MathDB

7

Part of 2019 Tournament Of Towns

Problems(3)

100 piles of 400 stones each, game with awarded no of points

Source: Tournament of Towns, Junior A-Level , Spring 2019 p7

5/13/2020
There are 100100 piles of 400400 stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is 00?
(Maxim Didin)
gamecombinatoricsgame strategy
a square 101 x101 with all cells except one corner cell are black

Source: Tournament of Towns, Junior A-Level , Fall 2019 p7

4/20/2020
Peter has a wooden square stamp divided into a grid. He coated some 102102 cells of this grid with black ink. After that, he pressed this stamp 100100 times on a list of paper so that each time just those 102102 cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square 101×101101 \times 101 such that all the cells except one corner cell are black?
(Alexsandr Gribalko)
gridColoringcombinatorics
Quantity phi(n)

Source: ToT 2019

12/25/2019
We color some positive integers (1,2,...,n)(1,2,...,n) with color red, such that any triple of red numbers (a,b,c)(a,b,c)(not necessarily distincts) if a(b−c)a(b-c) is multiple of nn then b=cb=c. Prove that the quantity of red numbers is less than or equal to φ(n)\varphi(n).
number theorycombinatorics