x PREFAC E T O TH E RUSSIA N EDITIO N

(1) Session s featur e vivid , spontaneou s communicatio n betwee n student s an d

teachers, i n which eac h studen t i s treated individually , i f possible.

(2) The process begins at a rather earl y age: usuall y durin g the 6th grad e (ag e

11-12), an d sometime s eve n earlier .

This boo k wa s written a s a guide especiall y fo r secondar y schoo l student s an d

for thei r teachers . Th e ag e o f the student s wil l undoubtedl y influenc e th e styl e of

the sessions . Thus , a few suggestions :

A) We consider i t wrong to hold a long session for younger student s devoted t o

only one topic. W e believe that i t i s helpful t o chang e th e directio n o f the activit y

even within on e session .

B) I t i s necessar y t o kee p goin g bac k t o materia l alread y covered . On e ca n

do thi s b y usin g problem s fro m olympiad s an d othe r mathematica l contest s (se e

Appendix A) .

C) I n discussin g a topic , tr y t o emphasiz e a fe w o f th e mos t basi c landmark s

and obtai n a complet e understandin g (no t jus t memorization! ) o f these fact s an d

ideas.

D) W e recommen d constan t us e o f non-standar d an d "gamelike " activitie s i n

the sessions , with complet e discussio n o f solutions an d proofs . I t i s important als o

to use recreational problem s an d mathematica l jokes. Thes e can be found i n [5â€”7,

16-18, 26-30] .

We must mentio n her e our predecessorsâ€”those wh o have tried earlie r to creat e

a sor t o f anthology fo r Leningra d mathematica l circles . Thei r book s [32 ] and [43] ,

unfortunately, di d no t reac h a larg e numbe r o f reader s intereste d i n mathematic s

education i n secondary school .

In 1990-91 the original version of the first par t of our book was published by the

Academy of Pedagogical Science s of USSR as a collection of articles [21] written b y

a number o f authors. W e would lik e to thank al l our colleague s whose materials we

used whe n workin g o n th e preparatio n o f the presen t book : Deni s G . Benua , Igo r

B. Zhukov , Ole g A . Ivanov , Alexe y L . Kirichenko , Konstanti n P . Kokhas , Nikit a

Yu. Netsvetaev , an d Ann a G . Prolova .

We also expres s ou r sincer e gratitud e t o Igo r S . Rubanov , whos e pape r o n in -

duction written especially for the second part o f the book [21] (but never published ,

unfortunately) i s included her e a s the chapte r "Induction" .

Our specia l thank s g o to Alexe y Kirichenk o whos e hel p i n th e earl y stage s o f

writing thi s boo k canno t b e overestimated . W e woul d als o lik e t o than k Ann a

Nikolaeva fo r drawin g th e figures.

Â§2. Structur e o f th e boo k

The boo k consist s o f this preface , tw o main parts , Appendi x A "Mathematica l

Contests", Appendix B "Answers , Hints, Solutions", and Appendix C "References" .

The first par t ("Th e Firs t Yea r of Education") begin s with Chapte r Zero , eon-

sisting o f test question s intende d mostl y fo r student s o f age s 10-11 . Th e problem s

of this chapter hav e virtually n o mathematical content , an d thei r mai n objective i s

to reveal the abilities of the students i n mathematics an d logic . Th e rest o f the first

part i s divide d int o 8 chapters. Th e first seve n o f thes e ar e devote d t o particula r

topics, an d th e eight h ("Problem s fo r th e first year" ) i s simpl y a compilatio n o f

problems o n a variet y o f themes .