MathDB

1

Part of 2015 European Mathematical Cup

Problems(2)

Travelling on a nxn board

Source: European Mathematical Cup, 2015, Junior, P1

12/30/2016
We are given an n×nn \times n board. Rows are labeled with numbers 11 to nn downwards and columns are labeled with numbers 11 to nn from left to right. On each field we write the number x2+y2x^2 + y^2 where (x,y)(x, y) are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following conditions is satisfied: [*] the numbers in those 22 fields give the same remainders when divided by nn, [*] those fields are point reflected with respect to the center of the board.Can all the fields be visited in case: [*] n=4n = 4, [*] n=5n = 5?
Josip Pupić
combinatoricsboardnumber theory
10 divides difference of products of powers

Source: European Mathematical Cup, 2015, Senior, P1

12/30/2016
A={a,b,c}A = \{a, b, c\} is a set containing three positive integers. Prove that we can find a set B⊂AB \subset A, B={x,y}B = \{x, y\} such that for all odd positive integers m,nm, n we have 10∣xmyn−xnym.10\mid x^my^n-x^ny^m.
Tomi Dimovski
number theorycombinatoricspigeonhole principle