loren c. larson - problem solving through problems [ocr]
TRANSCRIPT
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Problem Books in Mathemat ics
S e r i e s E d i t o r : P . R . H a l m o s
Unsolved Problems in Intuit ive Mathematics, Volume I:
Unsolved Problems in Number Theory
by Richard K. Guy
1981. xviii, 161 pages. 17 illus.
Theorem s and Problem s in Functional A nalysis
by A. A. Kiriliov andA .D. Gvishiani ( t r an s . Harold H . McFa dert)
1982. ix, 347 pages. 6 il lus.
Problems in Analysis
by Bernard Getbaum
1982. vii , 228 pages. 9 il lus.
A Problem Seminar
by Donald J. Newm an
1982. viii, 113 pages.
Problem-Solv ing Through Problems
by Loren C. Larson
1983. xi, 344 pages. 104 illus.
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To Elizabeth
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Preface
The pu rpose o f t h i s book i s t o i so l a t e and d raw a t t en t ion to t he mos t
i m p o r t a n t p r o b l e m - s o l v i n g t e c h n i q u e s t y p i c a l l y e n c o u n t e r e d i n u n d e r g r a d u -
a t e mathemat i cs and to i l l u s t r a t e t he i r u se by in t e res t ing examples and
p rob lems no t eas i ly found in o ther sou rces . Each sec t ion f ea tu res a s ing le
idea , t he pow er an d ver sa t i li t y o f wh ich is de m on s t r a t e d in the ex am ples
and re in fo rced in t he p rob lems . The book se rves as an in t roduc t ion and
gu ide to t he p rob lems l i t e ra tu re ( e .g . , a s found in t he p rob lems sec t ions o f
u n d e r g r a d u a t e m a t h e m a t i c s j o u r n a l s ) a n d a s a n e a s il y a c c e s s e d r e f e r e n c e of
e s s e n t i a l k n o w l e d g e f o r s t u d e n t s a n d t e a c h e r s o f m a t h e m a t i c s .
T h e b o o k is b o t h a n a n t h o l o g y o f p r o b l e m s a n d a m a n u a l o f i n s t r u c t i o n .
I t con ta in s over 700 p ro b le m s , over one- th i rd o f wh ic h a re work ed in de t a i l .
E a c h p r o b l e m i s c h o s e n f o r it s n a t u r a l a p p e a l a n d b e a u t y , b u t p r i m a r i l y f o r
i t s un ique cha l l enge . Each i s i nc luded to p rov ide the con tex t fo r i l l u s t r a t i ng
a g i v e n p r o b l e m - s o l v i n g m e t h o d . T h e a i m t h r o u g h o u t i s t o s h o w h o w a
bas i c se t o f s imp le t echn iques can be app l i ed in d iver se ways to so lve an
e n o r m o u s v a r i e t y o f p r o b l e m s . W h e n e v e r p o s s i b l e , p r o b l e m s w i t h i n s e c t i o n s
a r e c h o s e n t o c u t a c r o s s e x p e c t e d c o u r s e b o u n d a r i e s a n d t o t h e r e b y
s t r eng then the ev idence tha t a s ing le i n tu i t i on i s capab le o f b road app l i ca -
t i o n . E a c h s e c t i o n c o n c l u d e s w i t h " A d d i t i o n a l E x a m p l e s " t h a t p o i n t t o
o t h e r c o n t e x t s w h e r e t h e t e c h n i q u e i s a p p r o p r i a t e .
T h e b o o k i s w r i t t e n a t t h e u p p e r u n d e r g r a d u a t e l e v e l . I t a s s u m e s a
r u d i m e n t a r y k n o w l e d g e o f c o m b i n a t o r i c s , n u m b e r t h e o r y , a l g e b r a , a n a l y s i s ,
an d geo me t ry . M u c h o f t he con ten t is access ib l e t o s tu den t s wi th on ly a
year o f ca l cu lu s , and a s i zab le p ropo r t ion does no t even r equ i re t h i s .
