NUMBER SYSTEM In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, called di!its to re"resent any number# $%is is t%e decimal system w%ere we use t%e numbers 0 to # 0 is called insi!ni&icant di!it w%ereas 1, 2, 3, 4, 5, 6, 7, 8, are called si!ni&icant di!its# A !rou" o& o& &i!ures, denotin! denotin! a number number is called called a numeral# 'or a !i(en numeral, numeral, we start &rom e)treme ri!%t as *nit+s "lace, $en+s "lace, Hundred+s "lace and so on# e re"resent t%e number 50,78,6,324 as s%own below e read it as 'i&ty crores, se(enty -ei!%t lacs, si)ty nine t%ousands ,t%ree %undred and twenty &our# In t%is numeral. $%e /lace (alue o& 4 is 4 1 4 $%e "lace (alue o& 2 is 2 10 20 $%e "lace (alue o& 3 is 3 100 300 $%e "lace (alue o& is 1000 000 and so on# $%e "lace (alue o& is 6 10000 60000 and so on $%e &ace (alue o& a di!it in a number is t%e (alue itsel& w%ere(er it may be# $%us, t%e &ace (alue o& 7 in t%e abo(e numeral is 7# $%e &ace (alue o& 6 in t%e abo(e numeral is 6 and so on#
NATURAL NUMBERS ountin! numbers 1, 2, 3, 4, 5, ### are nown as natural numbers#
$%e set o& all natural numbers can be re"resented by 1, 2, 3, 4, 5, #### WHOLE NUMBERS
I& we include 0 amon! t%e natural numbers, t%en t%e numbers 0, 1, 2, 3, 4, 5, ### are called w%ole numbers# $%e set o& w%ole numbers can be re"resented by 0,1,2,3,4,5, ### learly, e(ery natural number is a w%ole number but 0 is a w%ole number w%ic% is not a natural number#
INTEGERS All countin! numbers and and t%eir ne!ati(es ne!ati(es includin! includin! ero are nown nown as inte!ers . $%e set o& inte!ers can be re"resented by or I ### 9 4, 4, :3, :2, :1, 0, 1, 2, 3, 4 ###### POSITIVE INTEGERS
$%e set I; 1, 2, 3, 4, #### is t%e set o& all "ositi(e inte!ers# learly, "ositi(e inte!ers and natural numbers are synonyms# NEGATIVE INTEGERS
$%e set :1, :2, : 3, ### is t%e set o& all ne!ati(e inte!ers# 0 is neit%er "ositi(e nor ne!ati(e# NON-NEGATIVE INTEGERS
$%e set 0, 1, 2, 3, ### is t%e set o& all non-ne!ati(e inte!ers# RATIONAL NUMBERS p
$%e numbers o& t%e &orm
q
, w%ere " and < are inte!ers and<
0, are nown
as rational numbers# $%e set o& all rational numbers is denoted by =# i#e#
Q
x
: x
p q
, p, q I , q
0
Since e(ery natural number >n? can be written as
e(ery natural number is a rational number# Since 0 can be written as e(ery non-ero inte!er >n? can be written as
n 1
0 1
n 1
and
, ie# e(ery inte!er is a rational
number# @(ery rational number %as a "eculiar c%aracteristic t%at w%en e)"ressed in decimal &orm is e)"ressible eit%er in terminatin! decimal e)"ansion or in nonterminatin! re"eatin! decimals# IRRATIONAL NUMBERS
$%ose numbers w%ic% w%en e)"ressed in decimal &orm are neit%er terminatin! nor re"eatin! decimals are nown as irrational numbers, e#!# ,
3
,
5
,
2
,
8
etc#
ote t%at t%e e)act (alue o& is not227, 227 is rational but is irrational#
,
227 is a""ro)imate (alue o& # Similarly, 3#14 is not an e)act (alue o& it#
REAL NUMBERS
