Mathematics and Youth Magazine Problems - Nov 2017, Issue 485

Let $a=n^{3}+2n$ and $b=n^{4}+3n^{2}+1$. For each $n\in\mathbb{N}$, find the greatest common divisor ($\gcd$) of $a$ and $b$.
Given an isoscesles triangle $ABC$ with $\widehat{A}=100^{0}$, $BC=a$, $AC=AB=b$. Outside $ABC$, we construct the isosceles triangle $ABD$ with $\widehat{ADB}=140^{0}$. Compute the perimeter of $ABD$ in terms of $a$ and $b$.
Find all pairs of intergers $x,y$ such that \[x