data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangul... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 38149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many subsets of the set $\\{1, 2, 3, 4, 5, 6\\}$ must contain the number 6 and at least one of the numbers 1 or 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the seven-digit number $854n526$ is divisible by $11$, what is $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangular prism $ABC A_{1} B_{1} C_{1}$ with the base $ABC$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 3. Segment $CD$ is a diameter of this sphere. Find the volume of the prism if $A D = 2 \\sqrt{6}$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{15}"
} | {
"split": "train",
"index": 38152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits $5,$ $6,$ $7,$ and $8,$ used exactly once to form four-digit integers, list these integers from least to greatest. For numbers starting with $7$ or $8,$ reverse the order of the last two digits. What is the $20^{\\text{th}}$ integer in the list? Let's think ste... | math | {
"style": "rule",
"ground_truth": "7865"
} | {
"split": "train",
"index": 38153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child whose age is between 13 and 19 writes his own age after his father's age, creating a four-digit number. The absolute difference between their ages is subtracted from this new number to obtain 4289. What is the sum of their ages?\n\n(Note: From the 22nd Annual USA High S... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 38154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a unique triplet of positive integers \\((a, b, c)\\) such that \\(a \\leq b \\leq c\\) and\n$$\n\\frac{25}{84}=\\frac{1}{a}+\\frac{1}{a b}+\\frac{1}{a b c}.\n$$\nDetermine \\(a + b + c\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 38156,
"solution": "We are tasked with finding how many integers between 1000 and 2013 have the property that their units digit is the sum of the other digits. We will consider two cases based on the thousands digit, which can be either 1 or 2.\n\n#### Case 1: Thousands digit is 1\nTh... |
deepscaler | [
{
"role": "user",
"content": "Let $g(x) = 2x^7 - 3x^3 + 4x - 8.$ If $g(6) = 12,$ find $g(-6).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-28"
} | {
"split": "train",
"index": 38157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its ... | math | {
"style": "rule",
"ground_truth": "683"
} | {
"split": "train",
"index": 38158,
"solution": "This method does not rigorously get the answer, but it works. As the bug makes more and more moves, the probability of it going back to the origin approaches closer and closer to 1/3. Therefore, after 10 moves, the probability gets close to $341.33/1024$. We can ei... |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 38159,
"solution": "We will count the number of it $< 2^{11}=2048$ instead of $2003$ (In other words, the length of the base-2 representation is at most $11$. If there are even digits, $2n$, then the leftmost digit is $1$, the rest, $2n-1$, has odd number of digits. In order for the b... |
deepscaler | [
{
"role": "user",
"content": "Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have\r\n\\[ a^{3}\\plus{}b^{3}\\plus{}c^{3}\\geq k\\left(a\\plus{}b\\plus{}c\\right)^{3}.\\] Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2} - 4}{2}"
} | {
"split": "train",
"index": 38160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n$$f(xf(x + y)) = yf(x) + 1$$\nholds for all $x, y \\in \\mathbb{R}^{+}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\frac{1}{x}"
} | {
"split": "train",
"index": 38161,
"solution": "\nTo solve the functional equation \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) given by\n\n\\[\nf(xf(x + y)) = yf(x) + 1\n\\]\n\nfor all \\( x, y \\in \\mathbb{R}^{+} \\), we aim to determine a function \\( f \\) that satisfies this relationship.\n\n###... |
deepscaler | [
{
"role": "user",
"content": "If $g(x) = 2x^2+2x-1$, what is the value of $g(g(2))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 38162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c=a+\\cos A\\frac{b}{\\cos B}$. \n$(1)$ Find the measure of angle $B$. \n$(2)$ If $b=4$ and $a+c=3\\sqrt{2}$, find the area of $\\triangle ABC$. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 38163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \\leq n \\leq 100$ , consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 38164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 38165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jane places six ounces of tea into a ten-ounce cup and six ounces of milk into a second cup of the same size. She then pours two ounces of tea from the first cup to the second and, after stirring thoroughly, pours two ounces of the liquid in the second cup back to the first. Wh... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin x\\cos x+2\\sqrt{3}\\cos^{2}x-\\sqrt{3}$.\n(1) Find the smallest positive period and the interval where the function is decreasing;\n(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, ... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 38167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N=123456789101112\\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 4\\qquad\\textbf{(C)}\\ 9\\qquad\\textb... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 38168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 38169,
"solution": "1. **Identify the relationships given by the problem:**\n - $PA = 3$ inches, $PD = 4$ inches, $PC = 5$ inches, and we need to find $PB = x$ inches.\n\n2. **Use the perpendicular distances from $P$ to the sides of the rectangle:**\n - Let $a, b, c, d$ be the per... |
