Difference between revisions of "1979 USAMO Problems/Problem 1"
(Wow, Number theory was easier way back when...) |
(→Problem) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | Determine all non-negative integral solutions <math>(n_1,n_2,\dots , n_{14})</math> if any, apart from permutations, of the Diophantine Equation <math>n_1^4+n_2^4+\cdots +n_{14}^4=1599</math> | + | Determine all non-negative integral solutions <math>(n_1,n_2,\dots , n_{14})</math> if any, apart from permutations, of the Diophantine Equation <math>n_1^4+n_2^4+\cdots +n_{14}^4=1599</math>. |
== Solution == | == Solution == | ||
Revision as of 22:40, 13 April 2011
Problem
Determine all non-negative integral solutions 
if any, apart from permutations, of the Diophantine Equation 
.
Solution
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
Recall that 
for all integers 
. Thus the sum we have is anything from 0-14 mod16. But 1599=15mod16, and thus there are no integral solutions to the given Diophantine equation.
See also
| 1979 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
