You have to use the TeX file available **here**. Download gPENv02n1eng.tex. **The deadline for your submission is 15th, July.**.

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]

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**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)

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This weblication is written by PEN team and contributors:

**Alexander Remorov** (Canada),

**Darij Grinberg** (Germany),

**Harun Siljak** (Bosnia and Herzegovina),

**Mari****n Misur** (Croatia).

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

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

**Here goes the table of contents: **

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 Dierent 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 Dierent 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 . . . . . .

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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 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

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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 consecutive integers it is always possible to find one which is relatively prime to all the rest.

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

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Here is the official solution file: pen18.pdf

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is expressed as a fraction, where is a prime, then divides the numerator.

**A24** Let be a prime number and . Prove that

is divisible by .

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is an integer.

Here is the official solution file: pen17.pdf.

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