However , mos t o f t he p rob lems a re a t a l eve l s l i gh t ly beyond the u sua l
co n ten t s o f t ex tb ook s . Th us , t he ma te r i a l i s e spec i a l ly app r op r i a t e fo r
s t u d e n t s p r e p a r i n g f o r m a t h e m a t i c a l c o m p e t i t i o n s .
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vii i
T h e m e t h o d s a n d p r o b l e m s f e a t u r e d i n t h is b o o k a r e d r a w n f r o m m y
e x p e r i e n c e of s o l v i n g p r o b l e m s a t t h is l e v e l . E a c h n e w is s u e of
The
American Mathematical Monthly ( a n d o t h e r u n d e r g r a d u a t e j o u r n a l s ) c o n -
t a i n s m a t e r i a l t h a t w o u l d b e j u s t right f o r i n c l u s i o n . B e c a u s e t h e s e i d e a s
c o n t i n u e t o f i n d n e w e x p r e s s i o n , t h e r e a d e r s h o u l d r e g a r d t h i s c o l l e c t i o n a s
a s t a r t e r s e t a n d s h o u l d b e e n c o u r a g e d t o c r e a t e a p e r s o n a l f i le of p r o b l e m s
a n d s o l u t i o n s t o e x t e n d th i s b e g i n n i n g i n b o t h b r e a d t h a n d d e p t h . O b v i -
o u s l y , w e c a n n e v e r h o p e t o d e v e l o p a " s y s t e m " f o r p r o b l e m - s o l v i n g ;
h o w e v e r , t h e a c q u i r i n g o f i d e a s is a v a l u a b l e e x p e r i e n c e a t a ll s t a g e s of
d e v e l o p m e n t .
M a n y o f t h e p r o b l e m s in t h is b o o k a r e o l d a n d p r o p e r r e f e r e n c i n g is v e r y
d i f f i c u l t . I h a v e g i v e n s o u r c e s f o r t h o s e p r o b l e m s t h a t h a v e a p p e a r e d m o r e
r e c e n t l y in t h e l i t e r a t u r e , c i ti n g c o n t e s t s w h e n e v e r p o s s i b l e . I w o u l d a p p r e c i -
a t e r e c e i v i n g e x a c t r e f e r e n c e s f o r t h o s e I h a v e n o t m e n t i o n e d .
I w i s h t o t a k e t h i s o p p o r t u n i t y t o e x p r e s s m y t h a n k s t o c o l l e a g u e s a n d
s t u d e n t s w h o h a v e s h a r e d m a n y h o u r s of e n j o y m e n t w o r k i n g o n t h e s e
p r o b l e m s . I n t h i s r e g a r d I a m p a r t i c u l a r l y g r a t e f u l t o O . E . S t a n a i t i s ,
P r o f e s s o r E m e r i t u s of S t . O l a f C o l l e g e . T h a n k s t o S t . O l a f C o l l e g e a n d t h e
M e l l o n F o u n d a t i o n f o r p r o v i d i n g t w o s u m m e r g r a n t s t o h e l p s u p p o r t t h e
w r i t i n g o f t h is m a n u s c r i p t . F i n a l l y , t h a n k s t o a ll i n d i v i d u a l s w h o c o n t r i b -
u t e d b y p o s i n g p r o b l e m s a n d s h a r i n g s o l u t i o n s . S p e c ia l a c k n o w l e d g e m e n t
g o e s t o M u r r a y s . K l a m k i n w h o f o r o v e r a q u a r t e r of a c e n t u r y h a s s t o o d
a s a g i a n t i n t h e a r e a of p r o b l e m - s o l v i n g a n d f r o m w h o s e p r o b l e m s a n d
s o l u t i o n s I h a v e l e a r n e d a g r e a t d e a l .
