data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "180\\pi"
} | {
"index": 34100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second hand on the clock pictured below is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 30 minutes? Express your answer in terms of $\\pi$.\n\n[asy]\ndraw(Circle((0,0),20));\nlabel(\"12\",(0,20),S);\nlabel(\"9\",(-20,0),E);\n... | math | {
"style": "rule",
"ground_truth": "360\\pi"
} | {
"index": 34101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The secret object is a rectangle measuring $200 \\times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the s... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 34102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segment \\( PQ \\) is divided into several smaller segments. On each of them, a square is constructed (see illustration).\n\nWhat is the length of the path along the arrows if the length of the segment \\( PQ \\) is 73? If necessary, round your answer to 0.01 or write the a... | math | {
"style": "rule",
"ground_truth": "219"
} | {
"index": 34103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{3}"
} | {
"index": 34104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segment connecting the centers of two intersecting circles is divided by their common chord into segments of 4 and 1. Find the length of the common chord, given that the radii of the circles are in the ratio $3:2$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"index": 34105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segments \\( AP \\) and \\( AQ \\) are tangent to circle \\( O \\) at points \\( P \\) and \\( Q \\), respectively. Moreover, \\( QE \\) is perpendicular to diameter \\( PD \\) of length 4. If \\( PE = 3.6 \\) and \\( AP = 6 \\), what is the length of \\( QE \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"index": 34106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segments connecting the feet of the altitudes of an acute-angled triangle form a right triangle with a hypotenuse of 10. Find the radius of the circumcircle of the original triangle."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The segments of two lines, enclosed between two parallel planes, are in the ratio of \\( 5:9 \\), and the acute angles between these lines and one of the planes are in the ratio of \\( 2:1 \\), respectively. Find the cosine of the smaller angle."
}
] | math | {
"style": "rule",
"ground_truth": "0.9"
} | {
"index": 34108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $ (a_n)$ is given by $ a_1\\equal{}1,a_2\\equal{}0$ and:\r\n\r $ a_{2k\\plus{}1}\\equal{}a_k\\plus{}a_{k\\plus{}1}, a_{2k\\plus{}2}\\equal{}2a_{k\\plus{}1}$ for $ k \\in \\mathbb{N}.$ \r\n\r\nFind $ a_m$ for $ m\\equal{}2^{19}\\plus{}91.$"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 34109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\\frac{1992}{n} \\cdot \\sum_{k=0}^{n-1} x_k$ for $n \\geq 1$ .\nFind $\\sum_{n=0}^{1992} 2^nx_n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"index": 34110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_i\\}_{i \\ge 1}$ is defined by $a_1 = 1$ and \\[ a_n = \\lfloor a_{n-1} + \\sqrt{a_{n-1}} \\rfloor \\] for all $n \\ge 2$ . Compute the eighth perfect square in the sequence.\n\n*Proposed by Lewis Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$ for every positive integer $n$ , where $a_0 = a_1 = 1$ . Calculate the sum $$ \\frac{a_0}{a_1} + \\frac{a_1}{a_2} + ... + \\frac{a_{2008}}... | math | {
"style": "rule",
"ground_truth": "2009 * 1005"
} | {
"index": 34112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\\geq 2$. Determine the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 34115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies\n\\[a_1 + a_2 + a_3 + \\dots + a_n = n^2 a_n\\]for all $n \\ge 2.$ If $a_{63} = 1,$ find $a_1.$"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 34116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac85a_n + \\frac65\\sqrt {4^n - a_n^2}$ for $n\\geq 0$. Find the greatest integer less than or equal to $a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "983"
} | {
"index": 34117,
"solution": "We can now simply start to compute the values $b_i$ by hand:\n\\begin{align*} b_1 & = \\frac 35 \\\\ b_2 & = \\frac 45\\cdot \\frac 35 + \\frac 35 \\sqrt{1 - \\left(\\frac 35\\right)^2} = \\frac{24}{25} \\\\ b_3 & = \\frac 45\\cdot \\frac {24}{25} + \\frac 35 \\sqrt{1 - \\left(\\frac ... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac{8}{5}a_n + \\frac{6}{5}\\sqrt{4^n - a_n^2}$ for $n \\geq 0$. Find $a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24576}{25}"
} | {
"index": 34118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 34119,
"solution": "We notice that by multiplying the equation from an arbitrary $a_n$ all the way to $a_1$, we get: \\[5^{a_n-a_1}=\\dfrac{n+\\tfrac23}{1+\\tfrac23}\\] This simplifies to \\[5^{a_n}=3n+2.\\] We can now test powers of $5$.\n$5$ - that gives us $n=1$, which is useless.\n$25$ - that gives a... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \\geq 3$). Find $a_{1000}$."
