under the coordination of lights and... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 39520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(7 + 5 + 3) \\div 3 - 2 \\div 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\frac{1}{3}"
} | {
"split": "train",
"index": 39521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers greater than 3999 can be formed such that the product of the middle two digits exceeds 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3480"
} | {
"split": "train",
"index": 39522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the three-dimensional Cartesian coordinate system, given points $A(2,a,-1)$, $B(-2,3,b)$, $C(1,2,-2)$.<br/>$(1)$ If points $A$, $B$, and $C$ are collinear, find the values of $a$ and $b$;<br/>$(2)$ Given $b=-3$, $D(-1,3,-3)$, and points $A$, $B$, $C$, and $D$ are coplanar, f... | math | {
"style": "rule",
"ground_truth": "a=1"
} | {
"split": "train",
"index": 39523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an old-fashioned bicycle the front wheel has a radius of $2.5$ feet and the back wheel has a radius of $4$ inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes $100$ revolutions? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 39524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling a... | math | {
"style": "rule",
"ground_truth": "179"
} | {
"split": "train",
"index": 39525,
"solution": "If it weren’t for the small tube, the larger tube would travel $144\\pi$. Consider the distance from which the larger tube first contacts the smaller tube, until when it completely loses contact with the smaller tube.\n Drawing the radii as shown in the diagram, no... |
deepscaler | [
{
"role": "user",
"content": "Let \\(p\\) and \\(q\\) be relatively prime positive integers such that \\(\\dfrac pq = \\dfrac1{2^1} + \\dfrac2{4^2} + \\dfrac3{2^3} + \\dfrac4{4^4} + \\dfrac5{2^5} + \\dfrac6{4^6} + \\cdots\\), where the numerators always increase by 1, and the denominators alternate between ... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 39526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their spe... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 39527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 125th digit beyond the decimal point in the decimal representation of $\\frac47$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8k-168}{k-12}"
} | {
"split": "train",
"index": 39529,
"solution": "1. **Calculate the total score of the first 12 students:** \n Given that the mean score of the first 12 students is 14, the total score for these students is calculated as:\n \\[\n 12 \\times 14 = 168\n \\]\n\n2. **Set up the equation for the mean of the e... |
deepscaler | [
{
"role": "user",
"content": "If the ratio of $2x-y$ to $x+y$ is $\\frac{2}{3}$, what is the ratio of $x$ to $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 39530,
"solution": "1. **Set up the equation from the given ratio:**\n Given that the ratio of $2x-y$ to $x+y$ is $\\frac{2}{3}$, we can write this as:\n \\[\n \\frac{2x-y}{x+y} = \\frac{2}{3}\n \\]\n\n2. **Cross-multiply to eliminate the fraction:**\n Cross-multiplying the ... |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\\frac{[CEF]}{[DBE]}.$\n\n[asy]\nunitsize(0.025 cm);\n\npair A, B, C, D, E, F;\n\nB = (0,0);\nC = (80,0);\nA = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));\nD = interp(A,B,38/115);\nF = interp(A... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{96}"
} | {
"split": "train",
"index": 39531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum and product of the solutions of the quadratic equation $9x^2 - 45x + 50 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{9}"
} | {
"split": "train",
"index": 39532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 39533,
"solution": "Obviously, all of the terms must be odd. The common difference between the terms cannot be $2$ or $4$, since otherwise there would be a number in the sequence that is divisible by $3$. However, if the common difference is $6$, we find that $5,11,17,23$, and $29$ fo... |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\(2011 \\times 20122012 \\times 201320132013-2013 \\times 20112011 \\times 201220122012\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39534,
"solution": "Both terms are equal to \\(2011 \\times 2012 \\times 2013 \\times 1 \\times 10001 \\times 100010001\\)."
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)4.7+\\left(-2.5\\right)-\\left(-5.3\\right)-7.5$;<br/>$(2)18+48\\div \\left(-2\\right)^{2}-\\left(-4\\right)^{2}\\times 5$;<br/>$(3)-1^{4}+\\left(-2\\right)^{2}\\div 4\\times [5-\\left(-3\\right)^{2}]$;<br/>$(4)(-19\\frac{15}{16})×8$ (Solve using a simple met... | math | {
"style": "rule",
"ground_truth": "-159\\frac{1}{2}"
} | {
"split": "train",
"index": 39535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A, B, and C start from the same point on a circular track with a circumference of 360 meters: A starts first and runs in the counterclockwise direction; before A completes a lap, B and C start simultaneously and run in the clockwise direction; when A and B meet for the first ti... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 39536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(11 \\cdot 14 n\\) is a non-negative integer and \\(f\\) is defined by \\(f(0)=0\\), \\(f(1)=1\\), and \\(f(n)=f\\left(\\left\\lfloor \\frac{n}{2} \\right\\rfloor \\right)+n-2\\left\\lfloor \\frac{n}{2} \\right\\rfloor\\), find the maximum value of \\(f(n)\\) for \... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $((2^2-2)-(3^2-3)+(4^2-4))$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39538,
"solution": "The problem requires us to evaluate the expression \\[(2^2-2)-(3^2-3)+(4^2-4).\\]\n\n1. **Calculate each term separately:**\n - For the first term, \\(2^2 - 2 = 4 - 2 = 2\\).\n - For the second term, \\(3^2 - 3 = 9 - 3 = 6\\).\n - For the third term, \\(4^2 -... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\\dfrac{\\sqrt{3}}{2}$, and it passes through point $A(2,1)$.\n\n(Ⅰ) Find the equation of ellipse $C$; \n(Ⅱ) If $P$, $Q$ are two points on ellipse $C$, and the angle bisector... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 39539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 39540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $((x^2+1)^2(x-1)^6)$, find the coefficient of the $x^3$ term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-32"
} | {
"split": "train",
"index": 39541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are functions $f(x)$ with the following properties: $f(x)=ax^{2}+bx+c$ for some integers $a, b$ and $c$ with $a>0$, and $f(p)=f(q)=17$ and $f(p+q)=47$ for some prime numbers $p$ and $q$ with $p<q$. For each such function, the value of $f(pq)$ is calculated. The sum of all... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 39542,
"solution": "Since $f(p)=17$, then $ap^{2}+bp+c=17$. Since $f(q)=17$, then $aq^{2}+bq+c=17$. Subtracting these two equations, we obtain $a(p^{2}-q^{2})+b(p-q)=0$. Since $p^{2}-q^{2}=(p-q)(p+q)$, this becomes $a(p-q)(p+q)+b(p-q)=0$. Since $p<q$, then $p-q \\neq 0$, so we divide ... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$ . Lines $AM$ and $BC$ intersect in point $N$ . What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$ ? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "3/8"
} | {
"split": "train",
"index": 39544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\left\\{a_{n}\\right\\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent triangles can be formed by selecting vertices from the ten points in the triangular array, where the bottom row has four points, the next row has three points directly above the gaps of the previous row, followed by two points, and finally one point at th... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers between 200 and 400 are divisible by 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.) Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39548,
"solution": "1. **Convert dimensions from feet to yards**: \n Given that there are 3 feet in a yard, we convert the dimensions of the room from feet to yards by dividing each dimension by 3:\n - Length in yards: \\( \\frac{12 \\text{ feet}}{3 \\text{ feet/yard}} = 4 \\text{... |
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