Tolpygo A.K.

Two players are involved. The first player has 1000 even cards (2, 4, ..., 2000), and the second player has 1001 odd cards (1, 3, ..., 2001). They take turns, with the first player starting. A move consists of the following: the player whose turn it is lays down one of their cards, and the other player, after looking at it, lays down one of their cards; the player with the higher number on their card scores one point, and both laid-down cards are discarded. In total, there are 1000 moves (one card of the second player is not used). What is the maximum number of points each player can guarantee for themselves (regardless of how their opponent plays)?
Let's call a move critical if the card 2001 is used. We will prove that the first player can score points on all of the second player's moves, except, perhaps, the critical one.

At any moment, we will denote the first player's cards as $a_{1}b_{k}$. We will prove that the first player can maintain this situation until the critical move.

Specifically, on their turn, the first player plays with $a_{1}$, and they are left with the cards $A_{k}=a_{k+1}$ ( $k=1,2, \ldots, n-1$ ); for any response, the remaining cards of the second player satisfy $B_{k} \leq b_{k+1}b_{k+1}$. Similarly, we verify that the first player can maintain this situation until the end of the game: on their turn, they give up $a_{1}$, and on the second player's turn, they place $a_{k}$ on $b_{k+1}$ (and $a_{1}$ on $\left.b_{1}\right)$.

Now, let's prove that the second player can score points on all of the first player's moves plus on their last move. In the same notation, at the beginning, the inequalities $b_{k+1}>a_{k}$ are satisfied. Acting according to the strategy outlined above (on their turn - place $b_{1}$, on the first player's turn - place $b_{k+1}$ on $a_{k}$), the second player maintains this situation and scores points on all of the first player's moves. Before the last move, the second player is left with the largest card at that moment, $b_{2}$.

By placing it, they score an additional point.

## Answer

First player - 499 points, second player - 501.