**07**. **PEN D2** (Putnam 1991/B4) Suppose that is an odd prime. Prove that

## Archive for March, 2008

## 07. A combinatorial congruence

Posted in Problem of the Bi-Week on March 27, 2008| Leave a Comment »

## S06 A historical divisibility.

Posted in PEN Solutions Archive on March 19, 2008| Leave a Comment »

## 06. A historical divisibility

Posted in Problem of the Bi-Week on March 12, 2008| Leave a Comment »

**06**. **PEN A3** ( IMO 1988 ) Let and be positive integers such that divides . Show that

is the square of an integer.

## S05 On the monotonicity of the divisor function.

Posted in PEN Solutions Archive on March 6, 2008| 1 Comment »

**05a**. *[Saint-Petersburg 1998]* Let denote the number of positive divisors of the number . Prove that the sequence does not become strictly monotonic from some point onwards.

**05b**. **PEN J11** Prove that does not become monotonic from any given point onwards.

Here is the official solution file: PEN05S

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