AoPS Academy
and 
, Ivica and Marica play the following game: Given two piles of cookies, on each turn a player takes 
cookies from one of the piles, of which he eats 
and puts of is arbitrary in every move. Players take turns alternatively, with Ivica going
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be a rectangle with 
and the center the point 
. Perpendicular from on 
intersects lines 
and 
in points 
and 
respectively. Points 
and 
are midpoints of segments ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
and ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
respectively. Prove that 
.
be a rectangle with and the center the point . Perpendicular from on intersects lines and in points and respectively. Points and are midpoints of segments and respectively. Prove that .
, thus 
and thence 
as homologous medians. But this is very easy, since 
and 
are cyclic, making 
.
midpoint of and to prove 
.
and 
are similar (then since 
this will imply 
as desired).
it will suffice to prove that 
.
and 
we find that 
and similarly 
.![\[\frac{AE}{CF}=\frac{b-BE}{BF-a}=\frac{b-\frac{d^2}{2b}}{\frac{d^2}{2a}-a}=\frac{a}{b} \cdot \frac{2b^2-d^2}{d^2-2a^2}=\frac{a}{b}\]](http://latex.artofproblemsolving.com/a/3/f/a3fd4d9a8e4e263a09004a53180356900ea7182c.png)
using that 
.