[/quote]
,
of the triangle 
the orthogonal projection of 
on 
.
and 
are such that 
is a paralellogram, find the locus of point 
when 
and 
.
![$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$](http://latex.artofproblemsolving.com/2/2/e/22e6ec2f98777db80e39333542664e903663bf81.png)
![$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$](http://latex.artofproblemsolving.com/f/c/a/fca343ee701eeff4c309b31beb3426ec96b49b55.png)
Let 
.
![$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$](http://latex.artofproblemsolving.com/6/9/5/695cc35986a391fdc15e4863d361ce373fbbf64c.png)
and 
is a circle tangent to 
The tangent
to 
If 
are the contact points of 
respectively, prove that 
,
and 
,
an obtuse triangle can be formed?
.
is an integer, and let 
be 
integers such that 
for any two integers 
and 
satisfying 
.
.
of the form 
with 
,
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
grid which has rows and columns numbered from 
to 
;
.
,
satisfying 
or 
are not in 
be real numbers such that 
and 
Find the value of 
and 
Find the value of 
for which there are infinitely many positive integers 
is not divisible by 
.
Equality holds when 
Equality holds when 
,
is a regular polygon with 
diagonals. Randomly select a triangle whose three vertices are the three vertices of polygon 
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
for some positive integer 
such that 
.
onto 
.
bisects 
.
Y by
Let 
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
and we are done.
