Feeds:
Posts

## 05. 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.

## S04 A hidden symmetry

04. PEN I11 (Korea 2000) Let $p$ be a prime number of the form $4k + 1$. Show that

$\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right\rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) =\frac{p-1}{2}.$

Here is the official solution file: PEN04S
You can also disscuss the problems here!

## 04. A hidden symmetry

O4. PEN I11 (Korea 2000) Let $p$ be a prime number of the form $4k + 1$. Show that

$\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right\rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) =\frac{p-1}{2}.$

03. PEN O53 (Schur Theorem) Suppose the set $M=\{1,2,\ldots,n\}$ is partitioned into $t$ disjoint subsets $M_1,\ldots,M_t$. Show that if $n\ge\lfloor t!\cdot e\rfloor$ then at least one class $M_z$ contains three elements $a,b,c$ with the property that $a+b=c$.