II picked the following problem from Professor Titu Andreescu’s literature).
Let k be an odd positive integer. Prove that
for all positive integers n.
We consider two cases:
Case I: In the first case, we assume that n is odd and write . Then,
. We have
, which, in turn equals,
Since k is odd, is a factor of . Hence, divides for . Consequently, divides . Likewise, we have
which is equal to
Hence, divides for . Consequently, divides . We have shown that each of and divides . Since , we conclude that divides .
Case II: In the second case, we assume that n is even. The proof is similar to that of the first case. We leave it to the reader.