2021 AIME II Problems/Problem 7
Problem
Let and be real numbers that satisfy the system of equations There exist relatively prime positive integers and such that Find .
Solution
From the fourth equation we get , substitute this into the third equation and you get . Hence . Solving we get or . From the first and second equation we get , if , substituting we get . If you try solving this you see that this does not have real solutions in , so must be . Since , . If , then the system and does not give you real solutions. So . From here you already know and , so you can solve for and pretty easily and see that . So the answer is .
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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