AoPS Academy
and 
pebbles, respectively. In the first round Barry chooses a positive integer 
and Andrew takes away 
pebbles from one of the two heaps (if is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game.
,
is the midpoint of side 
,
is the midpoint of side 
,
is the midpoint of side 
.The line(different from line ) that is tangent to the inscribed circle of triangle and passing through point intersect line 
at 
.Define 
similarly.Prove that 
are collinear.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
and natural number ,
are there such that 
and 
?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
and 
, then we can deduce 
\le 2 $ ?
, the apex angle 
is 
. Points 
and 
are marked on the sides 
and diagonal 
, respectively, so that the quadrilateral 
is a parallelogram. Prove that triangle 
is equilateral.
and 
.
Since 
you get 
Because 
,
Similar calculations give 
this proves the result