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2012 ToT Spring Senior A p3 2n-1| (...(((x^2+a_1)^2 +a_2)^2+...)^2+a_{n-1})^2

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March 5, 2020
number theorydividespolynomialdivisible

Problem Statement

Let nn be a positive integer. Prove that there exist integers a1,a2,...,ana_1, a_2,..., a_n such that for any integer xx, the number (...(((x2+a1)2+a2)2+...)2+an−1)2+an(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n is divisible by 2n−12n - 1.
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