M a r c h 2 1 , 1 9 8 3
L O R E N C . L A R S O N
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C o n t e n t s
Chapter 1. Heu ristics 1
1.1. Se arch fo r a Pa t te rn 2
1 .2 . D r a w a F i g u r e 9
1 .3 . F o r m u l a t e a n E q u i v a l e n t P r o b l e m 15
1 .4 . M o d i f y t h e P r o b l e m 2 2
1 .5 . C h o o s e E f f e c t i v e N o t a t i o n 2 5
1 .6 . Exp lo i t Sy m m et ry 30
1 .7 . Div ide i n to Ca ses 36
1 .8 . W o r k B a c k w a r d 4 0
1 .9 . A r g u e b y C o n t r a d i c t i o n 4 5
1 .10 . Pu rsu e Pa r i t y 47
1 .1 1. C o n s i d e r E x t r e m e C a s e s 5 0
1 .12. G ene ra l i ze 54
Chapter 2. Tw o Important Principles: Induction and Pigeo nh ole 58
2 . 1 . I n d u c t i o n : B u i l d o n
P{k)
58
2 . 2 . I n d u c t i o n : S e t U p
P(k
+ 1 ) 6 4
2 . 3 . S t r o n g I n d u c t i o n 6 7
2 . 4 . I n d u c t i o n a n d G e n e r a l i z a t io n 6 9
2 . 5 . R e c u r s i o n 7 4
2 .6 . P ige onh ole Pr inc ip l e 79
Chapter 3. Arithm etic
3 .1 . Grea tes t C om m on Div i sor
3 .2 . M odu lar Ar i thmet i c
8 4
8 4
91
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X Contents
3 .3 - U n i q u e F a c t o r i z a t i o n 1 00
3 . 4 . P o s i t i o n a l N o t a t i o n 1 06
3 .5 . A r i t h m e t i c o f C o m p l e x N u m b e r s 1 14
C h a pter 4 . A l g ebra 1 2 0
4 . 1 . A l g e b r a i c I d e n t i t i e s 1 21
4 . 2 . U n i q u e F a c t o r i z a t i o n o f P o l y n o m i a l s 1 25
4 . 3 . T h e I d e n t i t y T h e o r e m 1 32
4 . 4 . A b s t r a c t A l g e b r a 1 4 4
C h a p t e r 5 . S u m m a t i o n o f S e r i e s 1 5 4
5 . 1. B i n o m i n a l C o e f f i c i e n t s 1 5 4
5 . 2. G e o m e t r i c S e r i e s 1 6 4
5 . 3 . T e l e s c o p i n g S e r i e s 1 7 0
5 . 4 . P o w e r S e r i e s 1 7 6
C ha pter 6 . I n term edi a te R e a l A n a l y s i s 1 9 2
6 .1 . C o n t i n u o u s F u n c t i o n s 1 92
6 .2 . T h e I n t e r m e d i a t e - V a l u e T h e o r e m 1 98
6 . 3 . T h e D e r i v a t i v e 2 0 3
6 .4 . T h e E x t r e m e - V a l u e T h e o r e m 2 0 6
6 . 5 . R o l l e ' s T h e o r e m 2 1 0
6 .6 . T h e M e a n V a l u e T h e o r e m 2 1 6
6 . 7 . L ' H o p i t a l ' s R u l e 2 2 5
6 . 8 . T h e I n t e g r a l 2 2 7
6 .9 . T h e F u n d a m e n t a l T h e o r e m 2 3 4
C h a pter 7 . I nequ a l i t i e s 2 4 1
7 . 1 . B a s i c I n e q u a l i t y P r o p e r t i e s 2 4 1
7 .2 . A r i t h m e t i c - M e a n - G e o m e t r i c - M e a n I n e q u a l i t y 2 4 8
7 .3 . C a u c h y - S c h w a r z I n e q u a l i t y 2 5 4
7 .4 . F u n c t i o n a l C o n s i d e r a t i o n s 2 5 9
7 . 5 . I n e q u a l i t i e s b y S e r i e s 2 6 8
7 . 6. T h e S q u e e z e P r i n c i p l e 2 7 1
C h a p t e r 8 . G e o m e t r y 2 8 0
8 . 1 . C l a s s i c a l P l a n e G e o m e t r y 2 8 0
8 . 2 . A n a l y t i c G e o m e t r y 2 9 1
8 .3 . V e c t o r G e o m e t r y 2 9 9
8 .4 . C o m p l e x N u m b e r s i n G e o m e t r y 3 1 2
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Contents xi
G l o s s a r y o f S y m b o l s a n d D e f i n i t i o n s 3 1 7
S o u r c e s 3 1 9
I n d e x 3 3 1
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Chapter 1. Heuristics
St ra tegy o r t ac t i cs in p rob lem-so lv ing i s ca l l ed
heuristics.
