THE MINISTRY OF EDUCATION
"TRANSILVANIA" UNIVERSITY FROM BRASOV
THE FACULTY OF MATHEMATICS AND INFORMATICS
BRASOV SCHOOL INSPECTORATE
THE TECHNICAL "MIRCEA CRISTEA” FROM BRASOV
“ADY ENDRE” TEORETICAL HIGH SCHOOL, BUCHAREST
MEN
The Faculty of Mathematics and Informatics
" Transilvania" University Brasov
No.6240./10.11.2014
SSMR10/10.11.2014
The Brasov School Inspectorate
No. 15700/7.11.2014
The Technical College "Mircea “Ady Endre” Teoretical High School
Cristea” Brasov Bucharest
No. 2312/3.11.2014 No. 1204/13.11.2014
The "Christian Kertsch" Association The International Association József Wildt
No29/ 3.11.2014 No27/27.10.2014
Organizes:
THE INTERNATIONAL CONFERENCE:
MATHEMATICAL EDUCATION IN THE CURRENT EUROPEAN CONTEXT
FIFTH EDITION taking place under the aegis of

The Society of Mathematical Sciences from Romania, the Brasov branch

The Faculty of Mathematics and Informatics from the
"Transilvania" University from Brasov

The Brasov School Inspectorate

The Scientific Association József Wildt

The Christian Kertsch Association
Distinguished guests:
Professor Emil Stoica, PhD the chief executive of the Senate at "Transilvania"
University from Brasov
Teacher Bucur Ariana, General School Inspector at the Brasov School Inspectorate,
Teacher Mária Szabó, assistant to the General School Inspector at the Brasov School
Inspectorate
Scientific Committee:
University lecturer Eugen Păltănea, PhD at the Faculty of Mathematics and
Informatics, "Transilvania" University from Brasov
Teacher Vijoli Victor, assistant to the General School Inspector at the Brasov School
Inspectorate
Teacher Zubaşcu Andreica Florica, mathematics inspector, Braşov School
Inspectorate
Project team:
Coordinators:
Teacher Mihaly Bencze - The National College "Áprily Lajos” from Brasov
Teacher Virginia Marinescu - The Technical College "Mircea Cristea” Brasov
Organizers:

Principal Costea Gabriela- The Technical College "Mircea Cristea”
Teacher Stancu Diana - The Technical College "Mircea Cristea”
Teacher Moldovan Camelia - The Technical College "Mircea Cristea”
Teacher Goran Angelica- The Technical College "Mircea Cristea”
Teacher Felicia Mereuţă - The Technical College "Mircea Cristea”
Teacher Anca Popa Alexandru - The "Petru Rares” College
Teacher Cristina Centea - The Technical College "Mircea Cristea”
Teacher Daniela Andreiu - The Technical College "Mircea Cristea”
Muntean Dragoş- The Technical College "Mircea Cristea”
Bogdan Barbu
Organizers for Section IV b, high school students’ contest, grades IX-XII:
Teacher Negrea Camelia, The Technical College "Mircea Cristea”
Teacher Dinu Cristina, General School No. 5
Organizers theme exhibition with students’ drawings, grades V-XII
Coordinator: Ţucanu Daniela, technical disciplines inspector, Braşov School
Inspectorate
Teacher Marian Ioan - The Technical College "Mircea Cristea”
Teacher Ivan Alina
The motivation of the project:
„Learning Mathematics, one learns to think ”
Grigore C. Moisil
Romania's affiliation to the European Union involves harmonizing the Romanian educational system with the other European educational systems. The Program of Bologna traced common development directions for the university education and it is now necessary to find essential landmarks in the pre-university education as well as to form the young.
The suggested event will allow useful experience exchanges in the field of interactive teaching strategies and that of integrating multimedia technologies during the Mathematics classes. We aim at promoting the innovative teaching measures, at encouraging actions which allow the insurance of quality in education and the life-long constant professional training.
Mathematics is an essential subject in forming a young person. The actions dedicated to
" Mathematical education in the current European context 2013" allow certain ideas to be known.
They also allow the knowledge of certain teaching-learning approaches of this subject in schools and they should be known by our teachers and students.
The Objectives:

Knowing the place held by Mathematics in the curriculum of different schools

Promoting modern methods of teaching, learning and assessment (including the computer-assisted learning)

