[/quote]
,
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.
equals to a positive integer power of 
.
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. 
,
is similar to these two triangles.
be a triangle and let 
,
,
.
and 
.
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.
.
(other than 
be the midpoint of side 
intersects sides 
be the midpoint of segment 
,
be the foot of the perpendicular from 
.
is tangent to the circumcircle of triangle 
.
be a acute triangle with altitudes 
.
intersect 
,
with diameter 
.
.
with real coefficents, that satisfies 
for all real numbers 
be its incenter. A point 
intersects 
,
intersects 
.
intersects 
and 
intersect at 
and 
intersect at 
.
.
Y by Davi-8191, Adventure10, Mango247, Rounak_iitr
Let the circles 
and 
intersect in 
and 
. Tangent to circle at intersects in 
and tangent to circle at intersects in 
. Suppose that 
intersects and in 
and 
, respectively (assume that is between and ). The perpendicular to 
from intersects in point 
and perpendicular to 
from intersects in point 
(The points 
and lie on the same side of the line ). Prove that the points and are collinear.
Proposed by Mahdi Etesami Fard
and 
intersect in 
and 
. Tangent to circle at intersects in 
and tangent to circle at intersects in 
. Suppose that 
intersects and in 
and 
, respectively (assume that is between and ). The perpendicular to 
from intersects in point 
and perpendicular to 
from intersects in point 
(The points 
and lie on the same side of the line ). Prove that the points and are collinear.Proposed by Mahdi Etesami Fard
This post has been edited 1 time. Last edited by MRF2017, Sep 13, 2016, 5:10 AM
Reason: subject changed!
Reason: subject changed!
and let 
.
and 
to 
such that 
.
and 
and 
respectively, and 
we get 
.
.
.
.
.
.
is a cyclic quadrilateral with center 
.
.
.
will be collinear only if 
