This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

1990 Romania Team Selection Test, 1
Tags: infinite set , finite set , function , algebra
Let $f : N \to N$ be a function such that the set $\{k | f(k) < k\}$ is finite. Prove that the set $\{k | g(f(k)) \le k\}$ is infinite for all functions $g : N \to N$.

2009 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , integer , finite set
Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.