1979 USAMO Problems/Problem 1
Problem
Determine all non-negative integral solutions if any, apart from permutations, of the Diophantine Equation .
Solution 1
Recall that for all integers . Thus the sum we have is anything from 0 to 14 modulo 16. But , and thus there are no integral solutions to the given Diophantine equation.
Solution 2
In base , this equation would look like:
We notice that the unit digits of the LHS of this equation should equal to . In base , the only unit digits of fourth powers are and . Thus, the maximum of these terms is 14 or . Since is less than , there are no integral solutions for this equation.
Video Solution
https://youtu.be/zfChnbMGLVQ?t=4778
~ pi_is_3.14
See Also
1979 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.