deepscaler | [
{
"role": "user",
"content": "How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1845"
} | {
"split": "train",
"index": 38170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the two smallest prime factors of $250$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38171,
"solution": "1. **Find the prime factorization of 250**: \n To factorize 250, we start by dividing by the smallest prime number, which is 2. Since 250 is even, it is divisible by 2:\n \\[\n 250 \\div 2 = 125\n \\]\n Next, we factorize 125. Since 125 ends in 5, it is d... |
deepscaler | [
{
"role": "user",
"content": "Let the altitude of a regular triangular pyramid \\( P-ABC \\) be \\( PO \\). \\( M \\) is the midpoint of \\( PO \\). A plane parallel to edge \\( BC \\) passes through \\( AM \\), dividing the pyramid into two parts, upper and lower. Find the volume ratio of these two parts. ... | math | {
"style": "rule",
"ground_truth": "4/21"
} | {
"split": "train",
"index": 38172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( T \\) be a right triangle with sides having lengths 3, 4, and 5. A point \\( P \\) is called awesome if \\( P \\) is the center of a parallelogram whose vertices all lie on the boundary of \\( T \\). What is the area of the set of awesome points? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 38173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\\{0,1,2,\\ldots,9\\}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 38174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, \\dots , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. Let's th... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 38175,
"solution": "\nTo solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\\{1, 2, 3, \\ldots, 2015\\}$. We will use combinatorial arguments to derive the solution.\n\n#### Step 1: Counting subsets with a fixed least... |
deepscaler | [
{
"role": "user",
"content": "An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\\triangle ADE$, which is a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be positive integers, with \\( x < y \\). The leading digit of \\( \\lg x \\) is \\( a \\), and the trailing digit is \\( \\alpha \\); the leading digit of \\( \\lg y \\) is \\( b \\), and the trailing digit is \\( \\beta \\). They satisfy the condit... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 38177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=\\cos 2x+\\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\\left(x\\right)$, then $\\cos (x_{1}+x_{2})=$____. 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": 38178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hectoliter barrel is full of wine; 1 liter of liquid flows out in 24 hours, which is replaced by the inflow of 1 liter of water. After how much time will the contents of the barrel be half wine and half water? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 38179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( m = \\min \\left\\{ x + 2y + 3z \\mid x^{3} y^{2} z = 1 \\right\\} \\). What is the value of \\( m^{3} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 38180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A systematic sampling method is used to select 5 representatives from 752 students, after removing 2 students randomly. Calculate the probability that student A is selected. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{752}"
} | {
"split": "train",
"index": 38181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{8}{0}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function such that (i) For all $x, y \\in \\mathbb{R}$, $f(x)+f(y)+1 \\geq f(x+y) \\geq f(x)+f(y)$ (ii) For all $x \\in[0,1), f(0) \\geq f(x)$, (iii) $-f(-1)=f(1)=1$. Find all such functions $f$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "f(x) = \\lfloor x \\rfloor"
} | {
"split": "train",
"index": 38183,
"solution": "Plug $y \\rightarrow 1$ in (i): $$f(x)+f(1)+1 \\geq f(x+1) \\geq f(x)+f(1) \\Longleftrightarrow f(x)+1 \\leq f(x+1) \\leq f(x)+2$$ Now plug $y \\rightarrow-1$ and $x \\rightarrow x+1$ in (i): $$f(x+1)+f(-1)+1 \\geq f(x) \\geq f(x+1)+f(-1) \\Longleftrightarrow f(x) ... |
deepscaler | [
{
"role": "user",
"content": "Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \\frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\\triangle DEF$ to the area of quadrilateral $ABEF$ is: Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1: 5"
} | {
"split": "train",
"index": 38184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(81)^{\\frac12}=3^m$. Find $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 38185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? Let's think step... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 38186,
"solution": "1. **Identify the ratio of volumes between the actual water tower and the miniature model**: \n The actual water tower holds 100,000 liters, and Logan's miniature holds 0.1 liters. The ratio of the volumes is:\n \\[\n \\frac{100000 \\text{ liters}}{0.1 \\text... |
deepscaler | [
{
"role": "user",
"content": "If: (1) \\( a, b, c, d \\) are all elements of \\( \\{1,2,3,4\\} \\); (2) \\( a \\neq b, b \\neq c, c \\neq d, d \\neq a \\); (3) \\( a \\) is the smallest value among \\( a, b, c, d \\),\n\nthen, how many different four-digit numbers \\( \\overline{abcd} \\) can be formed? Let... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle. Let's... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $2x^{2}+3x^{2}$ when $x=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38189,
"solution": "When $x=2$, we obtain $2x^{2}+3x^{2}=5x^{2}=5 \\cdot 2^{2}=5 \\cdot 4=20$."
