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.
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
be an acute triangle and 
it's incenter. Let 
, 
and 
be the projections of onto 
, 
and 
respectively. 
and 
is the projection of onto 
. Let 
be the middle of the arc not containing 
and 
the second intersection of 
and the circumcircle of 
. If 
is the midpoint of 
, 
, 
and 
, prove that 
is cyclic.
is recursively defined such that 
is not a power of 
, and for all nonnegative integers 
:
is even, then 
is the largest odd factor of is odd, then 
where 
is the smallest prime factor of such that 
for all 
.
such that 
is the square of an integer.
let 
denote the midpoint of side . Let the circle passing through point and tangent to side at point intersect sides and at points and , respectively. Let the line segments 
and 
intersect in point 
. Let 
be the second point of intersection of the circumcircles of triangles 
and 
. Prove that points 
and are collinear.
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
, point is the midpoint of side and the centers of the - excircles of triangles 
and 
are 
and 
, respectively. The circumcircle of triangle 
intersects line at points 
and . The circumcircle of triangle 
intersects line at points 
and 
. Prove that 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
. is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist. be a triangle inscribed in circle 
such that points 
are fixed, while moves on major arc of . The tangents through and to intersect at . The circle with diameter 
intersects and at and , respectively. Prove that 
is tangent to a fixed circle whose radius is half the radius of .
and 
be the circumcenter, centroid and symmedian point of 
respectively. is an arbitrary point on Euler line 
Show that the orthocorrespondent of WRT 
falls on 
be a subset of 
such that if 
, then the residue obtained when we divide 
by 
also belongs to .
.
be a convex quadrilateral, and , , and be the midpoints of diagonals and 
, and side 
, respectively. Also, suppose that 
and that 
, 
. Calculate the perimeter of triangle 
.
be a convex quadrilateral, and , , and be the midpoints of diagonals and , and side , respectively. Also, suppose that and that , . Calculate the perimeter of triangle .
and 
Prove that
