2020 AMC 12B Problems/Problem 8
Contents
Problem
How many ordered pairs of integers satisfy the equation
Solution
Set it up as a quadratic in terms of y: Then the discriminant is This will clearly only yield real solutions when , because it is always positive. Then . Checking each one: and are the same when raised to the 2020th power: This has only has solutions , so are solutions. Next, if : Which has 2 solutions, so and
These are the only 4 solutions, so
Solution 2
Move the term to the other side to get . Because for all , then . If or , the right side is and therefore . When , the right side become , therefore . Our solutions are , , , . There are solutions, so the answer is
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.