AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a square colored like a chessboard.
. Let 
be the foot of the altitude of triangle issuing from the vertex 
, and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
Prove that 
When does equality hold?
be a parallelogram such that 
Let 
and 
be the midpoints of 
and 
respectively. Assuming that 
is a cyclic quadrilateral, find 
be a polynomial with integer coefficients. Define 
and 
for 
.
such that 
, then prove that either 
or 
.
have 
distinct real roots, which sum up to 
. If 
, find the values of 
and their corresponding roots.
be a positive integer greater than . Find all functions 
satisfying:
for all integers 
be a triangle, and let be an arbitrary point on line 
, where is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point . Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines and 
intersect at point . Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
+1 w be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
., inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals and 
. The circle with center and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
with bases and 
is given. Circles with centers 
and 
are inscribed in triangles and 
. Prove that line 
is perpendicular on .
with bases and is given. Circles with centers and are inscribed in triangles and . Prove that line is perpendicular on .
,
.
,
,
is a parallelogram, so 
,
.
Therefore,