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Polynomial roots and rational numbers squared

Source: 2015 Iberoamerican Olympiad. Problem 3

November 10, 2015
algebrapolynomial

Problem Statement

Let α\alpha and β\beta be the roots of x2−qx+1x^{2} - qx + 1, where qq is a rational number larger than 22. Let s1=α+βs_1 = \alpha + \beta, t1=1t_1 = 1, and for all integers n≥2n \geq 2:
sn=αn+βns_n = \alpha^n + \beta^n
tn=sn−1+2sn−2+⋅⋅⋅+(n−1)s1+nt_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n
Prove that, for all odd integers nn, tnt_n is the square of a rational number.
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