1985 AIME Problems/Problem 3
Problem
Find if , , and are positive integers which satisfy , where .
Solution
Expanding out both sides of the given equation we have . Two complex numbers are equal if and only if their real parts and imaginary parts are equal, so and . Since are integers, this means is a divisor of 107, which is a prime number. Thus either or . If , so , but is not divisible by 3, a contradiction. Thus we must have , so and (since we know is positive). Thus .
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=mw2A1Fa7APM
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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