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problem
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"In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point(...TRUNCATED) |
34
| "<think>\nOkay, let's try to tackle this geometry problem. It seems a bit complex at first, but I'll(...TRUNCATED) |
Continue
| ["\n\nTo solve the problem, we are given a triangle $ ABC $ with:\n\n- $ \\sin \\angle A = \\frac{4}(...TRUNCATED) |
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"In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point(...TRUNCATED) |
34
| "<think>\nOkay, let's try to tackle this geometry problem. It seems a bit complex at first, but I'll(...TRUNCATED) |
Terminate
| ["\n\nTo solve the problem, we are given a triangle $ ABC $ with $ \\angle A $ such that $ \\sin(\\a(...TRUNCATED) |
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"Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$(...TRUNCATED) |
-3
| "<think>\nOkay, let's try to tackle this problem. So, we have three distinct numbers a, b, c. The eq(...TRUNCATED) |
Continue
| ["\n\nTo solve the problem, we are given three distinct real numbers $ a, b, c $, and two pairs of q(...TRUNCATED) |
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"Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$(...TRUNCATED) |
-3
| "<think>\nOkay, let's try to tackle this problem. So, we have three distinct numbers a, b, c. The eq(...TRUNCATED) |
Terminate
| ["\n\nTo solve the problem, we are given three distinct real numbers $ a, b, c $, and the following (...TRUNCATED) |
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"There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ card(...TRUNCATED) |
3
| "<think>\nOkay, let me try to figure out this problem. So there are 7 boxes in a row, numbered 1 to (...TRUNCATED) |
Continue
| ["\n\nWe are given a sequence of 2015 cards to be placed into 7 boxes arranged in a row, numbered 1 (...TRUNCATED) |
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"There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ card(...TRUNCATED) |
3
| "<think>\nOkay, so there's this problem with 7 boxes arranged in a row, numbered 1 through 7. There (...TRUNCATED) |
Terminate
| ["\n\nTo determine the final box in which the 2015th card is placed, we analyze the pattern of card (...TRUNCATED) |
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"Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The val(...TRUNCATED) |
37
| "<think>\nOkay, so I need to find the sum of 1/(2^a 3^b 5^c) where (a, b, c) are non-negative intege(...TRUNCATED) |
Continue
| ["\n\nTo evaluate the sum\n\n$$\n\\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c}\n$$\n\nwhere $ S $ is(...TRUNCATED) |
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"Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The val(...TRUNCATED) |
37
| "<think>\nOkay, so I need to find the sum of 1/(2^a 3^b 5^c) where (a, b, c) are non-negative intege(...TRUNCATED) |
Terminate
| ["\n\nTo evaluate the sum\n\n$$\n\\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c}\n$$\n\nwhere $ S $ is(...TRUNCATED) |
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For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?
|
10
| "<think>\nOkay, so I need to figure out for which values of n the expression $ n^4 + 6n^3 + 11n^2 + (...TRUNCATED) |
Continue
| ["\n\nTo determine for which integer values of $ n $ the expression\n\n$$\nP(n) = n^4 + 6n^3 + 11n^2(...TRUNCATED) |
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For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?
|
10
| "<think>\nOkay, so I need to figure out for which values of n the expression $ n^4 + 6n^3 + 11n^2 + (...TRUNCATED) |
Terminate
| ["\n\nTo determine for which integers $ n $ the expression\n\n$$\nP(n) = n^4 + 6n^3 + 11n^2 + 3n + 3(...TRUNCATED) |
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