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INMO 2017 Question 6

Source: 2017 India National Olympiad

January 16, 2017
binomial coefficientsbinomial theoremgeometryalgebraIntegersHeron's formula

Problem Statement

Let n≥1n\ge 1 be an integer and consider the sum x=∑k≥0(n2k)2n−2k3k=(n0)2n+(n2)2n−2⋅3+(n4)2n−k⋅32+⋯.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+12x-1,2x,2x+1 form the sides of a triangle whose area and inradius are also integers.
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