Problem

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

Tags: algebra, polynomial, induction, function, modular arithmetic, algebra proposed



Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]