AoPS Academy
and 
. Prove that
Equality holds when 
or 
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
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
let 
and 
be the circles passing through 
and 
respectively such that 
is tangent to and define 
to be a point such that it lies on both the circles and prove that 
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 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?
with orthocenter 
. Extend 
to meet at 
. The angle bisector of 
meets the midpoint of 
. If 
, compute the area of 
.
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
such that 
. Denote the midpoints of 
as 
respectively, compute the area of 
given the area of is 
.
with 
. Point 
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of are on the same line, compute the length of .
be reals with 
and 
. Prove that
. Perform an inversion about 
with radius 
. Let the -median intersect the circle with diameter 
at 
and at 
. Note that this inversion simply swaps and , and and . Morever, 
inverts to the nine-point circle of . Finally, the line 
inverts to itself, so these three objects all pass through , a point on , so inverting back yields the desired conclusion.
be the foot of the perpendicular from to . Since the reflection of lies on the circumcircle of 
, it follows that 
and are congruent. In particular, the reflection of over and lie on . Denote the reflection of over and as and , respectively. By power of a point, 
. It follows that 
are concyclic. Since 
, it follows that 
as well.
be the foot of the altitude from to , and 
be the ortic triangle.
and 
are tangent to 
, 
is a harmonic quad. Project it through to line , and goes to , goes to 
, goes to 
, and goes to the harmonic conjugate of wrt. (say, 
). But by ceva-menelao, 
and concurr at . Moreover, we can see that is the radical center of , and 
and thus 
is the rad ax of and which implies is in , but since it's also in the median, we have that 
and we're done.
,
.
and let 
are midpoints of 
,respectively.We know that 
is cyclic or 
, so 
lies on radical axis of the circles 
and 
,hence 
or 
.But we have by 
lies also radical axis 
,so we are done !
,
.
and 
.
is a parallelogram 
and 
.
.
.
is cyclic 
