## SUBJECT II (7 p)

The numbers are given:

$S_{1}=1+2+3+4+\cdots+(n-1)+n$ and

$S_{2}=1-2+3-4+\cdots-(n-1)+n$, where $n$ is an odd natural number.

a) Determine the number $n$ knowing that $S_{1}=2015 \cdot S_{2}$;

b) What is the smallest non-zero natural number by which $S_{1}+S_{2}$ must be multiplied so that a perfect square is obtained;

c) Find the natural numbers $n$ and $m$, with $n$ odd, such that $S_{1}-S_{2}=2^{m}$.
Solution:
a) $n=2 k+1, k \in \mathbb{N}, S_{1}=(2 k+1)(k+1), S_{2}=k+1$

$$
(2 \mathrm{k}+1)(\mathrm{k}+1)=2015(\mathrm{k}+1) \Rightarrow \mathrm{n}=2015
$$

b) $S_{1}+S_{2}=2(k+1)^{2}$. The number sought is 2 ............................................ $2 \mathrm{p}$

c) $S_{1}-S_{2}=2 k(k+1), k \in \mathbb{N}$

$k(k+1)=2^{m-1}, k \in \mathbb{N}, m \in \mathbb{N}^{*}$

$k=1, m=2$ (justification) $\Rightarrow n=3$

$.2 \mathrm{p}$