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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 . . . . . .

## Winter Break

We will be back on February!

## PEN Team!

Recruiting is finally over. We now have 10 members:

Andrei Frimu (Moldova)
Yimin Ge (Austria)
Ofir Gorodetsky (Israel)
Daniel Kohen (Argentina)
Hojoo Lee (Korea)
Soo-Hong Lee (Korea)
Cosmin Pohoata (Romania)
Ho Chung Siu (Hong Kong)
Peter Vandendriessche (Belgium)

## One more letter

As you see, the current version of PEN extended does not contain the problems in year 2008.

Please, send us number theory problems in national math olympiads ( taken in between September ’07 and June ’08 ) and selection test for IMO 2008 from your country. The problems we receive will be included in the first edition of PEN extended with contributor’s name.  The TeX format is available HERE.

from PEN TEAM

Before submitting your TeX file, check out the following list. The national codes are from http://www.imo-official.org

Code
Country Who contributed
AFG Afghanistan
ALB Albania
ALG Algeria
ARG Argentina
ARM Armenia
AUS Australia
AUT Austria
AZE Azerbaijan
BAH Bahrain
BLR Belarus
BEL Belgium
BEN Benin
BOL Bolivia
BIH Bosnia and Herzegovina
BRA Brazil
BRU Brunei
BGR Bulgaria
KHM Cambodia
CMR Cameroon
CHI Chile
CHN People’s Republic of China
COL Colombia
CIS Commonwealth of Independent States
CRI Costa Rica
HRV Croatia
CUB Cuba
CYP Cyprus
CZE Czech Republic
CZS Czechoslovakia
DEN Denmark
DOM Dominican Republic
EST Estonia
FIN Finland
FRA France
GEO Georgia
GDR German Democratic Republic
GER Germany
HEL Greece
GTM Guatemala
HND Honduras
HKG Hong Kong
HUN Hungary
ISL Iceland
IND India
IDN Indonesia
IRN Islamic Republic of Iran
IRL Ireland
ISR Israel
ITA Italy Donald Arapi
JPN Japan
KAZ Kazakhstan
PRK Democratic People’s Republic of Korea
KOR Republic of Korea
KWT Kuwait
KGZ Kyrgyzstan
LVA Latvia
LIE Liechtenstein
LTU Lithuania
LUX Luxembourg
MAC Macau
MKD The Former Yugoslav Republic of Macedonia
MAS Malaysia
MLT Malta
MRT Mauritania
MEX Mexico
MDA Republic of Moldova
MNG Mongolia
MNE Montenegro
MAR Morocco
MOZ Mozambique
NLD Netherlands
NZL New Zealand
NIC Nicaragua
NGA Nigeria
NOR Norway
PAK Pakistan
PAN Panama
PAR Paraguay
PER Peru
PHI Philippines
POL Poland
POR Portugal
PRI Puerto Rico
ROU Romania Cosmin Pohoata
RUS Russian Federation
SAU Saudi Arabia
SEN Senegal
SRB Serbia
SCG Serbia and Montenegro
SGP Singapore
SVK Slovakia
SVN Slovenia
SAF South Africa
ESP Spain
LKA Sri Lanka
SWE Sweden
SUI Switzerland
SYR Syria
TWN Taiwan
TJK Tajikistan
THA Thailand
TUN Tunisia
TUR Turkey Fatih Kursat CANSU
NCY Turkish Republic of Northern Cyprus
TKM Turkmenistan
UKR Ukraine
UAE United Arab Emirates
UNK United Kingdom
USA United States of America
URY Uruguay
USS Union of the Soviet socialist republics
UZB Uzbekistan
VEN Venezuela
VNM Vietnam Nguyen Tho Tung
YUG Yugoslavia

## Big Picture

To PEN friends.

After about two months, we will weblish the first official edition of PEN extended with more problems. This means that PEN Problems Book has at least 1000 challenging Problems in Elementary Number theory.

THE NEXT STEP IS OBVIOUS: THE EFFICIENT USE OF DATABASE. It entirely depends on you, PEN friends! To complete PEN Solutions Book, we need your help and more PEN Team members.

We all wait your creative solutions(including generalizations and comments) for problems in PEN Problems Book or in PEN extended. We want young minds who proofread “with sharp eyes” the solutions in our supporting site at ML. We need more people who do the TeXing work on the PEN Problems Book. If you want to participate this forever Project, then please let us know!

The second season of Problems of Bi-Week Project will begin later, not starting in 1st July. At least after IMO, maybe around August. All I can say now is that it will take some time. I hope, at least before the second week of September. In the second season, we will meet many number theory problems from IMO Short List Book ’07.  See you soon!

from PEN Team