question stringlengths 26 715 | answer stringlengths 104 4.41k ⌀ | input_ids listlengths 717 30.1k | reply stringlengths 1.87k 84.8k | original_index int64 0 624 | claims listlengths 29 1.12k |
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The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction. | Since the difference of the first two terms is $-2y$, the third and fourth terms of the sequence must be $x-3y$ and $x-5y$. Thus \[
x-3y = xy \quad\text{and}\quad x-5y = \frac{x}{y},
\]so $xy - 5y^{2} = x.$ Combining these equations we obtain \[
(x - 3y) - 5y^{2}= x\quad\text{and, therefore, }\quad -3y - 5y^{2} = 0.
\]... | [
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Okay, let's try to solve this problem. So, we have an arithmetic sequence where the first four terms are x + y, x - y, xy, and x/y. We need to find the fifth term. Alright, let me recall what an arithmetic sequence is. An arithmetic sequence is a sequence where the difference between consecutive terms is consta... | 0 | [
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... |
"A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet ea(...TRUNCATED) | "Since the oil is $3$ feet deep, we want to find the ratio of the area of the part of the circle cov(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's try to figure out this problem step by step. So, we have a right cylindrical o(...TRUNCATED) | 1 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"claim_text":"\nOkay, let's try to figur(...TRUNCATED) |
"The sum of the squares of three consecutive positive even numbers is $12296$. Find the product of t(...TRUNCATED) | "If $n$ is the middle number of the three, the other two numbers are $n-2$ and $n+2$. Therefore, the(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's try to solve this problem. The question says that the sum of the squares of th(...TRUNCATED) | 2 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11],"claim_text":"\nOkay, let's try to solve this proble(...TRUNCATED) |
"In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is(...TRUNCATED) | "Note that $1.5^2 + 2^2 = 2.5^2,$ so $\\triangle PED$ has a right angle at $P.$ (Alternatively, you (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the area of quadrilateral AEDC in triangle ABC where medians AD an(...TRUNCATED) | 3 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Let $\\alpha$ and $\\beta$ be the roots of $x^2 + px + 1 = 0,$ and let $\\gamma$ and $\\delta$ are (...TRUNCATED) | "Since $\\alpha$ and $\\beta$ are the roots of $x^2 + px + 1 = 0,$\n\\[(x - \\alpha)(x - \\beta) = x(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find an expression for the product $(\\alpha - \\gamma)(\\beta - \\gamm(...TRUNCATED) | 4 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"The sides of a triangle have lengths $11, 15,$ and $k,$ where $k$ is a positive integer. For how ma(...TRUNCATED) | "The longest side of the triangle either has length $15$ or has length $k.$ Take cases:\n\nIf the lo(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to figure out for how many integer values of k the triangle with sides 11,(...TRUNCATED) | 5 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Three points are chosen uniformly at random on a circle. What is the probability that no two of the(...TRUNCATED) | "Let us call the circle's center $O$. We first note that if $A$ and $B$ are points on the circle, t(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the probability that when three points are chosen uniformly at ran(...TRUNCATED) | 6 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"For how many different digits $n$ is the three-digit number $14n$ divisible by $n$?\n\nNote: $14n$ (...TRUNCATED) | "We have to account for each possible value of $n$ here. First of all, we can quickly find that for (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's try to figure out this problem. The question is: For how many different digits(...TRUNCATED) | 7 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12],"claim_text":"\nOkay, let's try to figure out thi(...TRUNCATED) |
"What is the hundreds digit in the following product: $5 \\times 6 \\times 7 \\times 8 \\times 9 \\t(...TRUNCATED) | "The hundreds digit in the given expression is the same as the tens digit in the expression $5\\time(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's see. I need to find the hundreds digit of the product 5 × 6 × 7 × 8 × 9 ×(...TRUNCATED) | 8 | [{"aligned_token_ids":[1,2,3,4,5,6,7],"claim_text":"\nOkay, let's see.","sentence":"\nOkay, let's se(...TRUNCATED) |
How many positive integers less than or equal to 100 have a prime factor that is greater than 4? | "The easiest solution is to find the number of positive integers with only 2 and 3 as their prime fa(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's try to figure out how many positive integers up to 100 have a prime factor gre(...TRUNCATED) | 9 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"(...TRUNCATED) |
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