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## 03. A theorem on sum-free subsets

Here goes the problem:

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 $a+b=c$.

Next week, you will be presented two different solutions and several related results.

* Any comments are welcome! You can also disscuss the problem here!
* To get the current edition of PEN Problems Book, visit here!
* Next week, the solutions will be uploaded here in the pdf file.
* Plus, do not forget to join the group Project PEN in FACEBOOK!

## New co-author.

We are glad to announce you that starting this week, Andrei Frimu is co-author of Problem of the bi-week.

## S02 Three ways to reach a Diophantine equation

02. PEN H15 (Balkan Mathematical Olympiad 1998) Prove that there are no integers $x$ and $y$ satisfying $x^{2} = y^{5} -4$.

Here is the official solution file: PEN02S
You can also disscuss the problems here!

Acknowledgement. We would like to express our gratitude to Andrei Frimu who proofreaded the manuscript.

## 02. Three ways to reach a Diophantine equation

Here goes the problem:

02. PEN H15 (Balkan Mathematical Olympiad 1998) Prove that there are no integers $x$ and $y$ satisfying $x^{2} = y^{5} -4$.

The readers will meet three different solutions!

* Any comments are welcome! You can also disscuss the problems here!
* To get the current edition of PEN Problems Book, visit here!
* Next week, the solutions will be uploaded here in the pdf file.
* Plus, do not forget to join the group Project PEN in FACEBOOK!

## S01 Increasing multiplicative functions

01a. PEN K12 (Canada 1969) Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m, \, n\in \mathbb{N}$: $f(2) = 2$, $f(mn) = f(m)f(n)$, $f(n + 1) > f(n)$.

01b. Let $f:\mathbb{N} \to \mathbb{R}^{+}$ be a function satisfying the conditions:
(a) $f(mn) = f(m)f(n)$ for all relatively prime $m$ and $n$, and
(b) $f(n+1) \geq f(n)$ for all positive integers $n$.
Show that there is a constant $\alpha \in \mathbb{R}$ such that $f(n)=n^{\alpha}$ for all $n \in \mathbb{N}$.

Here is the official solution: PEN01S
An alternative solution is here: CMO 2003 (problem 4)
You can also disscuss the problems here!

References
[1] P. Erdos, On the distribution function of additive functions, Ann. of Math., 47(1946), 1-20
[2] E. Howe, A new proof of Erdos’s theorem on monotone multiplicative functions, Amer. Math. Monthly, 93(1986), 593-595
[3] L. Moser and J. Lambek, On monotone multiplicative functions, Proc. Amer. Math. Soc., 4(1953), 544-545

Acknowledgement. We would like to express our gratitude to Andrei Frimu who helped us preparing the solutions and Orlando Doehring who offered me the solution file of CMO 2003.

## 01. Increasing multiplicative functions

Happy new year, everyone! In this week, we propose two problems (the first one is easy and the second one is challenging):

01a. PEN K12 (Canada 1969) Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all
$m, \, n\in \mathbb{N}$: $f(2) = 2$, $f(mn) = f(m)f(n)$, $f(n + 1) > f(n)$.

01b. Let $f:\mathbb{N} \to \mathbb{R}^{+}$ be a function satisfying the conditions:
(a) $f(mn) = f(m)f(n)$ for all relatively prime $m$ and $n$, and
(b) $f(n+1) \geq f(n)$ for all positive integers $n$.
Show that there is a constant $\alpha \in \mathbb{R}$ such that $f(n)=n^{\alpha}$ for all $n \in \mathbb{N}$.

* Any comments are welcome! You can also disscuss the problems here!
* To get the current edition of PEN Problems Book, visit here!
* Next week, the solutions will be uploaded here in the pdf file.