MathDB

4

Part of 2023 JBMO TST - Turkey

Problems(2)

A game with balls and boxes

Source: Turkey JBMO TST 2023 Day 1 P4

4/30/2023
Initially, Aslı distributes 10001000 balls to 3030 boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
combinatorics
number of divisiors with divisibility

Source: Turkey JBMO TST 2023 Day 2 P4

4/30/2023
For a prime number pp. Can the number of n positive integers that make the expression n3+np+1n+p+1\dfrac{n^3+np+1}{n+p+1} an integer be 777777?
number theoryDivisibility