AoPS Academy
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
given that there exist positive integers 
satisfying
with 
.
for all 
then 
hence 
and we are done. From now on 
for some 
and also 
for some 
.
has the IVP we may assume 
and we may assume that has no roots in ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
. We assume 
, otherwise we have to make some minor adjustments.
if 
is a strictly increasing sequence with all 
very close to 
.
if 
is a strictly increasing sequence with all very close to 
.
to 
within the set of strictly increasing sequences. Then, by continuity, we also find a sequence 
with 
.
satisfying 
,
,
)
.
such that![\[
\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.
\]](http://latex.artofproblemsolving.com/4/4/e/44e5eb17b17531e36c362e062471d9bdffc8144c.png)
