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.
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 .
.
, such that for any convex polygon with area 
, there exist a point in the plane, such that the area of convex hull of 
is at most , where 
denotes the image of under central symmetry with respect to .
be positive integers such that 
is a square. Prove that 
is not prime.
with parallel sides 
and 
, assume that there exist points 
on line 
outside segment , and 
inside segment 
such that 
. Denote by 
the point of intersection of and 
, and by 
the point of intersection of and . Let 
be the midpoint of segment , assume it does not lie on line . Prove that belongs to the circumcircle of 
if and only if belongs to the circumcircle of 
.
prove that
,
is midpoint AP,BP. the perpendicular of 
from intersect 
at 
. we'll prove that is circumcenter of 
= 
=
=
then => PKO'H is cyclic quadrilateral => (QED)
be the perpendicular from 
to . Let 
,
be the midpoints of 
, . Let 
and 
meet line at and 
, respectively. Then since 
and 
, 
and 
are cyclic. Also, 
.
is cyclic, so by the radical axis theorem, 
,, and are collinear, so ,, and are collinear.
, so 
are collinear.
?
. Just note the following:
.
on substitution of values from above. 
. Observe that ![\[\measuredangle APH = \measuredangle CPE = 90^{\circ} - \measuredangle PCE = 90^{\circ} - \measuredangle ABD = \measuredangle APO,\]](http://latex.artofproblemsolving.com/d/c/e/dce88d35f45abbfbc47776f4c1097ad9ad87df97.png)
which is enough to show 
are collinear. 
of a cyclic quadrilateral 
intersect at 
be the circumcenter of triangle 
and 
be the orthocenter of triangle 
.
.
,
,
,
,
a
is a cyclic quadrilateral.
