AoPS Academy
satisfying 
Prove that 
When does equality hold?
be a point inside the triangle 
. 
intersects 
at 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to intersect BC at points 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points , , , , , lie on one conic
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?
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
be a triangle such that it’s circumcircle radius is equal to the radius of 
-excircle. Suppose that the -excircle touches 
at 
. Prove that 
(center of circumcircle) is the orthocenter of 
be a triangle such that it’s circumcircle radius is equal to the radius of -excircle. Suppose that the -excircle touches at . Prove that (center of circumcircle) is the orthocenter of 
where 
and 
.
![$ a,b,c\in [0,1] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
.
.
.
