and 
.
be the set of all positive integers. Determine all subsets 
of 
such that
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
to be a point such that it lies on both the circles 
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 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
,
.
.
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
Y by Adventure10
Given convex quadrilateral 
. 
and 
are the midpoints of 
and 
respectively. Segments 
and 
intersect at the point 
, and the segments 
and 
at the point 
. And quadrilateral 
is inscribed. Let the circumscribed circles of triangles 
and 
second time intersect at the point 
, and circumscribed circles of triangles 
and 
at the point 
. Prove that the areas of quadrilaterals of and 
are equal.
. 
and 
are the midpoints of 
and 
respectively. Segments 
and 
intersect at the point 
, and the segments 
and 
at the point 
. And quadrilateral 
is inscribed. Let the circumscribed circles of triangles 
and 
second time intersect at the point 
, and circumscribed circles of triangles 
and 
at the point 
. Prove that the areas of quadrilaterals of and 
are equal.
cut 
again at 
and let 
cut 
again at 
is parallelogram. Similarly 
is parallelogram 
is parallelogram. By Miquel theorem in 
it follows that 
and similarly 
Thus since 
due to 
are midpoints of 
But 
which means that 
and 
form the same angle 
with 
Therefore![$[PMLK]=\tfrac{1}{2}LP \cdot KM \cdot \sin \theta=\tfrac{1}{2}NQ \cdot KM \cdot \sin \theta=[QMNK],$](http://latex.artofproblemsolving.com/d/5/d/d5d609539b8fb187e0d6dc38319b953571007e5c.png)
![$[KLMN]=[KPMQ],$](http://latex.artofproblemsolving.com/2/f/0/2f009d69c2dae0cec2ee828381073ac85f504850.png)
as desired.