} |
deepscaler | [
{
"role": "user",
"content": "\nTo walk 2 km, ride 3 km by bicycle, and drive 20 km by car, Uncle Vanya needs 1 hour and 6 minutes. If he needs to walk 5 km, ride 8 km by bicycle, and drive 30 km by car, it takes him 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 4 km, ride 5 km by bicy... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 38190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the geometric series $3 - \\left(\\frac{3}{4}\\right) + \\left(\\frac{3}{4}\\right)^2 - \\left(\\frac{3}{4}\\right)^3 + \\dots$. 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{12}{7}"
} | {
"split": "train",
"index": 38191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x+ \\frac {1}{2}\\right)= \\frac {2x^{4}+x^{2}\\sin x+4}{x^{4}+2}$, calculate the value of $f\\left( \\frac {1}{2017}\\right)+f\\left( \\frac {2}{2017}\\right)+\\ldots+f\\left( \\frac {2016}{2017}\\right)$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 38192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38193,
"solution": "1. **Define the Sets:**\n Let $D$ be the set of students who have a dog and $C$ be the set of students who have a cat. We know:\n - $|D| = 20$ (students with a dog)\n - $|C| = 26$ (students with a cat)\n - Total number of students, $|D \\cup C| = 39$.\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset of $\\{1,2,3,\\ldots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "905"
} | {
"split": "train",
"index": 38194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38196,
"solution": "To solve this problem, we need to determine how many noncongruent triangles can be formed using any three of the six points $R$, $S$, $T$, $X$, $Y$, and $Z$, where $X$, $Y$, and $Z$ are the midpoints of the sides of equilateral triangle $\\triangle RST$.\n\n#### St... |
deepscaler | [
{
"role": "user",
"content": "How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 38197,
"solution": "To solve this problem, we need to determine how many distinct lines can be formed that pass through four distinct points of the form $(i, j, k)$, where $i, j, k$ are positive integers not exceeding 4.\n\n1. **Total Points Consideration**:\n Each of $i, j, k$ can ... |
deepscaler | [
{
"role": "user",
"content": "The sum of two sides of a rectangle is 11, and the sum of three sides is 19.5. Find the product of all possible distinct values of the perimeter of such a rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15400"
} | {
"split": "train",
"index": 38198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38199,
"solution": "1. **Understanding the problem**: We are given a cube with an edge length of $3$ cm, and it is divided into smaller cubes with whole number edge lengths. We need to determine the number of smaller cubes, $N$.\n\n2. **Analyzing possible edge lengths of smaller cubes... |
deepscaler | [
{
"role": "user",
"content": " $ABC$ is triangle. $l_1$ - line passes through $A$ and parallel to $BC$ , $l_2$ - line passes through $C$ and parallel to $AB$ . Bisector of $\\angle B$ intersect $l_1$ and $l_2$ at $X,Y$ . $XY=AC$ . What value can take $\\angle A- \\angle C$ ? Let's thin... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 38200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\frac{810 \\times 811 \\times 812 \\times \\cdots \\times 2010}{810^{n}}\\) is an integer, find the maximum value of \\(n\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 38201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside an isosceles triangle \\( ABC \\), a point \\( K \\) is marked such that \\( CK = AB = BC \\) and \\(\\angle KAC = 30^\\circ\\). Find the angle \\( AKB \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 38202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cyclists started a trip at the same time from the same location. They traveled the same route and returned together. Both rested along the way. The first cyclist rode twice as long as the second cyclist rested. The second cyclist rode four times as long as the first cyclist... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 38203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school arranges for five people, \\( A \\), \\( B \\), \\( C \\), \\( D \\), and \\( E \\), to enter into three classes, with each class having at least one person, and \\( A \\) and \\( B \\) cannot be in the same class. Calculate the total number of different arrangements. ... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"split": "train",
"index": 38204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "24\\%"
} | {
"split": "train",
"index": 38205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are $ \\begin{cases} x=2+2\\cos θ \\\ny=2\\sin θ\\end{cases} $ for the parameter $(θ)$, and in the polar coordinate system $(rOθ)$ (with the same unit length as the rectangular co... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 38206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\left( 1 - \\frac{1}{\\cos 23^\\circ} \\right) \\left( 1 + \\frac{1}{\\sin 67^\\circ} \\right) \\left( 1 - \\frac{1}{\\sin 23^\\circ} \\right) \\left( 1 + \\frac{1}{\\cos 67^\\circ} \\right).\\] Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle C that passes through points A(1, 4) and B(3, -2), and the distance from the center of the circle C to line AB is $\\sqrt{10}$, find the equation of circle C. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $(625^{\\log_5 2015})^{\\frac{1}{4}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 38209,
