1. Given a triangle 
, the tangent of the circumcircle at 
intersects with the line 
at 
. Let 
be the points of symmetry for across the lines 
respectively. Prove that the line intersects orthogonally with the line 
. 2. Find all functions 
such that 
for all 
3. Let
be prime. Find all possible integers 
such that for all integers 
, if 
is divisible by , then is divisible by 
as well.
such that for all 3. Let
be prime. Find all possible integers such that for all integers , if is divisible by , then is divisible by as well. 4. Given two triangles 
and 
such that 
and 
are collinear in this order respectively. The circle 
passing through 
intersects with the circle 
passing through 
at distinct points 
.
and such that and are collinear in this order respectively. The circle passing through intersects with the circle passing through at distinct points . Prove that the circumcenter of the triangle 
is the midpoint of the centers of 
.
is the midpoint of the centers of . 5. Given is a piece at the origin of the coordinate plane. Two persons 
act as following. First, marks on a lattice point, on which the piece cannot be anymore put. Then 
moves the piece from the point 
to the point 
or 
, a number of 
times 
. Note that we may not move the piece to a marked point. If wins when can't move any pieces, then find all possible integers 
such that will win in a finite number of moves, regardless of how moves the piece.
act as following. First, marks on a lattice point, on which the piece cannot be anymore put. Then moves the piece from the point to the point or , a number of times . Note that we may not move the piece to a marked point. If wins when can't move any pieces, then find all possible integers such that will win in a finite number of moves, regardless of how moves the piece.
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