Motivating teachers and students to use the interactive teaching methods and strategies which are specific to modern pedagogy

Motivating teachers and students for a serious scientific training in the field of professional competences.
Target group: teachers from elementary schools, from high schools, beginner teachers, students and senior students, people working on an M.A. and a Ph.D. within the
Mathematical faculties, high school students (section IV b)
Participating countries: Romania, Spain, China, U.K.
The suggested sections:

Section I: Relevant aims/competences and contents for the Mathematical education of the young in Romanian and European Schools

Section II: Computer assisted Mathematics lessons


Section III: Themes, problems, Mathematical lessons - innovative and interdisciplinary approaches (contents, strategies, methods and approaching/solving techniques)

Section IV: a)-for teachers - Mathematics and its connections with other fields of study
 b)-for students - Mathematics and its connections with other fields of study
Theme exhibition (drawings) – at the Technical College "Mircea Cristea” and
Braşov School Inspectorate
Location : The National College Mircea Cristea, 3, Turnului Street, Brasov
Date: 11 .12.2014
Participation Rules: a.
Sending the paper and the registration form (see Appendix 1, deadline 9.12.2014 for the papers and 5.12.2014 for the registration form) to the contact persons.
The contact persons are:
1. Teacher Stancu Diana, e-mail : diannnas@yahoo.com
, : (Section 1)
2. Teacher Anca Alexandru Popa, phone no.: 0040 723 068 360, e-mail: anca_popa4@yahoo.com
, (Section 2)
3. Teacher Angelica Goran, phone no.0
040 748053794, e-mail angelica_goran@yahoo.com
; (Section 3)
4. Teacher Mereuţă Felicia, phone no. 0040 729154752 / 004 0 729988888, e-mail adyfely@yahoo.com
(Section 4)
5. Teacher Moldovan Camelia, cameliamoldovan@yahoo.com
The papers, written in the format shown in the appendix, will be sent on the symposium email: international.conference.brasov@gmail.com or to the e-mail addresses of the organizers.
Valuable materials in electronic format, may contribute to the creation of an international ISSN volume, on this topic. Each paper can have maximum three authors. b.
The material will be sent in electronic format MS Word („doc” or “docx” format) in
Romanian (with abstract in English) or in English. The paper should have the following specifications: MsWord 2007, page format Custom size 6.73 ̎ and 9.37 ̎
(International B5) , Justifiy, font: Times New Roman, size 12, line spacing: single, with diacritics. The mirror margins will be 2 cm/0.79 , except the left/right one, that will be 2.5cm/0.98 . The paper must have an even number of pages, between 2 and
10. The figures will be drawn using Word tools and will be numbered with Fig. 1, 2 etc, the tables with Table 1, 2 etc and the formulae/demonstrations will be written using Equation and Symbol. Bibliographical references will be included when necessary
(see Appendix 2). The abstract in English will have a maximum of 15 lines, TNR font, size
11, Italics (see appendix 2, with the template for the paper). c.
For any question, please write to one of the contact persons. You can also write to Bencze
Mihaly,phone no. 0040 724 061 251 benczemihaly@yahoo.com
and Marinescu
Virginia, phone no. 004 0740 850 110, gini_l18@yahoo.com
d.
Assessment.
All papers will be mentioned in the catalogue of the Conference and/or in the international volume . The persons from other towns, who do not participate directly to the conference, will receive by mail the diploma, the portfolio with the schedule of the activities, the catalogue and the ISSN volume/CD (for the papers selected by the Scientific Committee).
The participants whose paper was selected and want their paper to be published in the

