CMOQR 2022

Let $n \geq 2$ be a positive integer. On a spaceship, there are $n$ crewmates. At most one accusation of being an imposter can occur from one crewmate to another crewmate. Multiple accusations are thrown, with the following properties: • Each crewmate made a different number of accusations. • Each crewmate received a different number of accusations. • A crewmate does not accuse themself. Prove that no two crewmates made accusations at each other.

Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

Consider n real numbers $x_0, x_1, . . . , x_{n-1}$ for an integer $n \ge 2$ . Moreover, suppose that for any integer $i$ , $x_{i+n} = x_i$ . Prove that $$\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) \ge 0.$$

For a non-negative integer $n$ , call a one-variable polynomial $F$ with integer coefficients $n$ -good if: (a) $F(0) = 1$ (b) For every positive integer $c$ , $F(c) > 0$ , and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$ -good for any $n$ .