and 
.
Equality holds when 
or 
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
a
.
Y by
Let 
be a right triangle, with the right angle at 
. The altitude from meets 
at 
and 
is the midpoint of the hypotenuse ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. On the legs, in the exterior of the triangle, equilateral triangles 
and 
are constructed. If 
is the intersection point of the lines 
and 
, prove that the angles 
and 
are equal.
Miguel Ochoa Sanchez and Leonard Giugiuc
be a right triangle, with the right angle at 
. The altitude from meets 
at 
and 
is the midpoint of the hypotenuse ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. On the legs, in the exterior of the triangle, equilateral triangles 
and 
are constructed. If 
is the intersection point of the lines 
and 
, prove that the angles 
and 
are equal.Miguel Ochoa Sanchez and Leonard Giugiuc
are isogonal conjugates through 
