MathDB

arithmetic progressions

Source: 2011 Romania TST Problem 2

2/4/2012

Prove that the set

S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}

contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
Vasile Pop

floor functionpigeonhole principlearithmetic sequenceRamsey Theorynumber theory proposednumber theory

Concurrent on the Euler line

Source: Gazeta Matematica - Romania TST 2011- Second exam - P2

3/5/2012

In triangle

ABC

, the incircle touches sides

BC,CA

and

AB

in

D,E

and

F

respectively. Let

X

be the feet of the altitude of the vertex

D

on side

EF

of triangle

DEF

. Prove that

AX,BY

and

CZ

are concurrent on the Euler line of the triangle

DEF

.

Eulergeometrysearchcircumcircleincentergeometric transformationhomothety

A matrix of 0s and 1s

Source: American Mathematical Monthly

4/9/2012

Given a prime number

p

congruent to

1

modulo

5

such that

2p+1

is also prime, show that there exists a matrix of

0

s and

1

s containing exactly

4p

(respectively,

4p+2

)

1

s no sub-matrix of which contains exactly

2p

(respectively,

2p+1

)

1

s.

linear algebramatrixnumber theory proposednumber theory

2