Prove that 9ab\left(a-b+c\right)+9bc\left(b-c+a\right)+9ca\left(c-a+b\right)\ge \left(a+b+c\right)^{3},\ \ a\ge 0\ge b\ge c: a+b+c\le 0.

Consider 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.

Let a,b,c\ge 0: a+b+c=3 then prove \frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\le \frac{27}{2}\cdot\frac{1}{2ab+2bc+2ca+3}.

In a triangle is the incentre and point is defined such that and
prove that is the perpendicular bisector of

Let and Prove that
Let and Prove that
Let and Prove that
Let and Prove that

Consider acute triangle with orthocenter . Extend to meet at . The angle bisector of meets the midpoint of . If , compute the area of .

Consider square with side length of . The point is selected on the diagonal such that . Denote the circumcenters of as . Find the length of

Consider convex quadrilateral such that . Denote the midpoints of as respectively, compute the area of given the area of is .

Solve the equetion:

Given 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?

One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,