MathDB

6

Part of 2023 Durer Math Competition Finals

Problems(2)

Nim-like game on four piles

Source: Dürer Competition Finals 2023/E+ 6

3/8/2023
Two players play a game on four piles of pebbles labeled with the numbers 1,2,3,41,2,3,4 respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers mm and nn with 1≤m<n≤41\leq m<n\leq 4, removes mm pebbles from pile nn, and places one pebble in each of the piles n−1,n−2,…,n−mn-1,n-2,\dots,n-m. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.
combinatoricsgame
many days of the year can be Adel&amp;rsquo;s birthday?

Source: (2022-) 2023 XVI 16th D&uuml;rer Math Competition Finals Day 2 E6

5/25/2024
In Eldorado a year has 2020 months, and each month has 2020 days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the 1919th day of the 44th month, she would say 4⋅19=764 \cdot 19 = 76.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
number theory