## SUBJECT IV

Matei and Paul play the following game:

"A number 16 is written on a board. Matei adds to the number 16 any number from 1 to 9 and writes the result of the sum on the board. Then, Paul adds to the number written by Matei on the board any number from 1 to 9 and writes the result on the board. The game will continue in the same way (each of the competitors adds to the last number written by the other on the board any number from 1 to 9). The winner is the one who reaches 2014 first."

What will be Matei's strategy to win the game?

## MATHEMATICS OLYMPIAD  LOCAL STAGE
## SUBJECT IV

| It is observed that $2014=10 \cdot 199+24$ | $\mathbf{2 p}$ |
| :--- | :---: |
| First, Matei will add 8 and write the number $16+8=24$. |  |
| If Paul adds $\mathrm{x} \in\{1,2, \ldots, 9\}$, then on the second move, Matei will add $10-\mathrm{x}$ and write on the board the number $24+\mathrm{x}+10-\mathrm{x}=34$ and so on. | $\mathbf{2 p}$ |
| At each stage, whenever Paul adds y, Matei will add 10 -y. | $\mathbf{1 p}$ |
| Thus, on the 199th move, Matei writes on the board the number $4+10 \cdot 200=2004$. Then, Paul will write the number $2004+y, y \in\{1,2, \ldots, 9\}$, and Matei will add 10 -y to it and write 2014 on the board. | $\mathbf{2 p}$ |
| TOTAL | $\mathbf{7 p}$ |