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Polynomial having at most n real roots

Source: Iran 3rd round 2012-Algebra exam-P1

September 20, 2012
algebrapolynomialinductionfunctionmodular arithmeticalgebra proposed

Problem Statement

Suppose 0<m1<...<mn0<m_1<...<m_n and mi≡i(mod  2)m_i \equiv i (\mod 2). Prove that the following polynomial has at most nn real roots. (∀1≤i≤n:ai∈R\forall 1\le i \le n: a_i \in \mathbb R). a0+a1xm1+a2xm2+...+anxmn.a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.
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