international volume will contribute with the sum of 50 + 5 lei (for those from other towns/counties) for sections I, II, III and IV a, at this address:
ASOCIAŢIA CHRISTIAN
KERTSCH, in the account RO76BRDE080SV84870500800 (please mention „for
Conference 2014”), or it can be paid on the conference day to Ms. Mereuţă Felicia. e.
Section IV b) Teachers Negrea Camelia and Dinu Cristina organize the students’ contest. The papers will have the same requirements as for the symposium, but they will be written by a maximum of 3 pupils, coordinated by a teacher. They will be presented first at a school/town or county contest, organized by the teachers involved. The results of these contests at school/town/county level will be sent to the organizers of our symposium, containing the names of the selected students, the title of the paper, the level, the coordinating teacher and the place occupied in the contest hierarchy (we recommend a maximum of 2 teams/class in the same school). The presence of the students from outside Braşov county is not mandatory. In this case, the papers can be sent in electronic format, with the same scientific and technical requirements as the teachers, the maximum number of pages being 8. The papers will be sent by e-mail to the organizers: Negrea Camelia, camelianegrea@yahoo.com, Dinu Cristina phone no. 0040
771 516 722, cristinadinu18@yhaoo.co.uk.
The best papers will be published in a volume, for a fee of 20+5 lei sent to the account of CHRISTIAN KERTSCH
ASSOCIATION:RO76BRDE080SV84870500800 (please mention „for Conference
2014”). The participant teams and the coordinating teachers will receive diplomas and a
CD/volume (if the paper will be published in the 2015 volumethrough the contribution of the participating schools or teachers/students).
The contest will take place at the Technical College “Mircea Cristea”, No. 3 Turnului
Street, Braşov, on 11.12.2014, at 2:00 p.m. Before the contest, the participants/teachers will be sent by e-mail a schedule for the ppt/pptx presentations.
The drawing exhibition “Art and Mathematics”
Teachers Vetró B. Sebestyén András, Marian Ioan and Ivan Alina will form the jury for the drawing contest that will take place on 11.12.2014, opening at 2:00 p.m. at the
Technica College „Mircea Cristea”. The most impressive drawings will be exhibited at the School Inspectorate.
The best drawings will be awarded prizes, and all the participants will receive diplomas, as well as the coordinating teachers. The drawings will be sent to the Technical College
“Mircea Cristea”, No. 3 Turnului Street, Braşov, together with an A4 envelope with their own address, for the participation diplomas. For further details you can write to the organizers at the following e-mail addresses: vetrobsa@gmail.com, ioanemarian@yahoo.com.
Observation : The accommodation and the meals will be paid by the participants.
The Schedule of the Activities
11.12.2014
, 14:00, section IV b, students’ contest, opening of the exhibition “Art and
Mathematics”
12.12.2014 – teachers’ conference
14.00-14.30 Receiving the participants and the guests. Presentation of the drawing exhibition.
14.30-17.00 Presenting the papers (sections I, II, III, IV a)
17.00-17.30 Conclusions and diploma handing

Dean of the Faculty of Mathematics and Informatics, the Transilvania University from Brasov
PhD. Marin Marin
SSMR
Conf. Univ. dr Paltanea Eugen
General School Inspector,
Teacher Bucur Ariana
Principal of the Technical College “M. Cristea”, Principal of the “Ady Endre” High School
Teacher Costea Gabriela Teacher Bencze Mihaly
The Christian Kertsch Association The International Association József Wildt
President: Teacher Marinescu Virginia President: Teacher Mihály Bencze

APPENDIX 1
REGISTRATION FORM FOR THE CONFERENCE
Teaching Mathematics Within The Actual European Framework
No. Author’s surname and first name,
1.
2.
3.
Specialization Phone number
E-mail
Address
(underline the address of the teacher who will receive the conference papers/volume)
Institution
Address
Teachers
11-12.12. 2014
Title of the paper Section:
I, II, III, IV
Direct participati on
Indirect/vi deoconfer ence
Computer/ laptop
Presentation of the paper
(maximum 10 min PPT/PPTX 2007)
Video projector
Smart board
Other
Please enclose the statement of compliance signed
, at this address: Colegiul Tehnic. Mircea Cristea, Str. Turnului Nr,.3 Brasov

REGISTRATION FORM FOR THE CONFERENCE
Teaching Mathematics Within The Actual European Framework
Students/Pupils
No.
Author’s surname and first name
1.
2.
3 Coordinating teacher
Specialization/c lass
Phone number
E-mail
Address
(underline the address of the student who will receive the conference papers/volume)
University and faculty/high school
Address
11-12.12. 2014
Title of the paper Section:
I, II, III, IVa, IV b
Direct participatio n
Indirect/v ideoconfe rence
Presentation of the paper
(maximum 10 min PPT/PPTX 2007)
Computer/ laptop
Video projector
Other
Please enclose the statement of compliance signed
, at this address: Colegiul Tehnic. Mircea Cristea, Str. Turnului Nr.3 Brasov

