<s>[INST] On the divisors of $a^b + (\frac{a-b}2)^a + (\frac{b-a}2)^b+ b^a $ Let $a$ and $b$ be two odd natural numbers. Show that $\frac{a+b}{2}$ divides $$a^b  + \left(\frac{a-b}2\right)^a + \left(\frac{b-a}2\right)^b+ b^a$$
 [/INST] Well, that's simple. You need just one fact (easy to prove by induction): $x+y$ divides $x^n+y^n$ if $n$ is odd.
Now, $a^b+\left({b-a\over2}\right)^b$ is divisible by $a+{b-a\over2}={a+b\over2}$, and $b^a+\left({a-b\over2}\right)^a$ is divisible by $b+{a-b\over2}$ which also happens to be ${a+b\over2}$, hence so must be the sum of the two.
</s>