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. Then, perform the sign change progress ( from sign to sign 
, from sign to sign )follow these steps in turn:
: signs in all squares on the line 
will be changed times,
: Signs in all squares on the column 
will be changed 
times, 
it is fixed positiv number.
, where 
are integer numbers .
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
. Given this information, determine all possible values of 
.
be a point in the plane of 
, and 
a line passing through . Let 
be the points where the reflections of lines 
with respect to intersect lines 
respectively. Prove that are collinear.
Now
and from there you get the x,y values for which you will have a telescopic sum, so the rest is easy.
Prove that
where degrees are used.
![\[\sum_{k=0}^{118}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\sqrt{3} \cdot \sin(n)}{2},\]](http://latex.artofproblemsolving.com/7/7/a/77a735ebe4eb693d82aba74e0bfda7d30826b2a8.png)
where angles are measured in degrees.