$%e rational and irrational numbers combined to!et%er are called real numbers, e#!136,12,
5
,45, etc# are real numbers# $%e set o& all real numbers is
denoted by B# EVEN NUMBERS
All t%ose numbers numbers w%ic% w%ic% are e)actly di(isible by 2 are called called e(en numbers, numbers, e#!# 2, 6, 8,10, etc#, are e(en numbers# ODD NUMBERS
All t%ose numbers numbers w%ic% w%ic% are not e)actly e)actly di(isible di(isible by 2 are called odd numbers, e#!# 1, 3, 5, 7 etc#, are odd numbers# PRIME NUMBERS
A natural number number ot%er ot%er t%an 1, is is a "rime number number i& it is di(isible di(isible by 1 and and itsel& only# 'or e)am"le, eac% o& t%e numbers 2, 3, 5, 7 etc#, are "rime numbers#
COMPOSITE NUMBERS
atural numbers !reater t%an 1 w%ic% are not "rime are nown as com"osite numbers# 'or e)am"le, eac% o& t%e numbers 4, 6, 8, , 12, etc#, are com"osite numbers# ote. 1# $%e number 1 is neit%er a "rime number nor a com"osite number# 2# 2 is t%e only e(en number w%ic% is "rime# 3# /rime numbers u" to 100 are. 2, 3, 5, 7, 11, 11, 13, 17, 1, 23, 2, 31, 37, 41, 43, 47, 53, 5, 61, 67, 71, 73, 7, 83, 8, 7, i#e# 25 "rime numbers between 1 and 100# 4# $wo numbers w%ic% %a(e only 1 as t%e common &actor are called co-"rimes or relati(ely "rime to eac% ot%er, e#!# 3 and 5 are co-"rimes# ote t%at t%e numbers w%ic% are relati(ely "rime need not necessarily be "rime numbers, e#!# 16 and 17 are relati(ely "rime alt%ou!% 16 is not a "rime number# 2, 3, 5, 7, 11, 13, 17, 1, 23, 2, 31, 37, 41, 43, 47, 53, 5, 61, 67, 71, 73, 7, 83, 8, 7, i#e# 25 "rime numbers between 1 and 100# 4# $wo numbers w%ic% %a(e only 1 as t%e common &actor are called co-"rimes or relati(ely "rime to eac% ot%er, e#!# 3 and 5 are co-"rimes# ote t%at t%e numbers w%ic% are relati(ely "rime need not necessarily be "rime numbers, e#!# 16 and 17 are relati(ely "rime alt%ou!% 16 is not a "rime number# TESTS OF DIVISIBILITY
Divisibility by 2: A number is di(isible by 2 i& t%e unit?s di!it is ero or di(isible
by 2# 'or e)am"le, 24,16,108, etc#, are all di(isible by 2# Divisibility by 3: A number is di(isible by 3 i& t%e sum o& di!its in t%e number is
di(isible by 3# 'or e)am"le, t%e number 6543 is di(isible by 3 since 6 ; 5 ; 4 ; 3 18, w%ic% is di(isible by 3# Divisibility by : A number is di(isible by 4 i& t%e number &ormed by t%e last two
di!its Cten?s di!it and unit?s di!itD is di(isible by 4 or are bot% ero# 'or e)am"le, t%e number 236 is di(isible by 4 since 36 is di(isible by 4# Divisibility by !: A number is di(isible by 5 i& t%e unit?s di!it in t%e number is 0
or 5# 'or e)am"le, 14820, 605, 850, 35, etc#, are all di(isible by 5# Divisibility by ": A number is di(isible by 6 i& t%e number is e(en and sum o& its
di!its is di(isible by 3# 'or e)am"le, t%e number 6324 is di(isible by 6 since it is e(en and sum o& its di!its 6 ; 3 ; 2; 4 15is di(isible by 3# Divisibility by #: $%e unit di!it o& t%e !i(en number is doubled and t%en it is