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"index": 34120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(x_n)$ is defined by $x_1 = 115$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{115}"
} | {
"index": 34121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dots.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{150}"
} | {
"index": 34122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $(x_n)$ is defined by $x_1 = 25$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{25}"
} | {
"index": 34123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\\ldots$ consists of $1$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is"
}
] | math | {
"style": "rule",
"ground_truth": "2419"
} | {
"index": 34124,
"solution": "1. **Understanding the Sequence**: The sequence given is $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \\ldots$ where each $1$ is followed by an increasing number of $2$s. Specifically, the $n$-th block of $2$s contains $n$ instances of $2$.\n\n2. **Summing Blocks**: The sum of each b... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $1, 4, 5, 16, 17, 20, 21 \\cdots$ consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $150^{\\mbox{th}}$ term of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "16660"
} | {
"index": 34125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $2, 7, 12, a, b, 27$ is arithmetic. What is the value of $a + b$?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 34126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, ...$ consists of $3$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. Calculate the sum of the first $1024$ terms of this sequence.\nA) $4166$\nB) $4248$\nC) $4303$\nD) $4401$"
}
... | math | {
"style": "rule",
"ground_truth": "4248"
} | {
"index": 34127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 34128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $6075, 2025, 675 \\ldots$, is made by repeatedly dividing by 3. How many integers are in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left(z_{n}\\right)$ of complex numbers satisfies the following properties: $z_{1}$ and $z_{2}$ are not real. $z_{n+2}=z_{n+1}^{2} z_{n}$ for all integers $n \\geq 1$. $\\frac{z_{n+3}}{z_{n}^{2}}$ is real for all integers $n \\geq 1$. $\\left|\\frac{z_{3}}{z_{4}}... | math | {
"style": "rule",
"ground_truth": "65536"
} | {
"index": 34130,
"solution": "All complex numbers can be expressed as $r(\\cos \\theta+i \\sin \\theta)=r e^{i \\theta}$. Let $z_{n}$ be $r_{n} e^{i \\theta_{n}}$. $\\frac{z_{n+3}}{z_{n}^{2}}=\\frac{z_{n+2}^{2} z_{n+1}}{z_{n}^{2}}=\\frac{z_{n+1}^{5} z_{n}^{2}}{z_{n}^{2}}=z_{n+1}^{5}$ is real for all $n \\geq 1$, s... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{ a_n \\right\\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.\n\n$(1... | math | {
"style": "rule",
"ground_truth": "(1)(3)(4)(5)"
} | {
"index": 34131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ consists of 9 terms, where $a_{1} = a_{9} = 1$, and for each $i \\in \\{1,2, \\cdots, 8\\}$, we have $\\frac{a_{i+1}}{a_{i}} \\in \\left\\{2,1,-\\frac{1}{2}\\right\\}$. Find the number of such sequences."
}
] | math | {
"style": "rule",
"ground_truth": "491"
} | {
"index": 34132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \\cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\left\\{a_{n}\\right\\}$. Calculate $S_{2016}$."
}
] | math | {
"style": "rule",
"ground_truth": "7066"
} | {
"index": 34133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ satisfies $a_{1} = 1$, and for each $n \\in \\mathbf{N}^{*}$, $a_{n}$ and $a_{n+1}$ are the roots of the equation $x^{2} + 3n x + b_{n} = 0$. Find $\\sum_{k=1}^{20} b_{k}$."
}
] | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"index": 34134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}_{n \\geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\\left\\{b_{n}\\right\\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integer... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 34135,
"solution": "We show by induction that $a_{n}=F_{4 n-2}$ and $b_{n}=F_{2 n-1}$, where $F_{k}$ is the $k$ th Fibonacci number. The base cases are clear. As for the inductive steps, note that $$F_{k+2}=F_{k+1}+F_{k}=2 F_{k}+F_{k-1}=3 F_{k}-F_{k-2}$$ and $$F_{k+4}=3 F_{k+2}-F_{k}=8 F_{k}+3 F_{k-2}=7 ... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\left\\{x_{n}\\right\\}$ satisfies $x_{1}=1$, and for any $n \\in \\mathbb{Z}^{+}$, it holds that $x_{n+1}=x_{n}+3 \\sqrt{x_{n}}+\\frac{n}{\\sqrt{x_{n}}}$. Find the value of $\\lim _{n \\rightarrow+\\infty} \\frac{n^{2}}{x_{n}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 34136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\lg 1000, \\lg \\left(1000 \\cos \\frac{\\pi}{3}\\right), \\lg \\left(1000 \\cos ^{2} \\frac{\\pi}{3}\\right), \\cdots, \\lg \\left(1000 \\cos ^{n-1} \\frac{\\pi}{3}\\right), \\cdots$, when the sum of the first $n$ terms is maximized, the value of $n$ is ( )."
