CMOQR 2017

Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all possible values for the first integer.

For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$ . $$(x+y)f(x-y) = f(x^2-y^2).$$

In this question we re-define the operations addition and multiplication as follows: $a + b$ is defined as the minimum of $a$ and $b$ , while $a * b$ is defined to be the sum of $a$ and $b$ . For example, $3+4 = 3$ , $3*4 = 7$ , and $$3*4^2+5*4+7 = \min(\text{3 plus 4 plus 4}, \text{5 plus 4}, 7) = \min(11, 9, 7) = 7.$$ Let $a, b, c$ be real numbers. Characterize, in terms of $a, b, c$ , what the graph of $y = ax^2+bx+c$ looks like.