Question 1
Let $a,b$ satisfy $a\ge b\gt 0$ and $a+b=1$.
(a) Prove that if $m,n$ are positive integers satisfying $m\gt n$, then $a^m-a^n\ge b^m-b^n\gt 0$.
(b) For every positive integer $n$, form the quadratic
$$f_n(x)=x^2-b^nx-a^n.$$
Prove that $f_n(x)$ has two distinct roots lying between $-1$ and $1$.
Editorial notice: The original assertions are retained verbatim in mathematical content. They contain defects; see the solution discussion.
(a) Prove that if $m,n$ are positive integers satisfying $m\gt n$, then $a^m-a^n\ge b^m-b^n\gt 0$.
(b) For every positive integer $n$, form the quadratic
$$f_n(x)=x^2-b^nx-a^n.$$
Prove that $f_n(x)$ has two distinct roots lying between $-1$ and $1$.
Editorial notice: The original assertions are retained verbatim in mathematical content. They contain defects; see the solution discussion.