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NT with quadratic equations

Source: Romanian District Olympiad 2024 9.3

March 10, 2024
number theoryquadratic equation

Problem Statement

Let nn be a composite positive integer. Let 1=d1<d2<⋯<dk=n1=d_1<d_2<\cdots<d_k=n be the positive divisors of n.n.{} Assume that the equations di+2x2−2di+1x+di=0d_{i+2}x^2-2d_{i+1}x+d_i=0 for i=1,…,k−2i=1,\ldots,k-2 all have real solutions. Prove that n=pk−1n=p^{k-1} for some prime number p.p.{}
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