1

Part of 2001 Hungary-Israel Binational

Problems(2)

on equation $2000x^2 + y^2 = 2001z^2.$

Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

Find positive integers

x, y, z

x > z > 1999 \cdot 2000 \cdot 2001 > y

2000x^{2}+y^{2}= 2001z^{2}.

Diophantine equationnumber theory proposednumber theory

graph has got a triangle whose sides have di erent colors

Source: 12-th Hungary-Israel Mathematical Competition 2001

G_{n}

denotes a simple undirected graph with

n

K_{n}

denotes the complete graph with

n

K_{n,m}

the complete bipartite graph whose components have

m

n

C_{n}

n

vertices. The number of edges in the graph

G_{n}

e(G_{n})

K_{n}(n \geq 3)

are colored with

n

colors, and every color is used. Show that there is a triangle whose sides have different colors.

inductiongraph theorycombinatorics proposedcombinatorics