Problem #PRU-116775

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms 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 \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.