subtracted &rom t%e number obtained a&ter omittin! t%e unit di!it# I& t%e remainder is di(isible by 7, t%en t%e !i(en number is also di(isible by 7#
'or e)am"le, consider t%e number 518# En doublin! t%e unit di!it 8 o& 448 we !et 16# $%en, 51 : 16 35# Since 35 is di(isible by 7, 518 is di(isible by 7# Divisibility by $. A number is di(isible by 8, i& t%e number &ormed by last 3
di!its in di(isible by 8# 'or e)am"le, t%e number 65784 is di(isible by 8 as t%e number &ormed by last t%ree di!its, i#e# 784 is di(isible by 8# Divisibility by %: A number is di(isible by i& t%e sum o& its di!its is di(isible by
# 'or e)am"le, t%e number 25785is di(isible by as t%e sum o& its di!its 2 ; 5 ; 7 ; 8 ; 5 27 is di(isible by # Divisibility by &': A number is di(isible by 10, i& it ends in ero#
'or e)am"le, t%e last di!it o& 630 is ero, t%ere&ore, 480 is di(isible by 10# Divisibility by &&: &&: A number is di(isible by 11, i& t%e di&&erence o& t%e sum o& t%e
di!its at odd "laces and sum o& t%e di!its at e(en "laces is eit%er ero or di(isible by 11# 'or e)am"le, in t%e number 51623, t%e sum o& t%e di!its at odd "laces is 5 ; 6 ; 3 14 and t%e sum o& t%e di!its at e(en "laces is 1;2 3# $%e di&&erence is 14: 3 11, so t%e number is di(isible by 11#
Divisibility by &2: A number is di(isible by 12 i& it is di(isible by 3 and 4#
Divisibility by &$: An e(en number satis&yin! t%e di(isibility test o& is di(isible
by 18# Divisibility by 2!: A number is di(isible by 25 i& t%e number &ormed by t%e last
two di!its is di(isible by 25 or t%e last two di!its are ero# 'or e)am"le, t%e number 83675 is di(isible by 25 as t%e number &ormed by t%e last two di!its is 75 w%ic% is di(isible by 25# Divisibility by &2!: A number is di(isible by 125 i& t%e number &ormed by t%e
last t%ree di!its is di(isible by 125 or t%e last t%ree di!its are ero# 'or e)am"le, t%e number 5250 is di(isible by 125 as 250 is di(isible by 125# Divisibility by $$: A number is di(isible by 88 i& it is di(isible by 11 and 8#
SEF@ IF/EB$A$ /II/G@S I *F@B SS$@F 1DI()*+ti,( Pi(+i.l/: Get $CnD. n N be t%e set o& statements , &or eac% natural number n# I& CiD $CaD is true &or some a J andCiiD $CD is true im"lies $C;1D is true &or all K a , t%en $CmD is true &or all n K a 2D T0/ 1/t/st i(t/1/ *(+ti,(: L M is de&ined by L)M t%e !reatest int!er not e)ceedin! ), &or e(ery real ) 3D Li(/ity .,./ty : I& ab and ac t%en a "b;
4D E*+li)4s Al1,it05: $%e a and b be two non-ero inte!ers# $%en Ca,bD L !cd o& a and bM e)istes and is uni
Get iD
a b mod m
a c
iiD
c d mod m
and
a b mod mod m
#
t%en
(b d ) mod m
mod m a c (b d ) mod
iiiD
mod m ac bd mod
i(D
pa qc
(D
a
n
b
n
( pb qd ) mod m
mod m &or
&or all inte!ers "and < #
all ;(e inte!ers m #
mod m &or e(ery "olynomial wit% inte!er co-e&&icients# (iD f (a) f (b) mod
6D Get be a ;(e inte!er !reaer t%an 1 , say a "b
r, ## are ;(e inte!ers # $%e no# o& ways in w%ic%
can be resol(ed into two &actores is
1 2
( p 1)(q 1)(r 1)....
