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such that, for all 
, 
, 
, and![\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]](http://latex.artofproblemsolving.com/4/e/c/4ec892df19b4ec97af9bb1e5d1954e6c26be3172.png)
for all integers 
. Determine all positive integers 
such that![\[
\frac{a_n}{n}
\]](http://latex.artofproblemsolving.com/3/d/1/3d1d012e7c41e1e09abd14626a79b1fb1cbab34d.png)
is an integer.
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following: and are non-empty subsets from , and are equal, and
is an empty subset.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
that satisfy the equation
denote the number of positive integers that divide 
, including 
and . For example, 
and 
. (This function is known as the divisor function.) Let ![\[f(n)=\frac{d(n)}{\sqrt[3]{n}}.\]](http://latex.artofproblemsolving.com/4/d/d/4dd9a77a6a0a84fc9e69910675fe6da63ffbba3a.png)
There is a unique positive integer such that 
for all positive integers 
. What is the sum of the digits of 
let 
be the midpoint of side 
. Draw circles with centers 
and radii 
respectively and let 
be their common chord. Similarly denote the segments 
and 
. Show that lines 
and are concurrent.
