id stringlengths 6 10 | year int64 1.98k 2.01k | problem_number int64 1 15 | Question stringlengths 215 1.3k | answer stringlengths 1 3 | text stringlengths 4.43k 84.5k |
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1983-1 | 1,983 | 1 | <|im_start|>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 $\log_zw$ .
<|im_end|>
<|im_start|>as... | 60 | <think>
Okay, so I need to find log_z w, given some logarithmic equations involving x, y, z, and w. Let me start by recalling some logarithm properties.
First, I remember that log_b a = c means that b^c = a. Also, there are change of base formulas, like log_b a = log_c a / log_c b. Maybe that will help here.
The pr... |
1983-2 | 1,983 | 2 | <|im_start|>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 $p \leq x\leq15$ .
<|im_end|>
<|im_start|>assistant
| 15 | <think>
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 me think about how to approach this.
First, I remember that absolute value functions can be tricky because they have different behaviors depending on the value... |
1983-3 | 1,983 | 3 | <|im_start|>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 $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ ?
<|im_end|>
<|im_start|>assistant
| 20 | <think>
Okay, so I need to find the product of the real roots of this equation: x² + 18x + 30 = 2√(x² + 18x + 45). Hmm, let me think about how to approach this.
First, maybe I can simplify the equation by substitution. Let me see... The equation has x² + 18x terms in both sides. Maybe if I let y = x² + 18x + somethin... |
1983-4 | 1,983 | 4 | <|im_start|>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 $2$ cm. The... | 26 | <think>
Okay, let's try to figure out this problem. So, there's a notched circle, and we need to find the square of the distance from point B to the center of the circle. The radius of the circle is √50 cm, AB is 6 cm, BC is 2 cm, and angle ABC is a right angle.
First, I need to visualize the diagram. The Asymptote c... |
1983-5 | 1,983 | 5 | <|im_start|>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?
<|im_end|... | 4 | <think>
Okay, so I need to find the largest real value that x + y can have, given that the sum of their squares is 7 and the sum of their cubes is 10. Hmm, let me think about how to approach this.
First, let me write down what's given:
Let x and y be complex numbers. Then:
x² + y² = 7
x³ + y³ = 10
And we need to ... |
1983-6 | 1,983 | 6 | <|im_start|>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 $49$ .
<|im_end|>
<|im_start|>assistant
| 35 | <think>
Okay, so I need to find the remainder when a₈₃ is divided by 49, where aₙ is defined as 6ⁿ + 8ⁿ. Hmm, let's see. First, I remember that when dealing with remainders, modular arithmetic can be really helpful. So, I need to compute (6⁸³ + 8⁸³) mod 49.
But calculating 6⁸³ and 8⁸³ directly seems impossible becaus... |
1983-7 | 1,983 | 7 | <|im_start|>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 slay a troubles... | 57 | <think>
Okay, so I need to find the probability that at least two of the three knights chosen are sitting next to each other at the round table. Then, express that probability as a reduced fraction and add the numerator and denominator. Let me think about how to approach this.
First, there are 25 knights seated around... |
1983-8 | 1,983 | 8 | <|im_start|>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 $n = {200\choose 100}$ ?
<|im_end|>
<|im_start|>assistant
| 61 | <think>
Okay, so I need to find the largest 2-digit prime factor of the binomial coefficient n = C(200, 100). Hmm, binomial coefficients can be really big, right? But maybe there's a way to figure out which primes divide them without calculating the entire number. Let me think.
First, I remember that for a prime numbe... |
1983-9 | 1,983 | 9 | <|im_start|>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 $0 < x < \pi$ .
<|im_end|>
<|im_start|>assistant
| 12 | <think>
Okay, so I need to find the minimum value of the function $ \frac{9x^2\sin^2 x + 4}{x\sin x} $ where $ 0 < x < \pi $. Hmm, let me think about how to approach this.
First, maybe I can simplify the expression. Let me write it down again:
$ \frac{9x^2\sin^2 x + 4}{x\sin x} $
I notice that the numerator is $ 9x... |
1983-10 | 1,983 | 10 | "<|im_start|>user\nPlease answer the following math question. You should provide your final answer i(...TRUNCATED) | 432 | "<think>\nOkay, so I need to figure out how many 4-digit numbers start with 1 and have exactly two i(...TRUNCATED) |
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