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.
line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands 
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
is odd. is even.
such that a positive integer power of 
equals to a positive integer power of 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
to the following system:
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
be a parallelogram and points 
be on its exterior. If triangles 
and 
are similar, i.e. 
, prove that triangle 
is similar to these two triangles.
be a triangle and let 
be a point not lying on any of the three lines 
, 
, or 
. Distinct points 
, 
, and 
lie on lines , 
, and , respectively, such that 
and 
. Show that there exists a point 
on the circumcircle of 
such that 
is similar to 
.![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
is inscribed in circle 
. The interior angle bisector of angle 
intersects side and at and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides and again at and (other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
be a acute triangle with altitudes 
. is the orthocenter of . The angle bisectors of 
intersect at 
, respectively. The circle 
with diameter intersects segment 
at 
. Show that is tangent to 
.
with real coefficents, that satisfies 
for all real numbers 
be a scalene triangle, and let 
be its incenter. A point is chosen on line , such that the circumcircle of triangle 
intersects at 
, and the circumcircle of triangle 
intersects at 
. Circumcircle of triangle 
intersects and at and , respectively. Lines 
and 
intersect at , and lines 
and 
intersect at . Prove that 
.
. Solve the equation: 
such that 
and a point 
outside the angle 
in 
and 
in 
such that the triangle 
has a given perimeter.
