Twentieth IMAR Mathematical Competition

As $a_0+a_1+\cdots +a_n=1$, the required inequality is equivalent to ${\sum }_{k=0}^{n}k{a}_k>m$. To prove this inequality, note that the exponential $t\mapsto x^t$, $t\in \mathbb{R}$, is convex and apply Jensen's inequality to write $x^{{\sum }_{k=0}^{n}k{a}_k}\le {\sum }_{k=0}^n{a}_k{x}^k<x^m$. As $0<x<1$, the desired inequality follows by comparing the exponents of $x$ at both ends.