MathDB

2016 Bulgaria JBMO TST

Part of Bulgaria JBMO Team Selection Test

Subcontests

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Operation on a grid of equilateral triangles

Given is equilateral triangle ABCABC with side length n≥3n \geq 3. It is divided into n2n^2 identical small equilateral triangles with side length 11. On every vertex of the triangles there is a number. In a move we can choose a rhombus and add or subtract 11 from all 44 numbers on the vertices of the rhombus. Let point DD have coordinates (3,2)(3,2) where 33 is the number of the row and 22 is the position on it from left to right. On the vertices A,B,C,DA,B,C,D there are 11's and on the other vertices there are 00's. Is it possible, after some operations, all the numbers to become equal?

Combinatorics

Given is a table 4x4 and in every square there is 0 or 1. In a move we choose row or column and we change the numbers there. Call the square "zero" if we cannot decrease the number of zeroes in it. Call "degree of the square" the number zeroes in a "zero" square. Find all possible values of the degree.
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