[/quote]
,
be a triangle with 
and the rea is 
Find the minimum of 
,
,
.
.
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
be the feet of the altitudes of the triangle 
,
is the orthocenter of the triangle 
has radius 
.
be the point such that 
,
is perpendicular to the line 
and 
similarly. Prove that 
are collinear.
be real numbers such that 
and 
.
.
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an 
,
.
and 
,
is acute, and let 
lie on lines 
,
and 
and 
and 
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
be all the positive integers smaller than 
such that
for all 
.
and 
.
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers 
with 
Y by Adventure10, Mango247
Let ![$[mn]$](//latex.artofproblemsolving.com/3/c/9/3c9b75cda1e8e356ec62e6a47910aced04a872a6.png)
be a diameter of the circle 
and ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
a chord with given length on this circle. neither coincides nor is perpendicular to ![$[MN]$](//latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
.
Let
be the orthogonal projections of 
and 
on and 
the midpoint of .
Prove that
does not depend on the chord .
![$[mn]$](http://latex.artofproblemsolving.com/3/c/9/3c9b75cda1e8e356ec62e6a47910aced04a872a6.png)
be a diameter of the circle 
and ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
a chord with given length on this circle. neither coincides nor is perpendicular to ![$[MN]$](http://latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
.Let
be the orthogonal projections of 
and 
on and 
the midpoint of .Prove that
does not depend on the chord .
be circle center, it is more straight forward to prove 
.
,
is an isosceles triangle.
,
,
is also an isosceles triangle.
,
is on a circle, from which we have 
.
,
,
