MathDB

3

Part of 2020 European Mathematical Cup

Problems(2)

Tiling a board with F tiles and Z tiles

Source: European Mathematical Cup 2020, Problem J3

12/22/2020
Two types of tiles, depicted on the figure below, are given. https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG
Find all positive integers nn such that an n×nn\times n board consisting of n2n^2 unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the n×nn\times n board. \\ Proposed by Art Waeterschoot
combinatoricsboardTilingemcColoring
game on sequence

Source: 9th EMC, 12th December 2020 - 20th December 2020. SENIOR league, P3.

12/22/2020
Let pp be a prime number. Troy and Abed are playing a game. Troy writes a positive integer XX on the board, and gives a sequence (an)n∈N(a_n)_{n\in\mathbb{N}} of positive integers to Abed. Abed now makes a sequence of moves. The nn-th move is the following:  Replace Y currently written on the board with either Y+an or Y⋅an.\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n. Abed wins if at some point the number on the board is a multiple of pp. Determine whether Abed can win, regardless of Troy’s choices, if a)p=109+7a) p = 10^9 + 7; b)p=109+9b) p = 10^9 + 9. Remark: Both 109+710^9 + 7 and 109+910^9 + 9 are prime.
Proposed by Ivan Novak
number theorygameprime numbers