There are seven piles with $2014$ pebbles each and a pile with $2008$ pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between $1$ and $8$ pebbles. The first player who cannot make a move loses. a) Who has a winning strategy? b) If there were seven piles with $2015$ pebbles each and a pile with $2008$ pebbles, who has a winning strategy?

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

There are $2024$ mathematicians sitting in a row next to the river Tisza. Each of them is working on exactly one research topic, and if two mathematicians are working on the same topic, everyone sitting between them is also working on it. Marvin is trying to figure out for each pair of mathematicians whether they are working on the same topic. He is allowed to ask each mathematician the following question: “How many of these 2024 mathematicians are working on your topic?” He asks the questions one by one, so he knows all previous answers before he asks the next one. Determine the smallest positive integer $k$ such that Marvin can always accomplish his goal with at most $k$ questions.

(a) Two people perform a card trick. The first performer takes $5$ cards from a $52$-card deck (previously shuffled by a member of the audience) , looks at them, and arranges them in a row from left to right: one face down (not necessarily the first one) , the others face up . The second performer guesses correctly the card which is face down. Prove that the performers can agree on a system which always makes this possible. (b) For their second trick, the first performer arranges four cards in a row, face up, the fifth card is kept hidden. Can they still agree on a system which enables the second performer to correctly guess the hidden card? (G Galperin)