AoPS Academy
are there such that 
and 
?
, where 
is a real number. Prove that for all positive , 
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
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?
that satisfy 
with orthocenter 
. Extend 
to meet at 
. The angle bisector of 
meets the midpoint of 
. If 
, compute the area of 
.
(or starting with the other diagonal, that won't change anything)
and 
are drawn, which intersect at a point 
. Assuming the square has side length 1 (
), the total length is 
. and , such that all angles formed are 
. One can show that a network can be constructed of length 
, which is less than .
![[asy]pair A=(0,1),B=(1,1),C=(1,0),D=(0,0),E=(1/2,1-1/(2*sqrt(3))),F=(1/2,1/(2*sqrt(3)));
draw(A--E--F--C);draw(B--E--F--D);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
draw(box(A,C),dotted);
label("$A$",A,NW);label("$B$",B,NE);label("$C$",C,SE);label("$D$",D,SW);label("$E$",E,SW);label("$F$",F,NW);[/asy]](http://latex.artofproblemsolving.com/c/6/5/c6580b2cdc37be4904525085188dc8d259a72873.png)
and are drawn, which intersect at a point . Assuming the square has side length 1 (), the total length is . and , such that all angles formed are . One can show that a network can be constructed of length , which is less than .
