id stringlengths 6 10 | year int64 1.98k 2.02k | problem_number int64 1 15 | Question stringlengths 215 1.67k | answer stringlengths 1 3 | text stringlengths 1.83k 71.5k |
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1983-1 | 1,983 | 1 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
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 $\log_zw$ .
<｜Assistant｜>... | 60 | Okay, so I've got this logarithm problem here, and I need to find log base z of w, given some other logs. Let me see. The problem says:
Let x, y, and z all exceed 1, and let w be a positive number such that log base x of w is 24, log base y of w is 40, and log base xyz of w is 12. We need to find log base z of w.
Hmm... |
1983-2 | 1,983 | 2 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
Let $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $0 < p < 15$ . Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$ .
<｜Assistant｜><think>
| 15 | Okay, so I have this function f(x) = |x - p| + |x - 15| + |x - p - 15|, and I need to find its minimum value when x is between p and 15. Hmm, let me try to figure this out step by step.
First off, I remember that functions involving absolute values can often be minimized by considering the points where the expressions... |
1983-3 | 1,983 | 3 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ ?
<｜Assistant｜><think>
| 20 | Okay, so I have this equation to solve: \( x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45} \). The question is asking for the product of the real roots of this equation. Hmm, that sounds like a quadratic equation but with a square root on one side. Let me try to figure out how to approach this.
First, I notice that both side... |
1983-4 | 1,983 | 4 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is... | 26 | Okay, so I need to find the square of the distance from point B to the center of the circle. Let me try to visualize this first. There's an asy diagram provided, but since I can't see images, I'll have to go off the description.
The shape is a notched circle, which probably means it's a circle with a part removed, cre... |
1983-5 | 1,983 | 5 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have... | 4 | Okay, so I have this problem here where the sum of the squares of two complex numbers x and y is 7, and the sum of their cubes is 10. I need to find the largest real value that x + y can have. Hmm, okay, let me think about how to approach this.
First, let me write down what I know. The problem states:
- \( x^2 + y^2 ... |
1983-6 | 1,983 | 6 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ .
<｜Assistant｜><think>
| 35 | Okay, so I need to find the remainder when \( a_{83} = 6^{83} + 8^{83} \) is divided by 49. Hmm, that seems like a modular arithmetic problem. Let me think about how to approach this.
First, I remember that when dealing with large exponents modulo some number, Euler's theorem or Fermat's little theorem might be helpfu... |
1983-7 | 1,983 | 7 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to sla... | 57 | Okay, so I have this problem where there are 25 knights sitting around a round table. We need to find the probability that if we randomly choose three knights, at least two of them were sitting next to each other. Then, express this probability as a reduced fraction and find the sum of the numerator and denominator.
H... |
1983-8 | 1,983 | 8 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ ?
<｜Assistant｜><think>
| 61 | Alright, so I have this problem here: I need to find the largest two-digit prime factor of the integer n, which is equal to the binomial coefficient "200 choose 100". That is, n = C(200, 100) = 200! / (100! * 100!). Hmm, okay.
First, let me recall what a binomial coefficient is. It's the number of ways to choose k el... |
1983-9 | 1,983 | 9 | <｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should provide your final answer in the format \boxed{YOUR_ANSWER}.
Question:
Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$ .
<｜Assistant｜><think>
| 12 | Okay, so I have this problem here: Find the minimum value of (9x² sin²x + 4)/(x sinx) for 0 < x < π. Hmm, that looks a bit complicated, but maybe I can break it down step by step.
First, let me write down the function clearly:
f(x) = (9x² sin²x + 4)/(x sinx)
I need to find its minimum value on the interval (0, π). S... |
1983-10 | 1,983 | 10 | "<｜begin▁of▁sentence｜><｜User｜>Please answer the following math question. You should prov(...TRUNCATED) | 432 | "Okay, so I've got this problem here where I need to find how many four-digit numbers start with 1 a(...TRUNCATED) |
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