CMOQR 2016

(a) Find all positive integers $n$ such that $11|(3^n + 4^n)$ .

(b) Find all positive integers $n$ such that $31|(4^n + 7^n + 20^n)$ .

Let $P = (7, 1)$ and let $O = (0, 0)$ .

(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$ -axis so that $P$ is on the line segment $ST$ , determine the minimum possible area of triangle $OST$ .

(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$ -axis so that $P$ is on the line segment $UV$ , determine the minimum possible perimeter of triangle $OUV$ .

Given an $n \times n \times n$ grid of unit cubes, a cube is good if it is a sub-cube of the grid and has side length at least two. If a good cube contains another good cube and their faces do not intersect, the first good cube is said to properly contain the second. What is the size of the largest possible set of good cubes such that no cube in the set properly contains another cube in the set?

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x + f(y)) + f(x - f(y)) = x.$$