1

Part of 2007 Bulgaria National Olympiad

Problems(2)

Inscribed circle (St. Petersburg, 2002)

Source: Bulgarian MO 2007, Day 1, Problem 1

The quadrilateral

ABCD

\angle BAD+\angle ADC>\pi

, is inscribed a circle with centre

I

. A line through

I

AB

CD

X

Y

respectively such that

IX=IY

AX\cdot DY=BX\cdot CY

geometrytrigonometrygeometric transformationreflectionrhombusinradiuscircumcircle

Good set for all positive integers

Source: Bulgarian MO 2007, Day 2, Problem 4

k>1

be a given positive integer. A set

S

of positive integers is called good if we can colour the set of positive integers in

k

colours such that each integer of

S

cannot be represented as sum of two positive integers of the same colour. Find the greatest

t

such that the set

S=\{a+1,a+2,\ldots ,a+t\}

is good for all positive integers

a

.
A. Ivanov, E. Kolev

combinatorics proposedcombinatorics