"solution": "1. **Express 625 as a power of 5**: \n \\[\n 625 = 5^4\n \\]\n Therefore, we can rewrite the expression \\(625^{\\log_5 2015}\\) as:\n \\[\n (5^4)^{\\log_5 2015}\n \\]\n\n2. **Use the power of a power rule**:\n \\[\n (a^m)^n = a^{m \\cdot n}\n \... |
deepscaler | [
{
"role": "user",
"content": "You are given that $3^{400}\\equiv 1\\pmod{1000}$. What are the last three digits of $3^{12000}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "001"
} | {
"split": "train",
"index": 38210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_{L}=\\sum_{n=1}^{L}\\left\\lfloor n^{3} / 9\\right\\rfloor$ for positive integers $L$. Determine all $L$ for which $T_{L}$ is a square number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "L=1 \\text{ or } L=2"
} | {
"split": "train",
"index": 38211,
"solution": "Since $T_{L}$ is square if and only if $9 T_{L}$ is square, we may consider $9 T_{L}$ instead of $T_{L}$. It is well known that $n^{3}$ is congruent to 0,1, or 8 modulo 9 according as $n$ is congruent to 0,1, or 2 modulo 3. Therefore $n^{3}-9\\left\\lfloor n^{3} / ... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ Let's t... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 38212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points \\( A(4,0) \\) and \\( B(2,2) \\) are inside the ellipse \\( \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1 \\), and \\( M \\) is a point on the ellipse, find the maximum value of \\( |MA| + |MB| \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "10 + 2\\sqrt{10}"
} | {
"split": "train",
"index": 38213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Daphne's four friends visit her every 4, 6, 8, and 10 days respectively, all four friends visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends will visit her. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 38214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $ n \\in \\mathbb{N}$ , let $ f(n)\\equal{}1^n\\plus{}2^{n\\minus{}1}\\plus{}3^{n\\minus{}2}\\plus{}...\\plus{}n^1$ . Determine the minimum value of: $ \\frac{f(n\\plus{}1)}{f(n)}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8/3"
} | {
"split": "train",
"index": 38215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \\sqrt {5}$ . Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {10} }{2}"
} | {
"split": "train",
"index": 38216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Q$ be a $(2n+1) \\times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \\times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "(2n+1)(n+1)"
} | {
"split": "train",
"index": 38217,
"solution": "\nLet \\( Q \\) be a \\( (2n+1) \\times (2n+1) \\) board. The problem states that every \\( 2 \\times 2 \\) sub-board of \\( Q \\) contains at most \\( 2 \\) black cells. We need to determine the maximum number of black cells that can be accommodated on the board u... |
deepscaler | [
{
"role": "user",
"content": "There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38218,
"solution": "To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.\n\n1. **Unders... |
deepscaler | [
{
"role": "user",
"content": "Find the ordered pair $(j,k)$ that satisfies the equations $5j-42k=1$ and $2k-j=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,-\\frac{1}{2})"
} | {
"split": "train",
"index": 38219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point \\((x, y)\\) is randomly selected such that \\(0 \\leq x \\leq 4\\) and \\(0 \\leq y \\leq 5\\). What is the probability that \\(x + y \\leq 5\\)? 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}{5}"
} | {
"split": "train",
"index": 38220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the coordinate plane, the graph of \\( y = \\frac{2020}{x} \\) is plotted. How many points on the graph have a tangent line that intersects both coordinate axes at points with integer coordinates? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle O with a radius of 6, the length of chord AB is 6.\n(1) Find the size of the central angle α corresponding to chord AB;\n(2) Find the arc length l and the area S of the sector where α is located. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 38222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days ... | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 38223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the poly... | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 38224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible sum of the digits in the base-nine representation of a positive integer less than $3000$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), side \\(AB = 6\\), \\(\\angle BAC = 30^\\circ\\), and the radius of the circumscribed circle is 5. Find side \\(AC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} + 4"
} | {
"split": "train",
"index": 38226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the four roots of\n\\[2x^4 + x^3 - 6x^2 + x + 2 = 0.\\]Enter the four roots (counting multiplicity), separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 1, -2, -\\frac{1}{2}"
} | {
"split": "train",