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{2n+1\\}$ ($n\\in\\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket ... | math | {
"style": "rule",
"ground_truth": "2072"
} | {
"index": 34138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a\\_n\\}$ is a geometric sequence with the first term $a\\_1=4$, and $S\\_3$, $S\\_2$, $S\\_4$ form an arithmetic sequence.\n(1) Find the general term formula of the sequence $\\{a\\_n\\}$;\n(2) If $b\\_n=\\log \\_{2}|a\\_n|$, let $T\\_n$ be the sum of the firs... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 34139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a\\_n\\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2017}{2}"
} | {
"index": 34140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, $a\\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 34141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ is an arithmetic sequence, and the sequence $\\{b_n\\}$ satisfies $b_n = a_na_{n+1}a_{n+2} (n \\in \\mathbb{N}^*)$. Let $S_n$ be the sum of the first $n$ terms of $\\{b_n\\}$. If $a_{12} = \\frac{3}{8}a_5 > 0$, then when $S_n$ reaches its maximum value,... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ is defined by \\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\] The sequence $\\{b_n\\}$ is defined by \\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ fo... | math | {
"style": "rule",
"ground_truth": "561"
} | {
"index": 34143,
"solution": "Rearranging the definitions, we have \\[\\frac{a_n}{a_{n-1}} = \\frac{a_{n-1}}{a_{n-2}} + 1,\\quad \\frac{b_n}{b_{n-1}} = \\frac{b_{n-1}}{b_{n-2}} + 1\\] from which it follows that $\\frac{a_n}{a_{n-1}} = 1+ \\frac{a_{n-1}}{a_{n-2}} = \\cdots = (n-1) + \\frac{a_{1}}{a_0} = n$ and $\\f... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 34144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=1$, $a_{n+2}=(1+\\sin^2 \\frac{n\\pi}{2})a_n+2\\cos^2 \\frac{n\\pi}{2}$. Find the sum of the first $20$ terms of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "1123"
} | {
"index": 34145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\\cos \\frac {2n\\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{b_n\\}$. Calculate $S_{24}$."
}
] | math | {
"style": "rule",
"ground_truth": "304"
} | {
"index": 34146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$."
}
] | math | {
"style": "rule",
"ground_truth": "5150"
} | {
"index": 34147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_n=13-3n$, $b_n=a_n⋅a_{n+1}⋅a_{n+2}$, $S_n$ is the sum of the first $n$ terms of $\\{b_n\\}$. Find the maximum value of $S_n$."
}
] | math | {
"style": "rule",
"ground_truth": "310"
} | {
"index": 34148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "1830"
} | {
"index": 34149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "3240"
} | {
"index": 34150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "1830"
} | {
"index": 34151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1}=(2|\\sin \\frac{n\\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\\{a_n\\}$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "2550"
} | {
"index": 34152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}_{n\\geq 0}$ of real numbers satisfies the relation:\n\\[ a_{m+n} + a_{m-n} - m + n -1 = \\frac12 (a_{2m} + a_{2n}) \\]\nfor all non-negative integers $m$ and $n$, $m \\ge n$. If $a_1 = 3$ find $a_{2004}$."
