ΗΜΕΡΑ 1η
1. Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that set; and a sociable set of girls is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)
(Poland) Marek Cygan
2. Given a non-isosceles triangle 
, let 
, and 
denote the midpoints of the sides 
, and 
respectively. The circle 
and the line 
meet again at 
, and the circle 
and the line 
meet again at 
. Finally, the lines 
and 
meet at 
. Prove that the centroid 
of the triangle lies on the circle 
.
, let , and denote the midpoints of the sides , and respectively. The circle and the line meet again at , and the circle and the line meet again at . Finally, the lines and meet at . Prove that the centroid of the triangle lies on the circle .(United Kingdom) David Monk
3. Each positive integer is coloured red or blue. A function 
from the set of positive integers to itself has the following two properties:
from the set of positive integers to itself has the following two properties:(a) if 
, then 
; and
, then ; and(b) if 
and 
are (not necessarily distinct) positive integers of the same colour and 
, then 
.
and are (not necessarily distinct) positive integers of the same colour and , then .Prove that there exists a positive number 
such that 
for all positive integers 
.
such that for all positive integers .(United Kingdom) Ben Elliott
(Russia) Valery Senderov
2. Given a positive integer 
, colour each cell of an 
square array with one of 
colours, each colour being used at least once. Prove that there is some 
or 
rectangular subarray whose three cells are coloured with three different colours.
, colour each cell of an square array with one of colours, each colour being used at least once. Prove that there is some or rectangular subarray whose three cells are coloured with three different colours.(Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov
3. Let be a triangle and let 
and 
denote its incentre and circumcentre respectively. Let 
be the circle through 
and 
which is tangent to the incircle of the triangle ; the circles 
and 
are defined similarly. The circles and meet at a point 
distinct from 
; the points 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.
be a triangle and let and denote its incentre and circumcentre respectively. Let be the circle through and which is tangent to the incircle of the triangle ; the circles and are defined similarly. The circles and meet at a point distinct from ; the points and are defined similarly. Prove that the lines and are concurrent at a point on the line .(Russia) Fedor Ivlev
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