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.
is harmonic where is the intersection of 
with 
where R is the intersection of the angle bisector of angle C with and is the intersection of 
with 
Since 
is 
degrees and 
bisects 
we are done.
are parallel, then the midpoints of and 
are collinear (which follows from similar triangles). to . Let be the -intouch point, 
the antipode on the incircle, 
the -excircle touch point, the -excircle antipode, and 
the reflection of 
over . By homotheties at the incircle and excircle, 
are collinear. Then this implies that 
are collinear since 
is collinear. Finally, since 
are collinear, it follows that since 
that 
are collinear as desired.
the antipodal of on 
then on 
invert we have 
and 
and combining I-E Lemma with the fact that the dimater makes a 90 degree we have 
which means that 
so as 
we are done.
,