}
] | math | {
"style": "rule",
"ground_truth": "4018021"
} | {
"index": 34153,
"solution": "\nWe are given the sequence \\( \\{a_n\\}_{n \\geq 0} \\) which satisfies the relation:\n\n\\[\na_{m+n} + a_{m-n} - m + n - 1 = \\frac{1}{2} (a_{2m} + a_{2n})\n\\]\n\nfor all non-negative integers \\( m \\) and \\( n \\) with \\( m \\ge n \\). We are also given that \\( a_1 = 3 \\), a... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_{n}\\}$ is an increasing sequence of integers, and $a_{1}\\geqslant 3$, $a_{1}+a_{2}+a_{3}+\\ldots +a_{n}=100$. Determine the maximum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $\\{a_{n}\\}$ satisfies $a_{1}+\\frac{1}{2}{a_2}+\\frac{1}{3}{a_3}+…+\\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\\in N^{*}$, and $a_{1}=1$.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>$(2)$ Let $S_{n}=a_{1}\\cdot a_{n}+a_{2}\\cdot a_{n-1}+a_{3}\\c... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"index": 34155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $a_1, a_2, \\ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\\log_8 a_1+\\log_8 a_2+\\cdots+\\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 34156,
"solution": "Using the above method, we can derive that $a^{2}r^{11} = 2^{1003}$. Now, think about what happens when r is an even power of 2. Then $a^{2}$ must be an odd power of 2 in order to satisfy the equation which is clearly not possible. Thus the only restriction r has is that it must be an... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $a_1,$ $a_2,$ $a_3,$ $\\dots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and for all $n \\ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$"
}
] | math | {
"style": "rule",
"ground_truth": "179"
} | {
"index": 34157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of real numbers satisfies the recurrence $a_{n+1}=\\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}{a_{n-1}+1}$. Given that $a_{1}=1$ and $a_{9}=7$, find $a_{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34158,
"solution": "Let $b_{n}=a_{n}+1$. Then the recurrence becomes $b_{n+1}-1=\\left(b_{n}^{2}-b_{n-1}\\right) / b_{n-1}=b_{n}^{2} / b_{n-1}-1$, so $b_{n+1}=b_{n}^{2} / b_{n-1}$. It follows that the sequence $\\left(b_{n}\\right)$ is a geometric progression, from which $b_{5}^{2}=b_{1} b_{9}=2 \\cdot 8... |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $b_1, b_2, \\ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\\log_4 b_1+\\log_4 b_2+\\cdots+\\log_4 b_{10} = 2010,$ find the number of possible ordered pairs $(b,s).$"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $b_1, b_2, b_3, \\dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \\ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 34160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence $b_1, b_2, b_3, \\dots$ satisfies $b_1 = 25,$ $b_{12} = 125,$ and for all $n \\ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 34161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \\sqrt { \\frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\\leqslant \\frac{m}{30}$ holds for any $n\\in\\mathbb{N}^{*}$, find the minimum value of the positive integer $m$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "3240"
} | {
"index": 34163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence 1,000,000; 500,000; 250,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "15625"
} | {
"index": 34164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence 1,3,1,3,3,1,3,3,3,1,3,3,3,3,1,3,... follows a certain rule. What is the sum of the first 44 terms in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"index": 34165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence 12, 15, 18, 21, 51, 81, $\\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\\mathrm{th}}$ term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "318"
} | {
"index": 34166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\( a_{0}, a_{1}, \\cdots, a_{n} \\) satisfies:\n\\[ \na_{0}=\\sqrt{3}, \\quad a_{n+1}=[a_{n}]+\\frac{1}{\\{a_{n}\\}} \n\\]\nwhere \\( [x] \\) denotes the greatest integer less than or equal to the real number \\( x \\), and \\( \\{x\\}=x-[x] \\). Find the value of... | math | {
"style": "rule",
"ground_truth": "3024 + \\sqrt{3}"
} | {
"index": 34167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\((b_n)\\) satisfies\n\\[kb_1 + kb_2 + kb_3 + \\dots + kb_n = n^2 kb_n\\] for all \\(n \\ge 2.\\) If \\(b_{70} = 2\\) and \\(k=3\\), find \\(b_1.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "4970"
} | {
"index": 34168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) is defined as follows:\n\\[ \n\\begin{align*}\na_{1} &= 0, \\\\\na_{2} &= 1, \\\\\na_{n+1} &= \\frac{\\sqrt{6}-\\sqrt{2}}{2} a_{n} - a_{n-1} \\quad \\text{for } n \\geq 2.\n\\end{align*}\n\\]\n\nDetermine the value of \\(a_{2019}\\)."... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{2}"
} | {
"index": 34169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies \\(a_{1} = 1\\) and \\(a_{n} = \\frac{1}{4 a_{n-1}} + \\frac{1}{n}\\) for \\(n \\geq 2\\). Find the value of \\(a_{100}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{101}{200}"
} | {