7Dumber o& ways in w%ic% a com"osite number can be resol(ed into &actors , w%ic% are "rime to eac% ot%er , is
2
n 1
, w%ere n is t%e no# o& distinct "rime "rime
&actors in t%e e)"ression &or 8D Get be ;(e inte!er !reater t%an 1 and let a "b
r, ## are ;(e inte!ers# $%en t%e sum o& all t%e
di(isors in t%e "roduct is e
b q 1 1 c r 1 1 . ........... b 1 c 1 O#
D $%e %i!%est "ower o& "rime " w%ic% is contained in nP is e
n n n 2 3 ..... p p p
%ere LM is t%e !reatest inte!er &unction #
106 E*l/4s T,ti/(t F*(+ti,( : Get be ;(e inte!er !reater t%an 1 # $%en t%e no# o& all t%e ;(e int!ers less t%an and "rime to it is denoted by ob(ious (3)
2
,
( 2)
1
,
( 4)
2
,
(5)
4
, (6) 2 O# $%e &uction
( N )
# It is
is called @uler?s $otient
'unction # 11D I& a, b, O are "rime to eac% ot%er , t%en ( ab )
( a ). (b )
or
(abcd ...)
(a). (b) (c ) (d )......
I& a"b
r, ## are ;(e
inte!ers t%en ( N )
n (1
1
a
)(1
1
b
)(1
1
c
).....
12D E*l/4s T0/,/5 : I& be any ;(e inte!er "rime to # $%en
x ( N ) 1(mod N )
13D F/5t4s Littl/ T0/,/5 : I& " is "rime and n is "rime to " t%en n p 1 1(mod p )
14D 7ILSON4s T0/,/5 : I& " is "rime , t%en
( p 1)!0(mod p )
MATHS OLYMPIAD SUMS EXERCISE 1.1(Number System 1. %. '. +.
C!"#u"!te $ %&'(m)* +1. ,-* t/e "!r0est "!r0est 2e -te0er -te0er su#/ t/!t '1&& -s *-2-s-b"e by 1& S/)3 t/!t %$$1 -s *-2-s-b"e by 11 S/)3 t/!t 114 %1451614 -s *-2-s-b"e by 14
$ (!. ,-* t/e *-77ere#e bet3ee t/e "!r0est
!* t/e sm!""est
umbers t/!t #! be 7)rme* 3-t/ 3 -t/ s-8 *-0-ts. (b T/e !2er!0e )7 -e #)se#ut-2e #)se#ut-2e !tur!" umbers umbers -s 91. ,-* t/e "!r0est )7 t/ese umbers. (# :/!t 3-"" be 44; )7 ! umber 3/)se $$;
-s %+&<
(* ,")3ers !re *r)==e* - ! b!s>et 3/-#/ be#)me *)ub"e !7ter e2ery m-ute. T/e b!s>et be#!me 7u"" - 1& m-utes. A7ter /)3 m!y m-utes t/e b!s>et 3!s /!"7 7u""< 6. A umber #)s-sts )7 ' *-0-ts 3/)se sum -s 4. T/e *-0-t !t t/e u-ts ="!#e -s t3-#e t/e *-0-t !t t/e te?s ="!#e. I7 %4 -s !**e* t) t/e umber@ t/e *-0-ts )7 t/e umber !re re2erse*. ,-* t/e umber. 4 (!:/e ! -te0er ? -s *-2-*e* by 1$.T/e rem!-*er -s 4$. :/!t -s t/e rem!-*er 3/e ? -s *-2-*e* by $4< (b,-* t/e m-ss-0 *-0-ts - t/e 7)"")3-0 mu"t-="-#!t-) sum '$4
$+1
9.,-* t/e "!r0est =r-me 7!#t)r )7 ' 1+'1'1% < !,-* t/e 0re!test umber )7 7)ur *-0-ts
3/-#/ 3/e *-2-*e* by
%@'@+@$@6@4 "e!2es ! rem!-*er rem!-*er 1 - e!#/ #!se. bH)3 m!y =r-me umbers bet3ee 1& !* rem!- =r-me 3/e t/e )r*er )7 t/e-r *-0-ts -s re2erse*< # E8!#t"y )e )7 t/e umbers %'+@%'+$@%'+$6@%'+$64@%'+$649@ %'+$649 -s ! =r-me. :/-#/ )e must -t 1& A t3)*-0-t umber -s su#/ t/!t -7 !
