Do there exist positive integers and such that both and are perfect squares?
Parallelogram has integer side lengths with . The bisectors at and intersect at a point which is outside of the parallelogram. What can be the length of ?
A teacher has a bag of 100 balls numbered from 1 to 100. She picks one ball at random, writes its number on the board, and puts it back in the bag. 100 students line up in alphabetical order, and each student takes a turn doing the same (pick a ball randomly, write its number on the board, return the ball to the bag). The first student to draw a ball whose number is already on the board wins a prize. Which student has the best chance of winning?
Alice and Bob are playing a game. At the beginning of the game, there are 2026 piles, consisting of 1, 2, 3, . . . , 2025, and 2026 blocks, respectively. The two players take turns, starting with Alice. Each player finds the pile with the most blocks (or one of such piles, if there are several) and proceeds to take any nonzero number of blocks from it. The player who takes the last block wins. Which player has a winning strategy?
The captain of a pirate ship wants to distribute pieces of a gold bar among his three crew mates. He first picks a number with 0 < < 1, cuts off a piece that is a proportion of the gold bar, and gives that piece to one of the three crew members. At every next step, the captain cuts a piece that is a portion of the gold bar remaining from the previous step, and gives that piece to one of the members. (for example: if = 0.234, then the captain always cuts off 23.4% of the remaining gold bar.) After a certain number of steps, the captain stops cutting pieces and saves the remaining gold bar for himself. Is it possible that at this point each of his three crew mates has received the same amount of gold?