"index": 34170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies \\(a_{1}=1, \\sqrt{\\frac{1}{a_{n}^{2}}+4}=\\frac{1}{a_{n+1}}\\). Let \\(S_{n}=\\sum_{i=1}^{n} a_{i}^{2}\\). If \\(S_{2 n+1}-S_{n} \\leqslant \\frac{t}{30}\\) holds for any \\(n \\in \\mathbf{N}^{*}\\), what is the smallest ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies: \\(a_1 = 1\\), and for each \\(n \\in \\mathbf{N}^{*}\\), \\(a_n\\) and \\(a_{n+1}\\) are the roots of the equation \\(x^2 + 3nx + b_n = 0\\). Find \\(\\sum_{k=1}^{20} b_k\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"index": 34172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies: \\(a_{1}=1\\), and for each \\(n \\in \\mathbf{N}^{*}\\), \\(a_{n}\\) and \\(a_{n+1}\\) are the two roots of the equation \\(x^{2}+3nx+b_{n}=0\\). Find \\(\\sum_{k=1}^{20} b_{k}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"index": 34173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\{a_n\\}\\) has consecutive terms \\(a_n\\) and \\(a_{n+1}\\) as the roots of the equation \\(x^2 - c_n x + \\left(\\frac{1}{3}\\right)^n = 0\\), with initial term \\(a_1 = 2\\). Find the sum of the infinite series \\(c_1, c_2, \\cdots, c_n, \\cdots\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 34174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\{a_n\\}\\) is a geometric sequence with a common ratio of \\(q\\), where \\(|q| > 1\\). Let \\(b_n = a_n + 1 (n \\in \\mathbb{N}^*)\\), if \\(\\{b_n\\}\\) has four consecutive terms in the set \\(\\{-53, -23, 19, 37, 82\\}\\), find the value of \\(q\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3}{2}"
} | {
"index": 34175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(\\{a_n\\}\\) is defined such that \\(a_1 = \\frac{\\pi}{6}\\) and \\(a_{n+1} = \\arctan \\left(\\sec a_n\\right)\\) for \\( n \\in \\mathbf{N}^{*}\\). Find the positive integer \\(m\\) such that\n\\[\n\\sin a_1 \\cdot \\sin a_2 \\cdots \\cdot \\sin a_m = \\frac{... | math | {
"style": "rule",
"ground_truth": "3333"
} | {
"index": 34176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(a_{0}, a_{1}, a_{2}, \\cdots, a_{n}\\) satisfies \\(a_{0}=\\sqrt{3}\\), \\(a_{n+1}=\\left\\lfloor a_{n}\\right\\rfloor + \\frac{1}{\\left\\{a_{n}\\right\\}}\\), where \\(\\left\\lfloor a_{n}\\right\\rfloor\\) and \\(\\left\\{a_{n}\\right\\}\\) represent the inte... | math | {
"style": "rule",
"ground_truth": "3024 + \\sqrt{3}"
} | {
"index": 34177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\(a_{n}\\) is defined as follows:\n\n\\[ a_{1} = 1, \\quad a_{n+1} = a_{n} + \\frac{2a_{n}}{n} \\text{ for } n \\geq 1 \\]\n\nFind \\(a_{100}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "5151"
} | {
"index": 34178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence \\\\(\\{a_n\\}\\) consists of numbers \\\\(1\\\\) or \\\\(2\\\\), with the first term being \\\\(1\\\\). Between the \\\\(k\\\\)-th \\\\(1\\\\) and the \\\\(k+1\\\\)-th \\\\(1\\\\), there are \\\\(2k-1\\\\) \\\\(2\\\\)s, i.e., the sequence \\\\(\\{a_n\\}\\) is \\\\... | math | {
"style": "rule",
"ground_truth": "3989"
} | {
"index": 34179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence consists of 19 ones and 49 zeros arranged in a random order. A group is defined as the maximal subsequence of identical symbols. For example, in the sequence 110001001111, there are five groups: two ones, then three zeros, then one one, then two zeros, and finally ... | math | {
"style": "rule",
"ground_truth": "2.83"
} | {
"index": 34180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence is $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{2}{3}$, $\\frac{1}{4}$, $\\frac{2}{4}$, $\\frac{3}{4}$, ... , $\\frac{1}{m+1}$, $\\frac{2}{m+1}$, ... , $\\frac{m}{m+1}$, ... Find the $20^{th}$ term."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"index": 34181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence is defined as \\( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\\frac{k}{a_{k}} \\) for \\( k=1,2, \\cdots, n-1 \\). Determine the value of \\( n \\) for which \\( a_{n}=0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 34182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence is defined recursively:\n\n\\[ x_{0} = 0, \\quad x_{n+1} = \\frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \\]\n\nFind \\( x_{8453} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "8453"
} | {
"index": 34183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence is formed by taking all positive multiples of 4 that contain at least one digit that is either a 2 or a 3. What is the $30^\\text{th}$ term of this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 34184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence of integers $ a_1 $ , $ a_2 $ , $ \\dots $ is defined as follows: $ a_1 = 1 $ and $ n> 1 $ , $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \\neq 3a_k $ for any $ i, j $ and $ k $ from $ \\{1, 2, \\dots, n + ... | math | {
"style": "rule",
"ground_truth": "3006"
} | {
"index": 34185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence of integers $\\{a_i\\}_{i = 0}^{\\infty}$ satisfies $a_0 = 3$ , $a_1 = 4$ , and\n\\[a_{n+2} = a_{n+1} a_n + \\left\\lceil \\sqrt{a_{n+1}^2 - 1} \\sqrt{a_n^2 - 1}\\right\\rceil\\]\nfor $n \\ge 0$ . Evaluate the sum\n\\[\\sum_{n = 0}^{\\infty} \\left(\\frac{a_{n... | math | {
"style": "rule",
"ground_truth": "0.518"
} | {
"index": 34186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 34187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$ ."
