Problem 1. Let $x$ and $y$ be two non-zero natural numbers such that $2015^{x}-2013=x+y$. Verify whether $y$ is a perfect square, given that:

$$
3 x+6 x+9 x+\ldots+102 x=3570
$$

Gratiela Popa, Slatina
## Solution and marking scheme

$3 x \cdot(1+2+\ldots+34)=3570$ ..... (1p)
$x=2$ ..... (2p)
$y=2014 \cdot 2015$ ..... (2p)
Justification that $y$ is not a perfect square ( $2014^{2}<y<2015^{2}$ or another method) ..... (2p)