[/quote]
,
be a triangle with circumcircle 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
be reals such that 
Prove that
be non-nagative real numbers such that 
.
.
,
at 
,
to 
are 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
.
and 
intersect at point 
,
is the point on segment 
such that 
.
is isosceles.
-
on the planes 
respectively. Let 
be the intersection point of the line 
and the 
-
.
moves on the perimeter of the circle with center 
, radius 1 on the plane 
,
and 
on the plane 
.
moves on the perimeter of the circle with center 
, radius 1 on the plane 
,
,
,
at 
,
,
intersects lines 
and 
at 
and 
be distinct points on side 
.
be the circumcircle of 
.
.
and 
be positive integers with 
such that 
divides 
.
and 
;
and 
.
wrt 
.
Prove that 
is the angle bisector of 
.
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
such that the following equations are both true:
Y by
Let 
be an integer. Alice and Bob play the following game: Alice chooses 
and draws a 
table, then he fills the 
cells of the first row with different numbers from 
. Then, Bob fills on the second row some of the cells (eventually none) with distinct numbers from , and the rest of them with 
. Finally, on each cell of the third row we write the sum of the two cells above. Show that regardless how Alice plays, Bob can guarantee that on the third row he can obtain, in some order, the terms of a non-constant arithmetical progression.
be an integer. Alice and Bob play the following game: Alice chooses 
and draws a 
table, then he fills the 
cells of the first row with different numbers from 
. Then, Bob fills on the second row some of the cells (eventually none) with distinct numbers from , and the rest of them with 
. Finally, on each cell of the third row we write the sum of the two cells above. Show that regardless how Alice plays, Bob can guarantee that on the third row he can obtain, in some order, the terms of a non-constant arithmetical progression.
