ID stringlengths 6 10 | Year int64 1.98k 2.02k | Problem Number int64 1 15 | Part stringclasses 2
values | problem stringlengths 37 2.66k | answer stringlengths 1 31 | completion sequencelengths 64 64 | completion_answer sequencelengths 64 64 | completion_correct sequencelengths 64 64 | completion_succ_rate float64 0 1 |
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1983-1 | 1,983 | 1 | null | Let $x$ , $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_xw=24$ , $\log_y w = 40$ and $\log_{xyz}w=12$ . Find $\log_zw$ . | 60 | [
"To solve the problem, we start by converting the logarithmic equations into exponential form and then manipulate them to find the desired logarithm.\n\n## Step 1: Convert the given logarithmic equations to exponential form\nGiven:\n\\[ \\log_x w = 24 \\]\n\\[ \\log_y w = 40 \\]\n\\[ \\log_{xyz} w = 12 \\]\n\nThese... | [
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1983-2 | 1,983 | 2 | null | "Let $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $0 < p < 15$ . Determine the minimum value taken by $f(x)$(...TRUNCATED) | 15 | ["To determine the minimum value of the function \\( f(x) = |x-p| + |x-15| + |x-p-15| \\) for \\( x (...TRUNCATED) | ["15","15","15","15","15","15","15","15","15","15","15","15","15","15","15","15","15","15","15","15"(...TRUNCATED) | [true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true(...TRUNCATED) | 1 |
1983-3 | 1,983 | 3 | null | "What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$ ?(...TRUNCATED) | 20 | ["## Step 1: Simplify the equation\nLet $y = x^2 + 18x + 30$. The equation becomes $y = 2 \\sqrt{y +(...TRUNCATED) | ["20","20","20","45","36","20","20","20","20","20","20","20","20","20","20","20","20","20","-114","2(...TRUNCATED) | [true,true,true,false,false,true,true,true,true,true,true,true,true,true,true,true,true,true,false,t(...TRUNCATED) | 0.84375 |
1983-4 | 1,983 | 4 | null | "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle i(...TRUNCATED) | 26 | ["To find the square of the distance from $B$ to the center of the circle, we need to follow these s(...TRUNCATED) | ["26","26","26","26","46","14","26","46","26","26","26","14","26","26","26","26","26","34","26","40"(...TRUNCATED) | [true,true,true,true,false,false,true,false,true,true,true,false,true,true,true,true,true,false,true(...TRUNCATED) | 0.6875 |
1983-5 | 1,983 | 5 | null | "Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cu(...TRUNCATED) | 4 | ["To solve this problem, we start by letting the two complex numbers be \\(x\\) and \\(y\\). We are (...TRUNCATED) | ["4","4","5","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4","4"(...TRUNCATED) | [true,true,false,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,true,tru(...TRUNCATED) | 0.953125 |
1983-6 | 1,983 | 6 | null | Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ . | 35 | ["To find the remainder when \\(a_{83} = 6^{83} + 8^{83}\\) is divided by 49, we can use properties (...TRUNCATED) | ["0","0","4","21","0","14","35","7","10","0","35","29","0","0","17","21","35","0","0","0","0","26","(...TRUNCATED) | [false,false,false,false,false,false,true,false,false,false,true,false,false,false,false,false,true,(...TRUNCATED) | 0.078125 |
1983-7 | 1,983 | 7 | null | "Twenty five of King Arthur's knights are seated at their customary round table. Three of them are c(...TRUNCATED) | 57 | ["## Step 1: Calculate the total number of ways to choose 3 knights out of 25\nTo find the total num(...TRUNCATED) | ["57","115","153","153","57","2857","115","113","2829","57","57","153","173","115","115","153","327"(...TRUNCATED) | [true,false,false,false,true,false,false,false,false,true,true,false,false,false,false,false,false,f(...TRUNCATED) | 0.109375 |
1983-8 | 1,983 | 8 | null | What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ ? | 61 | ["To find the largest 2-digit prime factor of \\( n = \\binom{200}{100} \\), we need to consider the(...TRUNCATED) | ["61","61","61","61","97","61","61","61","61","61","61","61","61","61","61","61","61","61","61","61"(...TRUNCATED) | [true,true,true,true,false,true,true,true,true,true,true,true,true,true,true,true,true,true,true,tru(...TRUNCATED) | 0.90625 |
1983-9 | 1,983 | 9 | null | Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$ . | 12 | ["To find the minimum value of the function \\( f(x) = \\frac{9x^2 \\sin^2 x + 4}{x \\sin x} \\) for(...TRUNCATED) | ["12","12","12","12","12","12","12","12","12","12","12","\\frac{9\\pi}{2} + \\frac{8}{\\pi}","12","1(...TRUNCATED) | [true,true,true,true,true,true,true,true,true,true,true,false,true,true,true,true,true,true,true,tru(...TRUNCATED) | 0.984375 |
1983-10 | 1,983 | 10 | null | "The numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginn(...TRUNCATED) | 432 | ["## Step 1: Identify the structure of the numbers\nThe numbers must be 4-digit, start with 1, and h(...TRUNCATED) | ["432","216","432","243","1512","216","216","432","243","216","153","243","270","216","459","288","2(...TRUNCATED) | [true,false,true,false,false,false,false,true,false,false,false,false,false,false,false,false,false,(...TRUNCATED) | 0.171875 |
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