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consecutive even numbers, such that the square of their product divides the sum of their squares.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
and 
in 
whose commutator is a five-cycle. (The commutator of 
and 
is 
.)
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. I think Mr. Barrington composed his permutations the wrong way. (Fortunately for complexity theory, the lemma is robust to this issue! :P)
, the angle at vertex 
is 
. It is known that 
and 
the midpoint of the side 
. Denote by 
the midpoint of the broken line and by 
the intersection point of the lines 
and 
. Find the measure of the angle 
.
, the angle at vertex is . It is known that and the midpoint of the side . Denote by the midpoint of the broken line and by the intersection point of the lines and . Find the measure of the angle .
