Olympiad problems

2020 Bundeswettbewerb Mathematik, 2

Konstantin moves a knight on a $n \times n$- chess board from the lower left corner to the lower right corner with the minimal number of moves. Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves. For which values of $n$ do they need the same number of moves?

1999 Nordic, 3

Tags: lattice points , integer sequence , chess knight , number theory

The infinite integer plane $Z\times Z = Z^2$ consists of all number pairs $(x, y)$, where $x$ and $y$ are integers. Let $a$ and $b$ be non-negative integers. We call any move from a point $(x, y)$ to any of the points $(x\pm a, y \pm b)$ or $(x \pm b, y \pm a) $ a $(a, b)$-knight move. Determine all numbers $a$ and $b$, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only $(a, b)$-knight moves.

2017 Baltic Way, 8

Tags: chess knight , chessboard , combinatorics

A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell. The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

2011 Armenian Republican Olympiads, Problem 6

Tags: chess knight , chessboard , combinatorics

Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.

2008 Bulgarian Autumn Math Competition, Problem 9.4

Tags: combinatorics , chessboard , cell , chess knight

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? [i]Nikolai Beluhov[/i]