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 a real number. Determine all natural numbers 
for which there exist distinct natural numbers 
, 
, 
, such that
denotes the greatest common divisor of natural numbers 
and 
).
with 
, point 
lies on the circumcircle of such that 
. The line through parallel to 
intersects 
in 
and 
in 
. Prove that the center of the circumcircle of triangle 
lies on the circumcircle of triangle 
.
, no two of which are 
. Prove that:
+1 w
and be arbitrary points on the sides 
and 
of triangle , respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point . Line 
meets line 
at point , while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
is the center of the inscribed circle and holds 
. Prove that 
.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
, 
, 
, and is the point such that 
and 
, where and 
lie on the same side of . Let be the point on such that 
, and let 
be the midpoint of 
. Prove that the line through parallel to 
passes through the midpoint of .
, points and lie on and respectively such that 
. Points 
and are the reflections of and with respect to the midpoints of sides and , respectively. Point lies on segment 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle . passes through the midpoint of segment 
.
be a triangle with a circle 
with center 
tangent to sides 
at 
respectively. Suppose the circle with diameter 
intersects the circumcircle of again at 
is the reflection of over . Suppose points 
lie on such that 
are parallel to . Prove that: The intersection of 
lie on the circumcircle of 
be an acute triangle with 
and let be its orthocenter. Let be a point on the perpendicular bisector of 
such that 
and and are on different sides of , a point on the perpendicular bisector of 
such that 
and a point on the perpendicular bisector of 
such that 
and 
lie on the opposite side of w.r.t . Prove that 
and are collinear.
be a triangle with points 
lie on the perpendicular bisector of such that 
lie on a circle. Suppose 
are perpendicular to sides 
at points 
The tangent lines from points to the circumcircle of 
intersects at point 
Prove that: 
are parallel.
and 
intersect each other at points and 
. Their external common tangent (closer to ) touches at and at . Let be the reflection of in line 
. Prove that 
.
.
concyclic and thus 
.
concyclic, we get , we are done. concyclic, we get , we are done.
and intersect each other at points and . Their external common tangent (closer to ) touches at and at . Let be the reflection of in line . Prove that .
is the 
point in 
, Anyways, here's a solution with just angle chasing,
is cyclic, 
, and 
.
and 
.
.
. is tangent to 
and 
at and respectively, we have 
and 
.
.
are concyclic.
, and too.
and we know 
is cyclic
hence done