}
] | math | {
"style": "rule",
"ground_truth": "81281"
} | {
"index": 34188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence of real numbers \\( a_1, a_2, \\cdots, a_n, \\cdots \\) is defined by the following equation: \\( a_{n+1} = 2^n - 3a_n \\) for \\( n = 0, 1, 2, \\cdots \\).\n1. Find an expression for \\( a_n \\) in terms of \\( a_0 \\) and \\( n \\).\n2. Find \\( a_0 \\) such that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 34189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 34190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "800000"
} | {
"index": 34191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$."
}
] | math | {
"style": "rule",
"ground_truth": "262"
} | {
"index": 34192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$."
}
] | math | {
"style": "rule",
"ground_truth": "262"
} | {
"index": 34193,
"solution": "Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for $b_2$. When we get to $b_2=9$ and $a_2=91$, we have $a_4=271$ and $b_4=729$, which works, therefore, the answer is $b_3+a_3=81+181=\\boxed{262}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The sequences of real numbers $\\left\\{a_{i}\\right\\}_{i=1}^{\\infty}$ and $\\left\\{b_{i}\\right\\}_{i=1}^{\\infty}$ satisfy $a_{n+1}=\\left(a_{n-1}-1\\right)\\left(b_{n}+1\\right)$ and $b_{n+1}=a_{n} b_{n-1}-1$ for $n \\geq 2$, with $a_{1}=a_{2}=2015$ and $b_{1}=b_{2}=2013$... | math | {
"style": "rule",
"ground_truth": "1+\\frac{1}{2014 \\cdot 2015}"
} | {
"index": 34194,
"solution": "First note that $a_{n}$ and $b_{n}$ are weakly increasing and tend to infinity. In particular, $a_{n}, b_{n} \\notin\\{0,-1,1\\}$ for all $n$. For $n \\geq 1$, we have $a_{n+3}=\\left(a_{n+1}-1\\right)\\left(b_{n+2}+1\\right)=\\left(a_{n+1}-1\\right)\\left(a_{n+1} b_{n}\\right)$, so $... |
DeepScaleR | [
{
"role": "user",
"content": "The sequences where each term is a real number are denoted as $\\left\\{a_{n}\\right\\}$, with the sum of the first $n$ terms recorded as $S_{n}$. Given that $S_{10} = 10$ and $S_{30} = 70$, what is the value of $S_{40}$?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 34195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set\n$$\nA=\\{\\sqrt[n]{n} \\mid n \\in \\mathbf{N} \\text{ and } 1 \\leq n \\leq 2020\\}\n$$\n\nhas the largest element as $\\qquad$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{3}"
} | {
"index": 34196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set $H$ is defined by the points $(x, y)$ with integer coordinates, $-8 \\leq x \\leq 8$, $-8 \\leq y \\leq 8$. Calculate the number of squares of side length at least $9$ that have their four vertices in $H$."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 34197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set $S = \\{1, 2, 3, \\ldots , 49, 50\\}$ contains the first $50$ positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34199,
"solution": "Since the average of the two smallest integers in $S$ is 5, their sum is $2 \\cdot 5=10$. Since the average of the two largest integers in $S$ is 22, their sum is $2 \\cdot 22=44$. Suppose that the other five integers in the set $S$ are $p<q<r<t<u$. (Note that the integers in $S$ are ... |
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