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FE in integers

Source: 2019 Belarus Team Selection Test 8.2

September 2, 2019
algebrafunctional equationfunction

Problem Statement

Let Z\mathbb Z be the set of all integers. Find all functions f:Z→Zf:\mathbb Z\to\mathbb Z satisfying the following conditions: 1. f(f(x))=xf(x)−x2+2f(f(x))=xf(x)-x^2+2 for all x∈Zx\in\mathbb Z; 2. ff takes all integer values.
(I. Voronovich)
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