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 |
|---|---|---|---|---|---|
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 have this logarithm problem here, and I need to find log base z of w, which is written as log_z w. The given information is that log base x of w is 24, log base y of w is 40, and log base xyz of w is 12. All variables x, y, z are greater than 1, and w is a positive number. Hmm, okay, let's try to figure this... |
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 need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x - p - 15| where 0 < p < 15 and x is between p and 15. Hmm, let's break this down step by step.
First, I remember that absolute value functions create V-shaped graphs, and their minima or maxima occur at certain points depending on... |
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've got this geometry problem here about a machine-shop cutting tool shaped like a notched circle. Let me try to visualize it based on the description and the Asymptote code provided. There's a circle with radius sqrt(50) cm. Points A, B, and C form a right-angled triangle at B, with AB being 6 cm and BC bein... |
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 need to find the largest real value of \( x + y \) where \( x \) and \( y \) are complex numbers satisfying two conditions: \( x^2 + y^2 = 7 \) and \( x^3 + y^3 = 10 \). Hmm, this seems a bit tricky, but let's try breaking it down step by step.
First, I remember some algebraic identities that might help her... |
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 have this problem where I need to find the remainder when \( a_{83} = 6^{83} + 8^{83} \) is divided by 49. Hmm, that's a pretty big exponent, so calculating it directly isn't feasible. I remember something about modular arithmetic and maybe using Euler's theorem or Fermat's little theorem could help here. Le... |
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 | Alright, so I've got this problem here: 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 denominato... |
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 | Okay, so I have this problem: I need to find the largest two-digit prime factor of the binomial coefficient n, which is equal to "200 choose 100". That is, n = C(200, 100). Hmm, let me remember what a binomial coefficient is. It's the number of ways to choose 100 elements from a set of 200, right? So, C(n, k) = n! / (k... |
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: find the minimum value of the function f(x) = (9x² sin²x + 4)/(x sinx) where x is between 0 and pi. Hmm, that looks a bit complicated, but maybe I can simplify it first.
Let me write it down again:
f(x) = (9x² sin²x + 4)/(x sinx)
I notice that both terms in the numerator are multiples o... |
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) |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 8