4. There are 2015 coins on the table. Two players play the following game: they take turns; on a turn, the first player can take any odd number of coins from 1 to 99, and the second player can take any even number of coins from 2 to 100. The player who cannot make a move loses. How many coins should the first player take on the first move to then guarantee a win with an unchanging strategy? (20 points)

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# Solution:

The strategy of the first player: he takes 95 coins, and then on each move, he takes (101-x) coins, where $\boldsymbol{x}$ is the number of coins taken by the second player. Since $\boldsymbol{x}$ is even (by the condition), 101-x is odd. Then $2015-95=1920$, since 101-x+x=101 coins will be taken per move, 1920/101=19 (remainder 1). In total, there will be 19 moves, in which 101 coins will be taken each time. The second player will be left with 1 coin and will not be able to make a move.

Answer: 95.

| Points | Criteria for evaluating the solution to problem № 4 |
| :---: | :--- |
| $\mathbf{2 0}$ | A correct and justified sequence of all steps of the solution is provided.  All transformations and calculations are performed correctly.  The correct answer is obtained. |
| $\mathbf{1 4}$ | A correct sequence of all steps of the solution is provided.  There are gaps in the justification of the first player's strategy or  a computational error or typo that does not affect the further course of the solution.  As a result of this error or typo, an incorrect answer may be obtained. |
| $\mathbf{7}$ | The problem is not solved, but its solution is significantly advanced, i.e.:  - a significant part of the solution is performed correctly, possibly inaccurately (for example,  the first move of the first player is indicated);  - another part is either not performed or performed incorrectly, possibly even with  logical errors (for example, the first player's strategy is incorrectly determined).  At the same time, the solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the criteria for scoring 7 and 14 points. |