MathDB

4

Part of 2019 Final Mathematical Cup

Problems(2)

a grasshopper moves along a nxn square net

Source: 1st Final Mathematical Cup 2019 FMC , juniors p4 / seniors p2

10/6/2020
Let n≥2n \ge 2 be a positive integer. A grasshopper is moving along the sides of an n×nn \times n square net, which is divided on n2n^2 unit squares. It moves so that а) in every 1×11 \times 1 unit square of the net, it passes only through one side b) when it passes one side of 1×11 \times1 unit square of the net, it jumps on a vertex on another arbitrary 1×11 \times 1 unit square of the net, which does not have a side on which the grasshopper moved along. The grasshopper moves until the conditions can be fulfilled. What is the shortest and the longest path that the grasshopper can go through if it moves according to the condition of the problem? Calculate its length and draw it on the net.
combinatorics
2 sheets of paper, n numbers on the one, m numbers on the other, sums

Source: 1st Final Mathematical Cup 2019 FMC ,seniors p4

10/6/2020
On two sheets of paper are written more than one positive integers. On the first paper nn numbers are written and on the second paper mm numbers are written. If one number is written on any of the papers then on the first paper is written also the sum of that number and 1313, and on the second paper the difference of that number and 2323. Calculate the value of mn\frac{m}{n}. .
combinatorics