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## 07. A combinatorial congruence

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

$\sum_{j = 0}^{p}\binom{p}{j}\binom{p + j}{j}\equiv 2^{p} + 1 \pmod{p^{2}}.$

## S06 A historical divisibility.

06. PEN A3 ( IMO 1988 ) Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that

$\frac{a^{2}+b^{2}}{ab+1}$

is the square of an integer.

Here is the official solution file: PEN06S
You can also disscuss the problems here!

## 06. A historical divisibility

06. PEN A3 ( IMO 1988 ) Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that

$\frac{a^{2}+b^{2}}{ab+1}$

is the square of an integer.

## S05 On the monotonicity of the divisor function.

05a. [Saint-Petersburg 1998] Let $d(n)$ denote the number of positive divisors of the number $n$. Prove that the sequence $d(n^2+1)$ does not become strictly monotonic from some point onwards.

05b. PEN J11 Prove that $d((n^2+1)^2)$ does not become monotonic from any given point onwards.

Here is the official solution file: PEN05S
You can also disscuss the problems here!