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!
, point 
lies on side 
such that 
. Point 
lies on side 
such that 
. 
is the midpoint of segment 
. Point 
is the foot of the perpendicular from 
to 
. Given 
and 
, find the size of 
.
then 
are tangent to the circumcircle 
of 
Let 
cut again at 
Moreover 
Hence
where 
and 
.
![$ a,b,c\in [0,1] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
.
.
,