Olympiad problems

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2010 IFYM, Sozopol, 4

Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.

2015 Indonesia MO Shortlist, A2

Tags: polynomial , functional equation , Natural Number , algebra

Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: number theory , Divisors , Natural Number , algebra , polynomial

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

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