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be 
. The tangents to at 
meet at point 
. For a point 
on line 
which is not on the segment , let the midpoint of 
be 
. Lines 
meet again at points 
respectively. Let 
be the midpoint of . Prove that the points 
lie on a circle.
red cards, 
white cards, and 
blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
, point 
lies on side and point 
lies on side 
. Let 
and 
be points on segments 
and 
, respectively, such that 
is parallel to 
. Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.
is the incenter of the right triangle , the ray 
crosses the circumscribed circle 
for the second time at the point 
. The circle passing through the points 
, , , intersects the ray 
again at 
. Prove that 
.
is the incenter of the right triangle , the ray crosses the circumscribed circle for the second time at the point . The circle passing through the points , , , intersects the ray again at . Prove that .
be the midpoint of and the perpendicular bisector of intersect at 
. Obviously, it suffices to show 
is cyclic.
. Now, the Incenter-Excenter Lemma yields 
