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Found problems: 1

1990 Romania Team Selection Test, 1
Tags: finite set , Infinite 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 $g : N \to N$.