MathDB

3

Part of 2020 Final Mathematical Cup

Problems(2)

Shading a Square Grid

Source: 2nd Final Mathematical Cup Junior Division P3 (2020)

10/1/2020
Let kk,nn be positive integers, k,n>1k,n>1, k<nk<n and a n×nn \times n grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer nn , such that they can obtain a configuration in which each row and column contains exactly kk black cells. Draw one example.
combinatoricssquare grid
Constant Sum of Numbers in a Book

Source: 2nd Final Mathematical Cup Senior Division P3 (2020)

10/1/2020
Given a paper on which the numbers 1,2,3…,14,151,2,3\dots ,14,15 are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say xx and yy) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of xy(x+y)xy(x+y) in his book. They were so bored that they both performed the operation until only 11 number remained. Then Bobby adds up all the numbers he wrote in his book, let’s call kk as the sum. aa. Prove that kk is constant which means it does not matter how they perform the operation, bb. Find the value of kk.
combinatoricsinvariant