be<
*e#-m!" =)-t -s ="!#e*
bet3ee -ts t3) *-0-ts@ t/e resu"t-0 umber -s )e u!rter )7 t/e sum )7 t3) *-0-ts. :/!t -s t/e )r-0-!" umber< 11 11 ,-* t/e 0re!test umber )7 7-2e *-0-ts 3/-#/
-s *-2-s-b"e by $6@
4%@ 9+ !* 6 "e!2es rem!-*ers +9@ 6+@ 46 !* 99 res=e#t-2e"y. 1% :/-#/ -s 0re!ter < '1 11 )r 14 1+ 1' S/)3 t/!t -s e8!#t"y *-2-s-b"e by 1 %' +$ -s e8!#t"y *-2-s-b"e by $ 1+ ,-* t/e umber )7 =er7e#t #ubes bet3ee 1 !* 1&&&&&1 3/-#/ !re e8!#t"y *-2-s-b"e by 4 < 1$ H)3 m!y umbers 7r)m 1 t) $& !re *-2-s-b"e by e-t/er $ )r 4@ !* /!2e e-t/er e-t/er $)r 4 !s ! *-0-t. *-0-t. 16T/e su!re )7 ! umber )7 t3) *-0-ts -s 7)ur t-mes t/e umber )bt!-e* by re2ers-0 -ts *-0-ts . ,-* t/e umber. 14 ,-* t/e sum )7 t/e *-0-tes - % %&&& . $%&&+
19 Arr!0e t/e 7)"")3-0 - !s#e*-0 )r*er % $$$$@ '''''@ 6%%%%. 1 ,-* !"" t/e =)s-t-2e =er7e#t =er7e#t #ubes t/!t *-2-*e %& ,- ,-** !"" !"" t/e t/e -te -te0er 0erss #")s #")sese ese t) 1&&(1% 1&&(1%F1 F1+' +' %1 (1%'+$6% 1%'+$6 1%'+$4 -s t/e su!re )7 5.. %% H)3 m!y m!y 7)ur 7)ur *-0-t *-0-t umb umber erss #! #! be 7)rme* 7)rme* us-0 us-0 t/e t/e *-0-t *-0-tss 1@ % )"y s) t/!t e!#/ )7 t/ese *-0-ts -s
use* !t "e!st )#e <
%' ,-* t/e 0re!tes 0re!testt umber umber )7 7)ur *-0-ts *-0-ts 1 -s e8!#t"y
3/-#/ 3/-#/ 3/e -#re!s -#re!se* e* by
*-2-s-b"e by %@' @+@$@6 !* 4 <
%+ ,-* t/e "!st t3) (te?s !* u-t?s *-0-t
)7 (%&&' %&&'
%$ ,-* t/e umber )7 =er7e#t #ubes bet3ee 1
!* 1&&&&& 3/-#/
!re e8!#t"y *-2-s-b"e by . %6 ,-* t/e umber )7 =)s-t-2e -te0ers "ess t/! )r
eu!" t) '&&
t/!t !re mu"t-="es )7 ' )r $@ but !re )t mu"t-="es )7 1& )r 1$. %4 T/e =r)*u#t )7 t/e *-0-ts )7 e!#/ )7 t/e t/ree *-0-t umbers 1'9@ %6% !* +'% -s %+. :r-te *)3 !"" t/ree *-0-t umbers /!2-0 %+ !s t/e =r)*u#t )7 t/e *-0-ts. %9. (! ,-* t/e umber )7 *-0-ts - t/e umber %%&&$ .$%&&& 3/e 3r-tte - 7u"". (b ,-* t/e rem!-*er 3/e % %&&$ -s *-2-*e* by 1' % ,-* t3) umbers b)t/ "y-0 bet3ee 6& *-2-*es %+91
!* 4&@ e!#/ )7 3/-#/
'& A umber 3/e *-2-*e* by 4@11 !* 1'(t/e =r-me 1&&1 su##ess-2e"y "e!2e t/e rem!-*ers
7!#t)rs )7
6@1& !* 1% res=e#t-2e"y.
