2

Part of 1991 Irish Math Olympiad

Problems(2)

Functional equations problem

Source:

Problem: Find all polynomials satisfying the equation

f(x^2) = (f(x))^2

for all real numbers x. I'm not exactly sure where to start though it doesn't look too difficult. Thanks!

algebrapolynomialalgebra unsolved

Sequence and Inequality

Source: 1991 IrMO Paper 2 Problem 2

Let a_n=\frac{n^2+1}{\sqrt{n^4+4}}, n=1,2,3,\dots and let

b_n

be the product of

a_1,a_2,a_3,\dots ,a_n

\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},

and deduce that

\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}

for all positive integers

n