Problem

Source: 2017 India National Olympiad

Tags: binomial coefficients, binomial theorem, geometry, algebra, Integers, Heron's formula



Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.