MathDB

1

Part of 2018 Dutch IMO TST

Problems(3)

a pawn moves on 2m x 2n grid, a snail wants to avoid the pawn's square

Source: Dutch IMO TST 2018 day 1 p1

8/30/2019
Suppose a grid with 2m2m rows and 2n2n columns is given, where mm and nn are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to fi nish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?
combinatoricsgrid
c(a^3+b^3)= a(b^3+c^3)= b(c^3+a^3), a(a^3+b^3)= b(b^3+c^3)= c(c^3+a^3)

Source: Dutch IMO TST 2018 day 2 p1

8/30/2019
(a) If c(a3+b3)=a(b3+c3)=b(c3+a3)c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3) with a,b,ca, b, c positive real numbers, does a=b=ca = b = c necessarily hold? (b) If a(a3+b3)=b(b3+c3)=c(c3+a3)a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3) with a,b,ca, b, c positive real numbers, does a=b=ca = b = c necessarily hold?
system of equationsalgebra
nice set of lines if every lines intersects odd no of other lines of set

Source: Dutch IMO TST 2018 day 3 p1

8/30/2019
A set of lines in the plan is called nice i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer k≥0k \ge 0 having the following property: for each 20182018 distinct lines ℓ1,ℓ2,...,ℓ2018\ell_1, \ell_2, ..., \ell_{2018} in the plane, there exist lines ℓ2018+1,ℓ2018+2,...,ℓ2018+k\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k} such that the lines ℓ1,ℓ2,...,ℓ2018+k\ell_1, \ell_2, ..., \ell_{2018+k} are distinct and form a nice set.
combinatorial geometrycombinatoricslines