Difference between revisions of "1976 USAMO Problems/Problem 3"

Latest revision as of 23:00, 9 October 2020

Problem

Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$ .

Solution

Either $a^2=0$ or $a^2>0$ . If $a^2=0$ , then $b^2=c^2=0$ . Symmetry applies for $b$ as well. If $a^2,b^2\neq 0$ , then $c^2\neq 0$ . Now we look at $a^2\bmod{4}$ :

$a^2\equiv 0\bmod{4}$ : Since a square is either 1 or 0 mod 4, then all the other squares are 0 mod 4. Let $a=2a_1$ , $b=2b_1$ , and $c=2c_1$ . Thus $a_1^2+b_1^2+c_1^2=4a_1^2b_1^2$ . Since the LHS is divisible by four, all the variables are divisible by 4, and we must do this over and over again, and from infinite descent, there are no non-zero solutions when $a^2\equiv 0\bmod{4}$ .

$a^2\equiv 1\bmod{4}$ : Since $b^2\neq 0\bmod{4}$ , $b^2\equiv 1\bmod{4}$ , and $2+c^2\equiv 1\bmod{4}$ . But for this to be true, $c^2\equiv 3\bmod{4}$ , which is an impossibility. Thus there are no non-zero solutions when $a^2\equiv 1\bmod{4}$ .

Thus the only solution is the solution above: $(a,b,c)=0$ .

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 15: / Line 15: @@
 {{alternate solutions}}
 == See Also ==
 {{USAMO box|year=1976|num-b=2|num-a=4}}
+{{MAA Notice}}
 [[Category:Olympiad Number Theory Problems]]

1976 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "1976 USAMO Problems/Problem 3"

Latest revision as of 23:00, 9 October 2020

Problem

Solution

See Also