[/quote]
,
with the smaller base 
and 
.
.
such that
for all real numbers 
satisfying the equation
?
has circumcenter 
.
and 
intersect at 
,
and 
intersect at 
and 
and 
intersect at 
.
are similar(with vertices in that order), than 
candidates and 
judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most 
candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
have the property that 
is a perfect square for infinitely many positive integers 
?
functions such that
for all real numbers 
and 
with equal radii are given. Let 
,
and 
are on 
,
and 
.
as the other intersection of 
and 
.
are concurrent.
such that 
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
be real numbers such that : 
. Prove that : 
and some point 
at 
at 
does not depend on the choice of 
Y by Adventure10, Mango247
Given is triangle 
with an inscribed circle with center 
and radius 
.
The tangent to this circle parallel to
intersects 
in 
and 
in 
.
The tangent to this circle parallel to intersects in 
and in 
.
The tangent to this circle parallel to intersects in 
and in 
.
Name the centers of the inscribed circles of triangle
, triangle 
and triangle 
successively 
and the rays successively 
and 
.
Prove that
.
with an inscribed circle with center 
and radius 
.The tangent to this circle parallel to
intersects 
in 
and 
in 
.The tangent to this circle parallel to
intersects in 
and in 
.The tangent to this circle parallel to
intersects in 
and in 
.Name the centers of the inscribed circles of triangle
, triangle 
and triangle 
successively 
and the rays successively 
and 
.Prove that
.