,-* t/e rem!-*er rem!-*er -7 t/e umber -s *-2-*e* by 1&&1. '1 ,-* t/e 0re!test umber )7 7)ur
*-0-ts 3/-#/ 3/e *-2-*e* by
'@ $@ 4@ "e!2es rem!-*ers 1@ '@ $@ 4 res=e#t-2e"y. '% A =r-ter umbers t/e =!0es )7 ! b))> st!rt-0 3-t/ 1. He uses '19 *-0-ts - !"". H)3 m!y
=!0es *)es t/e b))> /!2e<
'' ,-* t/e "!r0est =r-me =r-me 7!#t)r 7!#t)r )7 ' 1% %1% %.66 '+ ,-* t/e 2!"ue )7 S 1 % %% '%+%55559%% '$ ,- ,-** t/e t/e sm!" sm!""e "est st mu" mu"t-= t-="e "e )7 )7 1$ su#/ su#/ t/!t t/!t e!#/ e!#/ *-0*-0-tt )7 t/e t/e mu"t-="e -s e-t/er&?)r 9?. '6 A umber umber X? "e!2es "e!2es t/e s!me rem!-* rem!-*er er 3/-"e *-2-*-0 *-2-*-0 $91+@ $+'&@ $$9. :/!t -s t/e "!r0est "!r0est =)ss-b"e 2!"ue )7 X?. '4 C)s-*er t/e 7)"")3-0 mu"t-="-#!t-) mu"t-="-#!t-) - *e#-m!" )t!t-)s (. (!b# *e71'%@ *eterm-e t/e *-0-ts !@b@#@*@e@7. '9 I7 -s -s ! =)s-t-2e =)s-t-2e -te0er -te0er su#/ t/!t 91& &.*%$*%$5 &.*%$*%$5 3/ere 3/ere * -s -s ! s-0"e *-0-t - *e#-m!" b!se. ,-* ?. ' Let 8 be be t/e LCM )7 ' %&&%1 !* '%&&%1. ,-*
t/e "!st *-0-t )7 8.
+& Let 7& (X1(1X !* 7 (8 7 &(7 1(8 :/ere 1@%@'5.C!"#u"!te 1@%@'5.C!"#u"!te 7 %&& %&&(%&&
EXERCISE 1.%(GEOMETRY 1.%(GEOMETRY 1. I7 !@ b@# !re me!sures me!sures 3/-#/ 7)rm ! tr-!0"e 7)r !"" %@'@+ et#@ =r)2e t/!t
n
a, n b , n c
!"s) 3-""
7)rm ! tr-!0"e
%. G-2e t/e 2erte8 A @ t/e )rt/)#etre H !* t/e #etr)-* G @#)stru#t t/e tr-!0"e. Just-7y y)ur #)stru#t-). '. A su!re s/eet )7 =!=er ABCD ABCD -s s) 7)"*e* t/!t t/e =)-t B 7!""s ) t/e m-* =)-t M )7 CD. Pr)2e Pr)2e t/!t t/!t t/e #re!se #re!se 3-"" *-2-*e *-2-*e BC - t/e r!-) r!-) $'. +. I KABC @ t/e !re! -s t) BC . Pr)2e t/!t
1 2
bc
s.u-ts ( - t/e usus!" )t!t-) . AD -s ! me*-!
