ID stringlengths 8 10 | Year int64 2k 2k | Problem Number int64 1 15 | Part stringclasses 2
values | problem stringlengths 75 920 | answer stringlengths 1 3 | completion sequencelengths 16 16 | completion_answer sequencelengths 16 16 | completion_correct sequencelengths 16 16 | completion_succ_rate float64 0 1 |
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2000-I-7 | 2,000 | 7 | I | Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 5,$ and $y + \frac {1}{x} = 29.$ Then $z + \frac {1}{y} = \frac {m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 5 | [
"Okay, let's see. I have this problem where there are three positive numbers x, y, and z. They satisfy the equations:\n\n1. xyz = 1\n2. x + 1/z = 5\n3. y + 1/x = 29\n\nAnd I need to find z + 1/y, which is equal to m/n where m and n are coprime positive integers, and then find m + n.\n\nHmm. So, first, let's write d... | [
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2000-I-8 | 2,000 | 8 | I | "A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radiu(...TRUNCATED) | 52 | ["</think>\n\nAlright, let's tackle this problem step by step. We have a right circular cone that is(...TRUNCATED) | ["41"," no","..",", same no. is","52","52","52"," Wait","52",", which wait is","26",", wait is is","(...TRUNCATED) | [
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2000-I-9 | 2,000 | 9 | I | "The system of equations \\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \(...TRUNCATED) | 25 | ["Okay, so I have this system of logarithmic equations to solve. Let me try to understand what's goi(...TRUNCATED) | [" is....","25","2"," is","25","25","25",".","25","\\boxed.\\) wait\\\\)\\\n\n\\\\Wait.\n\n\\^\n\n\\(...TRUNCATED) | [
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2000-I-10 | 2,000 | 10 | I | "A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer (...TRUNCATED) | 173 | ["Okay, so I have this problem about a sequence of numbers \\( x_1, x_2, x_3, \\ldots, x_{100} \\). (...TRUNCATED) | [
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2000-I-11 | 2,000 | 11 | I | "Let $S$ be the sum of all numbers of the form $\\frac{a}{b}$ , where $a$ and $b$ are relatively pri(...TRUNCATED) | 248 | ["</think>\n\nLet's solve the problem step by step.\n\n**Problem Statement:**\nLet \\( S \\) be the (...TRUNCATED) | ["100","\\ \\]\n\ntimestext.\n\ntimestext \\}\n\n1..\n\n \\sum \\times \\ ?\n\n^ \\ \\ \\ \\ \\ \\ \(...TRUNCATED) | [
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2000-I-12 | 2,000 | 12 | I | "Given a function $f$ for which \\[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\\] holds for all re(...TRUNCATED) | 177 | ["</think>\n\nTo solve the problem, we analyze the given functional equation:\n\n\\[f(x) = f(398 - x(...TRUNCATED) | [
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2000-I-13 | 2,000 | 13 | I | "In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular (...TRUNCATED) | 731 | ["</think>\n\nOkay, let's try to solve this problem step by step. So, we have a firetruck stationed (...TRUNCATED) | ["26","8","26","26","29","29"," is is is","26","699","26","101","\\60.2iscal =.\n\n11boxedThus^ \\ a(...TRUNCATED) | [
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2000-I-14 | 2,000 | 14 | I | "In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\\(...TRUNCATED) | 571 | ["Alright, so I'm trying to solve this geometry problem. Let me read it again to make sure I underst(...TRUNCATED) | [".","1000","= is",".",".","2000","750","1539","2000","3600","515"," is.","1000",",","angle cos Wait(...TRUNCATED) | [
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2000-I-15 | 2,000 | 15 | I | "A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers o(...TRUNCATED) | 927 | ["Okay, so I'm trying to solve this problem about the stack of cards. Let me read it again to make s(...TRUNCATED) | ["1998","1998","1998","1998","1998","1998","1","The number of cards above the card labeled **1999** (...TRUNCATED) | [
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2000-II-2 | 2,000 | 2 | II | "A point whose coordinates are both integers is called a lattice point. How many lattice points lie(...TRUNCATED) | 98 | ["Alright, so I have this problem: I need to find how many lattice points lie on the hyperbola given(...TRUNCATED) | ["32","98",". as.","50"," by a. because.but","100",".but.","98","98","98",",?). but","98","perhaps p(...TRUNCATED) | [
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