2

Part of 2017 Bulgaria EGMO TST

Problems(2)

Challenging geometry with incircles

Source: Bulgaria EGMO TST 2017 Day 1 Problem 2

ABC

be a triangle with incenter

I

AI

BC

and the circumcircle of

ABC

T

S

, respectively. Let

K

L

be the incenters of

SBT

SCT

, respectively,

M

be the midpoint of

BC

P

be the reflection of

I

with respect to

KL

. a) Prove that

M

T

K

L

are concyclic. b) Determine the measure of

\angle BPC

geometryincenterincircleexcirclecircumcirclegeometric transformationreflection

Polychromatic colouring of a table

Source: Bulgaria EGMO TST 2017 Day 2 Problem 2

n

be a positive integer. Determine the smallest positive integer

k

such that for any colouring of the cells of a

2n\times k

n

colours there are two rows and two columns which intersect in four squares of the same colour.

combinatoricscellscolouring