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## PEN Global

Anyone who wants to contribute the PEN Global, the translation work of Problems in Elementary Number Theory 2 [ 2009 ] No. 1, please submit your translation to Yimin Ge at $\text{\small yimin.ge@gmx.at}$ together with your LaTeX file, its compiled PDF file and fonts file.

Before you contact Yimin Ge, please have a look at the PEN Global page at https://projectpen.wordpress.com/global .

Notice: We’ve already find translators: in Vietnamese,  Greek, Spanish, Bangla, Bosnian, Chinese, Croatian, Serbian [Latin] and Serbian [Cyrillic]

## Editors-in-Chief for 2010

As you know, Problems in Elementary Number Theory 2 [ 2009 ] No. 1 is NOW online. The Editors-in-Chief of the next issue are

Daniel Kohen (Argentina)
Cosmin Pohoata (Romania)
Harun Šiljak (Bosnia and Herzegovina)
Peter Vandendriessche (Belgium)

Anyone who wants to be one of the Editors-in-Chief of the fourth and fifth issue of Problems in Elementary Number Theory (2010), APPLY NOW.

Please, send an email to us at pen@problem-solving.be with your brief C.V.

You need to know basic LaTeX skills and of course you have to be fluent with Olympiad-style problems from Elementary number theory.

We are planning to recruit two or three Editors-in-Chief for Problems in Elementary Number Theory 2010. The deadline for application is 31th, July.

from Hojoo Lee (the founder of project PEN)

## Problems in Elementary Number Theory 2 [ 2009 ] No. 1

Is now available here !

This weblication is written by PEN team and contributors:

Darij Grinberg (Germany),
Harun Siljak (Bosnia and Herzegovina),
Marin Misur (Croatia).

We also need to thank  two Editors-in-Chief:

Daniel Kohen (Argentina) and Cosmin Pohoata (Romania).

1 Problems 1
2 Articles 3
2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3
2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Di erent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15
2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Fractions Mod p and Wolstenholme’s theorem . . . . . . . . . . . . . . . . . . . . . 29
2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Sequences of Consecutive Integers . . . . . .
1 Problems 1
2 Articles 3
2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3
2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Di erent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15
2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Fractions Mod p and Wolstenholme’s theorem . . . . . . . . . . . . . . . . . . . . . 29
2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Sequences of Consecutive Integers . . . . . .

## 20S. Sequences of Consecutive Integers

A37 If $n$ is a natural number, prove that the number

$(n + 1)(n + 2)\cdots(n + 10)$

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

Here is the official solution file: PEN A9 A37 O51

## 20. Sequences of Consecutive Integers

A37 If $n$ is a natural number, prove that the number

$(n + 1)(n + 2)\cdots(n + 10)$

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

## 19S. Binomial Sum Divisible by Primes

E16 Prove that for any prime $p$ in the interval $\left]n, \frac {4n}{3}\right]$, $p$ divides

$\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.$

Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.

## 19. Binomial Sum Divisible by Primes

E16 Prove that for any prime $p$ in the interval $\left]n, \frac {4n}{3}\right]$, $p$ divides

$\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.$