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.

**Proof:**

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.

—-Nalin Pihwa

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