3

Part of 2000 Romania Team Selection Test

Problems(3)

TST-Romania, 2000

Source: binomial coefficient

Prove that for any positive integers

n

k

there exist positive integers

a>b>c>d>e>k

n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}

number theory proposednumber theory

Romania 2000 TST

Source:

Determine all pairs

(m,n)

of positive integers such that a

m\times n

rectangle can be tiled with L-trominoes.

geometryrectanglecombinatorics proposedcombinatorics

Mapping of interior points of spehere and circle

Source: Romanian TST 2000

S

be the set of interior points of a sphere and

C

be the set of interior points of a circle. Find, with proof, whether there exists a function

f:S\rightarrow C

d(A,B)\le d(f(A),f(B))

for any two points

A,B\in S

d(X,Y)

denotes the distance between the points

X

Y

.
Marius Cavachi

geometry3D geometryspherefunctioncalculustopologyanalytic geometry