Problem #PRU-116775

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Algorithm Theory Theory of algorithms (other)

13–16 4.0

Problem

We are given a polynomial $P (x)$ and numbers $a_{1}$ , $a_{2}$ , $a_{3}$ , $b_{1}$ , $b_{2}$ , $b_{3}$ such that $a_{1} a_{2} a_{3} \neq 0$ . It turned out that $P (a_{1} x + b_{1}) + P (a_{2} x + b_{2}) = P (a_{3} x + b_{3})$ for any real $x$ . Prove that $P (x)$ has at least one real root.

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