"index": 38227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 38228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many cubic feet are in one cubic yard? One yard is equal to three feet.\n\n[asy]import three;\ncurrentprojection=orthographic(1/2,1/2,1);\ndraw((0,0,0)--(10,0,0)--(10,-10,0)--(0,-10,0)--cycle);\ndraw((0,0,10)--(10,0,10)--(10,-10,10)--(0,-10,10)--cycle);\ndraw((0,0,0)--(0,0,... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 38229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equilateral triangle has sides of \\(2x\\) and \\(x+15\\) as shown. Find the perimeter of the triangle in terms of \\(x\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 38230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of all positive integer divisors of $144,000$. Calculate the number of numbers that are the product of two distinct elements of $T$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "451"
} | {
"split": "train",
"index": 38231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a non-empty set of numbers, the sum of its maximum and minimum elements is called the \"characteristic value\" of the set. $A\\_1$, $A\\_2$, $A\\_3$, $A\\_4$, $A\\_5$ each contain $20$ elements, and $A\\_1∪A\\_2∪A\\_3∪A\\_4∪A\\_5={x∈N^⁎|x≤slant 100}$, find the minimum val... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"split": "train",
"index": 38232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RR... | math | {
"style": "rule",
"ground_truth": "157"
} | {
"split": "train",
"index": 38233,
"solution": "Assume without loss of generality that the first card laid out is red. Then the arrangements that satisfy Kathy’s requirements are RRRRR, RRRRG, RRRGG, RRGGG, and RGGGG. The probability that Kathy will lay out one of these arrangements is \\[\\frac49\\cdot\\frac38\... |
deepscaler | [
{
"role": "user",
"content": "120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each have been stockpiled at the quarry. A railroad platform can hold up to 40 tons. What is the minimum number of platforms required to transport all the slabs? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2\\sqrt{3}\\sin^2(x) + 2\\sin(x)\\cos(x) - \\sqrt{3}$, where $x \\in \\left[ \\frac{\\pi}{3}, \\frac{11\\pi}{24} \\right]$:\n\n1. Determine the range of the function $f(x)$.\n2. Suppose an acute-angled triangle $ABC$ has two sides of lengths equal to ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(x-y)^{8}(x+y)$, the coefficient of $x^{7}y^{2}$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular hexagon \\( A B C D E F \\) with a side length of 1, calculate \\((\\overrightarrow{A B}+\\overrightarrow{D C}) \\cdot(\\overrightarrow{A D}+\\overrightarrow{B E})\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 38237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a three-dimensional Cartesian coordinate system, there is a sphere with its center at the origin and a radius of 3 units. How many lattice points lie on the surface of the sphere? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total volume and the total surface area in square feet of three cubic boxes if their edge lengths are 3 feet, 5 feet, and 6 feet, respectively? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 38239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two distinct numbers \\(a\\) and \\(b\\) such that \\(\\frac{a}{b} + a = \\frac{b}{a} + b\\), find \\(\\frac{1}{a} + \\frac{1}{b}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \\mathrm{~cm}^{2}$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in the ratio $7: 3$. On Angela's portion of the ... | math | {
"style": "rule",
"ground_truth": "11: 9"
} | {
"split": "train",
"index": 38242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse and a hyperbola have the same foci $F\\_1(-c,0)$, $F\\_2(c,0)$. One endpoint of the ellipse's minor axis is $B$, and the line $F\\_1B$ is parallel to one of the hyperbola's asymptotes. If the eccentricities of the ellipse and hyperbola are $e\\_1$ and $e\\_2$, respec... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 38243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One TV was sold for a 12% profit and the other for a 12% loss at a selling price of 3080 yuan each. Determine the net profit or loss from these transactions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-90"
} | {
"split": "train",
"index": 38244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3,-2,2)"
} | {
"split": "train",
"index": 38245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the lines $l_1: ax+2y-1=0$ and $l_2: x+by-3=0$, where the angle of inclination of $l_1$ is $\\frac{\\pi}{4}$, find the value of $a$. If $l_1$ is perpendicular to $l_2$, find the value of $b$. If $l_1$ is parallel to $l_2$, find the distance between the two lines. Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 38246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a deck of cards marked with 1, 2, 3, and 4, two cards are drawn consecutively. The probability of drawing the card with the number 4 on the first draw, the probability of not drawing it on the first draw but drawing it on the second, and the probability of drawing the numb... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs \\((a, b)\\) of positive integers such that \\(a\\) and \\(b\\) both divide \\(20^{19}\\), but \\(ab\\) does not. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "444600"
} | {
"split": "train",
"index": 38248,
"solution": ""
} |
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