MathDB

6

Part of 2020 Durer Math Competition Finals

Problems(4)

counting up to m game

Source: 2020 Dürer Math Competition Finals E1. 6

11/29/2020
(Game) Károly and Dezso wish to count up to mm and play the following game in the meantime: they start from 00 and the two players can add a positive number less than 1313 to the previous number, taking turns. However because of their superstition, if one of them added xx, then the other one in the next step cannot add 13−x13-x. Whoever reaches (or surpasses) mm first, loses. Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.
gamecombinatorics
modified version of Pascal's triangle

Source: 2020 Dürer Math Competition Finals Day2 E6 https://artofproblemsolving.com/community/c1622639_2020_

1/7/2022
We build a modified version of Pascal’s triangle as follows: in the first row we write a 22 and a 33, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a 22 and end with a 33). In the 1313th row, what is the 55th number from the left? https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png
algebraSum
2player game with 4 piles of paper disks onto the table

Source: 2020 Dürer Math Competition Finals E+ 1. 6

11/30/2020
(Game) At the beginning of the game the organisers place 44 piles of paper disks onto the table. The player who is in turn takes away a pile, then divides one of the remaining piles into two nonempty piles. Whoever is unable to move, loses.
Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.
combinatoricsgamegame strategy
a divides b + c, b divides a + c , c divides a + b

Source: 2020 Dürer Math Competition Finals Day2 E+6 https://artofproblemsolving.com/community/c1622639_2020_

1/7/2022
Positive integers a,ba, b and cc are all less than 20202020. We know that aa divides b+cb + c, bb divides a+ca + c and cc divides a+ba + b. How many such ordered triples (a,b,c)(a, b, c) are there?
Note: In an ordered triple, the order of the numbers matters, so the ordered triple (0,1,2)(0, 1, 2) is not the same as the ordered triple (2,0,1)(2, 0, 1).
number theorydivides