APPENDIX 2 (paper model)
ABOUT AM-HM INEQUALITY
Mihály Bencze
Abstract
In [1] Bencze, M. introduced the following notation.
where a i
> 0
(i = 1, 2, ..., n) and by mathematical induction proved that
G(m) G(m − 1) ≥ ... ≥ G(1) ≥ G(0) = n 2 .
In this paper we generalize this result and we give some refinements.
Main result
Theorem 1.
If a, b, c > 0, then
Proof. This is a classical inequality, but now we give an elementary proof.
If x, y, z > 0, then
Using this inequality, we obtain the following
Theorem 2 . If a i
> 0 ( i = 1 , 2 , ..., n ) , then
Proof. We have which is equivalent with
, therefore :
2000 Mathematics Subject Classification. 26D15.
Key words and phrases. Means, AM-GM-HM inequality
References
[1] Bencze, M., Egyenl ő tlenségekr ő l, (In Hungarian), Matematikai Lapok (Kolozsvár),
Nr. 2, 1976, pp.49-54.
[2] Octogon Mathematical Magazine (1993-2009)
[3] Bencze, M., A new proof of CBS inequality, Octogon Mathematical Magazine, Vol.
10, Nr. 2, October 2002, pp. 841-842.

Str. Hărmanului 6, 505600 Săcele-Négyfalu, Jud. Braşov, Romania
E-mail: benczemihaly@yahoo.com

ON SOME PROPERTIES OF CYCLIC QUADRILATERAL
Nicuşor Minculete
Abstract
The aim of this paper is to present another simple proof of the Japanese Theorem and two
interesting relations.
In (see [5]), W. Reyes gave a proof of the Japanese Theorem (see 2, pp. 193) using a result due to the French geometer Victor Thébault. Reyes mentioned that a very long proof of this theorem can be found in (see [1], pp.125-128). In (see [6], pp. 154), P.
Yiu found a simple proof of the Japanese Theorem. Other two proofs of it can be found in books (see [3], pp. 92) and (see [4], pp. 110).
The Japanese Theorem. Let ABCD be a convex quadrilateral inscribed in a circle.
Denote by I a
  a
, I b
  b
, I c
  c
, I d
  d
the incircles of the triangles BCD, CDA, DAB and ABC. Prove the following:
(i) The incenters form a rectangle;
(ii) r a
 r c
 r b
 r d
.
In [3] D. Mihalca, I. Chiţescu and M. Chiriţă use the identity cos A
 cos B
 cos C
Panaitopol r a
 r c

R
 cos x

1
 r
, which is true in any triangle ABC , and in [4] M. E.

R and cos y
 cos z
L.
 cos u

2

 r b

Panaitopol r d
, show that where R is the circumradius, m
 
AB

2 x , m

 
BC

2 y , m

 
CD

2 z and m

 
AD

2 u .
We will give below another simple proof of it.
Proof. (i) In the cyclic quadrilateral ABCD we construct the bisectors which give the incenters of the triangles ABD , ABC and BCD (see figure 1).
2010 Mathematics Subject Classification: 51M04
Key words and phrases: cyclic quadrilateral, Japanese Theorem
References
[1] H. Fukagawa and D. Pedoe, Japanese Temple Geometry , Charles Babbage Research
Centre, Manitoba, Canada, 1989.
[2] R. A. Johnson, Advanced Euclidean Geometry , 1925, Dover reprint.
[3] D. Mihalca, I. Chiţescu and M. Chiriţă,
Geometria patrulaterului , Editura Teora,
Bucureşti, 1998 (in Romanian).
[4] M. E. Panaitopol and L. Panaitopol, Probleme de geometrie rezolvate trigonometric ,
Editura GIL, Zalău, 1994 (in Romanian).

[5] W. Reyes,
An Application of Thébaults Theorem
, Forum Geometricorum, Volume
2(2002), 183-185.
[6] P. Yiu, Euclidean Geometry , Florida Atlantic University Lecture Notes, 1998.
Dimitrie Cantemir University of Braşov, Department of REI, Braşov, Romania
E-mail address: minculeten@yahoo.com

STATEMENT OF COMPLIANCE
The undersigned .................. teacher/student/pupil at ...................... declare on my sole risk that the paper .........................., which I will present at the International Symposium
“Mathematical Education In The Current European Context” in Brasov, 11-12.12.2014 is original and all the sources used are mentioned in the Bibliography .
Participant’s signature, Section responsible teacher,
........................ ...................................
Brasov, December 11th-12 th 2014