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Archive for January, 2008

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!

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

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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.

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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!

Read Full Post »

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.

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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.

Read Full Post »