MathDB

Problem 3

Part of 1998 Ukraine National Math Olympiad

Problems(3)

moving piece on nxn board (Ukraine 1998 Grade 8 P3)

Source:

6/3/2021
On the lower-left corner of an n×nn\times n board there is a piece. In a move a player can move the piece to the adjacent cell up, right, or up-right. Two players alternate moving the piece, and the winner is the one who places the piece on the upper-right corner. Who has a winning strategy?
combinatorics
1998 segments, some intersect, show existence of points

Source: Ukraine 1998 Grade 10 P3

6/5/2021
A finite set of segments on a line has the following property: In any subset of 19981998 segments there are two having a common point. Show that there exist 19971997 points on the line such that each segment contains at least one of these points.
geometrycombinatoricscombinatorial geometry
sum of x/(1+y+zx) in [0,1]

Source: Ukraine 1998 Grade 11 P3

6/6/2021
For any numbers 0<x,y,z≤10<x,y,z\le1 prove the inequality x1+y+zx+y1+z+xy+z1+x+yz≤3x+y+z.\frac x{1+y+zx}+\frac y{1+z+xy}+\frac z{1+x+yz}\le\frac3{x+y+z}.
inequalities