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n^k + mn^l + 1 divides n^{k+l }- 1

Source: 2017 Saudi Arabia IMO Training Test p1

September 4, 2020
number theorydivides

Problem Statement

Let m,n,km, n, k and ll be positive integers with n≠1n \ne 1 such that nk+mnl+1n^k + mn^l + 1 divides nk+l−1n^{k+l }- 1. Prove that either m=1m = 1 and l=2kl = 2k, or l∣kl | k and m=nk−l−1nl−1m =\frac{n^{k-l} - 1}{n^l - 1}.
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