Mathematical Problems and Puzzles from the Polish Mathematical Olympiads

379 Pages·2010·20.96 MB·English

POPULAR LECTURES IN MATHEMATICS

S. Straszewicz

MATHEMATICAL
PROBLEMS AND
PUZZLES

from the Polish
Mathematical Olympiads

Pergamon Press

PWN-Polish Scientific Publishers
POPULAR LECTURES 18 MATHEMATICS
Pra: 1, Seeonos wom M Seas,

VOLUME 12

Mazkematical Problems and Puzzles
from the
Polish Mathematical Olympiads
TITLES IS THE POPULAR LECTUUMS IN
‘MATHEMATICA SERIES

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Mathematical Problems

and Puzzles

from the Polish
Mathematical Olympiads

TRASZEWICK

JT SMOLEKA

PERGAMON PREAS

ORFORD - LONDON RDINBURETE. REW YORK
PARIS BRANKFERT:

PwN—POLisH

PIIG PUBLISHERS
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PREFACE

‘Pars hove tate of the wovond Role ection, published
in Gin, fs nBiee vars inpeorenmabe were rae,

‘The evatel for seuwndary sda! pupil Lrg se ae Mache
tind Olaapad bev best hell Poland ney yor sins
Toiniie, dee cogeuined: by the, Dosh Nathootaloal rn
undcr cho superrbion of ths Miulatey of Huveation, hich pa
“lave ts teary anvil mesos. Laser contol of the coutest
iin the hare of ental Conpmitivo in Ware an of Re
iatal Grinittns bn Rowen nies: townse Ts nash re
TUitioe waheaniis profntow eallaborsie with seoonary ascol
teacher

‘ecbipstion i eho contre ie colaranp. ‘hey ane open
all muondseyoehest pif the sonrery hut in pete
andiatee foe the Digenpit are mart foes the xt serie
formas (9g 16 ty 18

“Tho Oljuiad compris thee sg.

Stamm ore (Hoparatary) late tacughons Osteber, Novoror
sant Drowenton ht the begaring ef cash rent te Caen
cae sor crt ty eles chasm ey nh
OF problem wlth pgs aor wswstsel ty seve at cually
‘working ab Beane, wlan Ube mits ia question. No ends 36
Inede ts ote whither they wor exnzely on ter owe, Tho pape
SUldinie ay then mated by ther school the oppor
een con sens, lune cy nen toarand The wat the
fst ocltuhs are ten igus tv He ttyad aps ional)
ontest. These tale plow i Bosch—un the sot dey #0 all
fie rere, moponal eautace, andthe mnce provlen's vee at
40 coal the wuneaearbe hye fer pec le
eoret rego stm, The eet lcte 200. dave, Ech tay
Fave problems hsvo v2 be solved ir thr alain tine al foe
ait The wundhinty. werk ovr min wer si pois
of mvabare nf the nol soghmal omaltte: (hey a 72t
‘logo to comupicete ith one azothor crt rveiom Tp
iow the prafearn Thi chun ines Gre pablest Ww
‘th pielibins wars ‘ol fle mist galanrngs vopunied for
ile paupars after the cantor.

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‘The penbiae sau tae eortte reine ory koe
of sus mthematies (vo. dlamonisry alpera. peometey el
‘igonomecy) bus age on cho whelo moro dilalt than iho usa
‘elnalvatrebrs, Ther doar of lt, however, de novnierah,
Fee come dtealntht oly He eno pe
Dab ako Ghee of average abilty should with etain et
risuage to solve say of the pmoboms aud gains nuaahor of
Bovaty me the ronda,

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ena bean prepared ip order ta provide secondary cebu pups
‘wih ealtable topics fo be worked oat ou thet ove, indians
I sollte, train nt extra thre Rovwinige of ute
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ap claims of mest the poblona hae booa given ino eal
‘of iy, ths sender atfetion beng drow to various dcts af
he woreuing: To tin, ever pron In er ord
STAN saumareewitlhog goose 1 arr Sar
ae f the Lopes in suet, Incusing vations capplawteataty
date of clrventary macboreti:ontide the scape ot the soa
‘sion, "Thre oommentanen a given separa fe ra
‘Pravda Bthoeiag Use onal ‘This serigetent ul toe
femteras hae besp afopted in order to ancke eho Doel cacy to
oa even ftv aduteed pupil, wha eve skip ov He mori
‘orton in the amare ory lv eudy ees
‘el Tad in Ue itouetive examples of mathemat al ceasing
lish may situate their oma bine i) pocing yavbleme
nd aveling ratone, kx rao wanes 9 te oe
foram Taye been vou, tsk aig pect seta eupendicus
$Bheains, howeven be bon to show tha elution of problom
tou nent Thou shock svetl sonsalione ade cate
Uitte difrenc'ncraoa of tazening, Be wil bo cbeerred
‘hie pero uf the sokations icon rts hose ara don fy the
file Unoacdsen. Tha fr astanee ploy 13 wat wivad Uy
enue of gromtaie Lnemtien metic ALL) by ane of the
Devtripa the Ted fig wad hm orig sorted TT)
[Steuvang pollen 134 eas ad Dy s gertcptat ta the Fourch
Ulpepinds Hb hae Sereatiy orm nt the Ulgmpice that
[popilehavwe prevent difiregt molations roa. Uns expected
Der the Counties,

8. Smasuemee
PART ONE

Arithmetic and Algebra