2017 USAJMO Problems/Problem 1

Problem

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$ .

Solution 1

Let $a = 2n-1$ and $b = 2n+1$ . We see that $(2n \pm 1)^2 = 4n^2-4n+1 \equiv 1 \pmod{4n}$ . Therefore, we have $(2n+1)^{2n-1} + (2n-1)^{2n+1} \equiv 2n + 1 + 2n - 1 = 4n \equiv 0 \pmod{4n}$ , as desired.

(Credits to laegolas)

Solution 2

Let $x$ be any odd number above 1. We have $x^2-1=(x-1)(x+1).$ Since $x-1$ is even, $x^2-1 \equiv 0 \pmod{2x+2}.$ This means that $x^{x+2}-x^x \equiv 0 \pmod{2x+2},$ and since x is odd, $x^{x+2}+(-x)^x \equiv 0 \pmod{2x+2},$ or $x^{x+2}+x+2^x \equiv 0 \pmod{2x+2}.$ This means for any odd x, the ordered triple $(x,x+2)$ satisfies the condition. Since there are infinitely many values of $x$ possible, there are infinitely many ordered triples, as desired.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2017 USAJMO Problems/Problem 1

Contents

Problem

Solution 1

Solution 2

See also