1. Let 
and 
be two circles touching each other externally at 
Let 
and 
be the centres of and 
respectively. Let 
be a line which is tangent to at 
and passing through 
and let 
be the line tangent to at 
and passing through 
Let 
If 
then prove that the triangle 
is equilateral.
and be two circles touching each other externally at Let and be the centres of and respectively. Let be a line which is tangent to at and passing through and let be the line tangent to at and passing through Let If then prove that the triangle is equilateral. 
4. Let 
be an integer greater than 
and let 
be the number of non empty subsets 
of 
with the property that the average of the elements of is an integer.Prove that 
is always even.
be an integer greater than and let be the number of non empty subsets of with the property that the average of the elements of is an integer.Prove that is always even.
5. In an acute triangle 
let 
be its circumcentre, centroid and orthocenter. Let 
and 
Let 
be the midpoint of 
If the triangles 
have the same area, find all the possible values of 
let be its circumcentre, centroid and orthocenter. Let and Let be the midpoint of If the triangles have the same area, find all the possible values of
6. Let 
be six positive real numbers satisfying
be six positive real numbers satisfying
and 
Further, suppose that 
and 
Prove that 
and 
and Prove that and
Πηγή: artofproblemsolving
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