MathDB

6

Part of 2010 Thailand Mathematical Olympiad

Problems(2)

f(-x) = f(x) if f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)

Source: 2010 Thailand Mathematical Olympiad day 1 p6

3/12/2021
Let f:R→Rf : R \to R be a function satisfying the functional equation f(3x+y)+f(3x−y)=f(x+y)+f(x−y)+16f(x)f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x) for all reals x,yx, y. Show that ff is even, that is, f(−x)=f(x)f(-x) = f(x) for all reals xx
algebrafunctional equationfunctional
no triples of primes p, q, r satisfy p > r, q > r, and pq | r^p + r^q

Source: Thailand Mathematical Olympiad 2010 day 2 p6

8/17/2020
Show that no triples of primes p,q,rp, q, r satisfy p>r,q>rp > r, q > r, and pq∣rp+rqpq | r^p + r^q
primesnumber theoryDivide