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The Return of Triangle Geometry

Source: 2023 ISL A7

July 17, 2024

Problem Statement

Let NN be a positive integer. Prove that there exist three permutations a1,…,aNa_1,\dots,a_N, b1,…,bNb_1,\dots,b_N, and c1,…,cNc_1,\dots,c_N of 1,…,N1,\dots,N such that ∣ak+bk+ck−2N∣<2023\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023 for every k=1,2,…,Nk=1,2,\dots,N.
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