AoPS Academy
+1 w
, where 
is a non-negative integer, 
is an odd number, and let 
. Prove that for any positive integer 
and for any positive odd number 
, 
is a multiple of 
.
to job 
and has a better ability in job than candidate B, but A is assigned job and B is assigned job , then A will complain. Also, if it is possible to assign each job to a candidate with a higher ability, the director will complain. Show that the boss can assign the jobs so as to avoid any complaints.
be a prime and suppose 
is prime. Prove that 
is prime
This is a special case of Pocklington's theorem. A proof of this special case is required.
and 
satisfying 
for all , and both series![\[
\sum \frac{a_n}{b_n}, \quad \sum \frac{b_n}{a_n}
\]](http://latex.artofproblemsolving.com/0/7/c/07ccd53f23c5e89c173f6dc3ead98aecf02a2ca1.png)
are convergent. Prove that the series![\[
\sum \frac{a_n}{a_n + b_n}
\]](http://latex.artofproblemsolving.com/5/9/5/595b976b8e81c8c7efa58469a63b1128f122e5ca.png)
also converges.
be assertion 
is obviously surjective
, comparaison of 
with 
implies 
and so is injective.
, and so 
for some constant 
and so 
for some constant 
in original equation, we get 
and
, which indeed fits
