2

Part of 2019 Iran RMM TST

Problems(1)

Nice polynomial with gcd greater than 1

Source: Iran RMM TST 2019,day1 p2

n >1

be a natural number and

T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0

.\\ Assume that for each nonzero real number

t

T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n}

.\\ Prove that for each

0\le i \le n-1

gcd (a_i,n) >1

.
Proposed by Morteza Saghafian

polynomialalgebrachebyshev polynomial