ABC
1 2
ADC
$. Pr)2e - !y KABC @ -7 )e !0"e -s 1%& & @ t/e !0"e 7)rme* by t/e 7eet )7 t/e !0"e b-se#t)rs -s r-0/t !0"e*. 6. I KABC @ t/e -#-r#"e t)u#/es t/e s-*es BC @ CA@ AB res=e#t-2e"y res=e#t-2e"y !t D@ E@ , res=e#t-2e"y . I7 t/e r!*-us )7 t/e -#-r#"e -s + u-ts !* -7 BD@ CE@ A, !re #)se#ut-2e -te0ers @ 7-* t/e s-*es )7 KABC. 4. T/r)u0/ ! =)-t P@ 3-t/- ! KABC@ str!-0/t "-es !re *r!3 7r)m t/e !0u"!r =)-ts A@B A@B !* C t) #ut t/e )==)s-te s-*es - D@ E !* , res=e#t-2e"y. Pr)2e t/e 7)"")3-0 - --
PD
AD AP AD
PE
PF
BE
BP BE
CF
CP CF
1
2
9. T/e =!r!""e" s-*es )7 ! tr!=e)-* !re ' #ms !* #ms . T/e ) =!r!""e" s-*es !re + #m !* 6 #m. A "-e =!r!""e" t) t/e b!se *-2-*es t/e tr!=e-)* -t) t3) tr!=e)-*s )7 eu!" eu!" =er-meteres. =er-meteres. ,-* t/e t/e r!t-) -t) 3/-#/ 3/-#/ e!#/ e!#/ )7 t/e ) ) =!r!""e" s-*es -s *-2-*e*. . Y)u !re 0-2e t/ree =!r!""e" "-es. C)stru#t ! eu"!ter!" tr!-0"e ABC su#/ t/!t A 3-"" be ) "-e "1 @ B 3-"" be ) "-e "% !* C 3-"" be ) t/-r* "-e "'. Just-7y y)ur #)stru#t-). ( T/e t/ree =!r!""e" "-es !re )t )7 eu!" eu!" /e-0/t /e-0/t 1&. O -s t/e #-r#um#etre )7 KABC !* M -s t/e m-**"e =)-t )7 t/e me*-! t/r)u0/ A . J)- OM !* =r)*u#e -t t) N s) t/!t OM MN . Pr)2e t/!t N
"-es ) t/e !"t-tu*e t/r)u0/ A. 11. T3) #-r#"es C1 !* C% -terse#t !t t3) *-st-#t =)-ts P !* - ! ="!e. Let ! "-e =!ss-0 t/r)u0/ P meet t/e #-r#"es C1 !* C% - A !* B res=e#t-2e"y. Let Y be t/e m-**"e =)-t )7 AB @ "et Y meet t/e #-r#"es C 1 !* C% - X !* res=e#t-2e"y . Pr)2e t/!t Y -s /te m-* =)-t )7 X !"s). 1%. AB -s ! *-!meter )7 ! #-r#"e !* P -s !=)-t - -ts e8ter-)r. Us-0 )"y ! um!r>e* ru"er !* ! =e#-" @ e8="!- /)3 y)u #)stru#t ! =er=e*-#u"!r 7r)m P t) AB. Just-7y u)ur #)stru#t-). #)stru#t- ). 1'. T/e m-* =)-t )7 t/e /y=)teuse t/r)u0/ M - su#/ ! 3!y t/!t t/e =)s-t-) )7 -t "y-0 -s-*e t/e tr-!0"e -s ' #ms ")0 !* )uts-*e t/e tr-!0"e u=t) t/e )t/er s-*e -s #ms. # ms. ,-* t/e "e0t/ )7 t/e /y=)teuse. 1+. I KABC @
A 2 B . Pr)2e - usu!" )t!t-) t/!t ! % b ( b #
1$. L -s ! =)-t ) t/e s-*e R )7 KPR .
