MathDB

2

Part of 2016 Czech-Polish-Slovak Match

Problems(2)

Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P2,day 1

7/12/2016
Let m,n>2m,n > 2 be even integers. Consider a board of size m \times n whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to fi nd the parity of the number of adjacent cell-pairs whose colors are diff erent. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
combinatorics
Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P2,day 2

7/12/2016
Prove that for every non-negative integer nn there exist integers x,y,zx, y, z with gcd(x,y,z)=1gcd(x, y, z) = 1, such that x2+y2+z2=32nx^2 + y^2 + z^2 = 3^{2^n}.(Poland)
number theory