## Problem 4

The smallest angle of a convex polygon measures $132^{\circ}$. Find the number of sides of the polygon, knowing that the measures of its angles are numbers in an arithmetic progression with a common difference of $2^{\circ}$. (It is known that the sum of the measures of the angles of a convex polygon with $n$ sides, $n \geq 3$, is $(n-2) \cdot 180^{\circ}$.

## Note

- Effective working time 3 hours.
- All problems are mandatory.
- 7 points are awarded for each correctly solved problem.

## National Mathematics Olympiad  Local stage -14.02.2015

## Grade IX $\mathrm{M}_{1}$

Solutions and marking schemes
## Problem 4

$a_{1}=132^{\circ}, r=2^{\circ}, S_{n}=\frac{\left[2 a_{1}+(n-1) r\right] \cdot n}{2}=n\left(n+131^{\circ}\right)$

$4 p$

Given $S_{n}=(n-2) \cdot 180^{\circ} \Rightarrow(n-2) \cdot 180^{\circ}=n\left(n+131^{\circ}\right)$ $.2 p$

Thus $n_{1}=1<3$ is not valid and $n_{2}=40 \geq 3$ is the solution.... $.1 p$