LM !* LN !re *r!3 =!r!""e" t) PR
!* P meet-0 P @ PR !t M@ N res=e#t-2e" res=e#t-2e"yy . MN =r)*u#e* =r)*u#e* meets R - T. T/e @ =r)2e t/!t @ LT LT -s t/e 0e)metr-# me! bet3ee RT !* T. T. 16. I KABC @ M -s t/e m-*=)-t )7 BC. P -s !y =)-t ) AM !* PE @ P, !re =er=e*-#u"!rs =er=e*-#u "!rs t) AB AB @ AC res=e#t-2e"y. I7 E, BC @ =r)2e t/!t !0"e )r
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14. T/ree eu!" #-r#"es )7 r!*-us r t)u#/ e!#/ )t/er. Pr)2e t/!t t/e "-e t/r)u0/ t/e #etres )7 !y t3) #-r#"es meets !y )e )7 t/e #-r#"es !t t/e =)t 3/-#/ -s !t ! *-st!#e )7
7
r 7r)m t/e #etre )7 t/e rem!--0 #-r#"e.
19. A #-r#"e )7 r!*-us % #ms 3-t/ #etre O #)t!-s t/ree sm!""er #-r#"es. T3) )7 t/em t)u#/ t/e )uter #-r#"e !* t)u#/ e!#/ )t/er !t O. T/e t/-r* #-r#"e t)u#/es e!#/ )7 t/e )t/er t/ree #-r#"es . ,-* t/e r!*-us )7 t/e t/-r* #-r#"e. 1. I7 u #)t %% & '&? !* 2
1
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@ =r)2e t/!t u s!t-s7-es ! u!*r!t-# !* 2 !
u!*r!t-# eu!t-) 3-t/ -te0r!" #)e77-#-ets !* 3-t/ "e!*-0 #)e77-#-ets !s u-ty . Use ! 0e)metr-#!" #)stru#t-) t) =r)2e t/-s =r)=)s-t-). %&. T/e me*-! AD )7 KABC-s =er=e*-#u"!r t) AB . Pr)2e t/!t t!A % t!B &. %1. I KABC AB $ @ BC 6 !* AC 4 . P)-ts P !* !re ")#!te* ) AB !* AC res=e#t-2e"y su#/ t/!t PA A eu!"s /!"7 t/e =er-meter )7 KABC. T/e !re! )7 KAP -s /!"7 )7 t/e !re! )7 KABC. I7 PB 8 @ =r)2e t/!t @ P s!t-s7-es t/e u!*r!t-# eu!t-) %8 % %8 $ &. %%. T/e #-r#um7ere#e )7 ! u-t #-r#"e -s *-2-*-e* -t) e-0/t eu!" !r#s by =)-ts A@ B@ C@ D@ E@ ,@ G@ H. C/)r*s #)e#t-0 =)-t A t) e!#/ )7 t/e )t/er =)-ts !re Q)-e*. ,-* t/e =r)*u#t =r)*u#t )7 t/e "e0t/s )7 t/ese se2e #/)r*s. Geer!"-e y)ur resu"t. %'. O -s t/e )rt/)#etre )7 KABC !* @L@M !re t/e m-rr)r -m!0es ) t/e t/ree s-*es .S/)3 t/!t t/e tr!-0"e LM /!s t/e s!me #-r#um#etre #-r#um#etre )7 KABC. %+. T/e s-*es )7 ! tr-!0"e !re )7 "e0t/ !@ b !* # @ 3/ere !@b@# !re -te0ers !* ! b. T/e !0"e )==)s-te t) t/e s-*e # -s 6& &. Pr)2e t/!t ! #!)t be ! =r-me umber. %$. C)stru#t ! re0u"!r /e8!0) us-0 ru"er !* #)m=!ss )"y . Use t/-s #)stru#t-) t) *r!3 t3) #-r#"es 3/-#/ 3-"" -terse#t )rt/)0)!""y. %6. A tr-!0"e /!s s-*es )7 "et0/ts 6@ 9@1& . C!"#u"!te t/e *-st!#e bet3ee -ts -#etre !* #-r#um#etre. %4. Let ABCD be ! #)2e8 u!*r-"!ter!" - 3/-#/ CBD'&&
!*
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t/e me*-!s A !* CL !re mutu!""y =er=e*-#u"!r. =er=e*-#u "!r. '&. I ! ) *e0eer!te tr-!0"e KABC @
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