In th i s chap ter we
wi l l be concerned wi th the heu r i s t i cs o f so lv ing mathemat ica l p rob lems .
T h o s e w h o h a v e t h o u g h t a b o u t h e u r i s t i c s h a v e d e s c r i b e d a n u m b e r o f b a s i c
ideas tha t a re typ ica l ly u sefu l . Among these , we sha l l focus on the fo l low-
ing:
(1 ) Search fo r a pa t t e rn .
(2 ) Draw a f igu re .
( 3 ) F o r m u l a t e a n e q u i v a l e n t p r o b l e m .
( 4 ) M o d i f y t h e p r o b l e m .
(5 ) Choose e f fec t ive no ta t ion .
(6 ) Exp lo i t symmet ry .
(7 ) Div ide in to cases .
( 8 ) W o r k b a c k w a r d .
( 9 ) A r g u e b y c o n t r a d i c t i o n .
(10 ) Pu rsue par i ty .
(11 ) Cons ider ex t reme cases .
(12 ) Genera l i ze .
Our in te res t in th i s l i s t o f p rob lem-so lv ing ideas i s no t in the i r descr ip -
t ion bu t in the i r imp lemen ta t ion . By look ing a t examples o f how o thers
have u sed these s imp le bu t powerfu l ideas , we can expec t to improve ou r
p rob lem-so lv ing sk i l l s .
Befo re beg inn ing , a word o f adv ice abou t the p rob lems a t the end o f the
sec t ions : D o no t be over ly conc erne d abo u t u s ing the heu r i s t i c t rea t ed in
tha t s ec t ion . Al though the p rob lems a re chosen to g ive p rac t i ce in the u se
o f the heu r i s t i c , a narro w focu s ma y be p sycho log ica l ly deb i l i t a t in g . ' A
s ing le p rob lem usua l ly admi t s severa l so lu t ions , o f t en emp loy ing qu i t e
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2 t. Heuristics
d i f f e r e n t h e u r i s t ic s . T h e r e f o r e , i t i s b e s t to a p p r o a c h e a c h p r o b l e m w i t h a n
o p e n m i n d r a t h e r t h a n w i t h a p r e c o n c e i v e d n o t i o n a b o u t h o w a p a r t i c u l a r
h e u r i s t i c s h o u l d b e a p p l i e d . I n w o r k i n g o n a p r o b l e m , s o l v i n g i t i s w h a t
m a t t e r s . I t is t h e a c c u m u l a t e d e x p e r i e n c e of a l l t h e i d e a s w o r k i n g t o g e t h e r
t h a t w il l r e s u l t i n a h e i g h t e n e d a w a r e n e s s o f t h e p o s s i b i l i t i e s i n a p r o b l e m .
1.1. Search for a Pattern
V i r t u a l l y a l l p r o b l e m s o l v e r s b e g i n t h e i r a n a l y s i s b y g e t t i n g a f e e l f o r t h e
p r o b l e m , b y c o n v i n c i n g t h e m s e l v e s o f t h e p l a u s i b i l i t y o f t h e r e s u l t . T h i s i s
b e s t d o n e b y e x a m i n i n g t h e m o s t i m m e d i a t e s p e c i a l c a s e s ; w h e n t h i s
e x p l o r a t i o n is u n d e r t a k e n in a s y s t e m a t i c w a y , p a t t e r n s m a y e m e r g e t h a t
w i ll s u g g e s t i d e a s f o r p r o c e e d i n g w i t h t h e p r o b l e m .
1 . 1 . 1 . P r o v e t h a t a s e t o f n ( d i f f e r e n t ) e l e m e n t s h a s e x a c t l y 2 " ( d i f f e r e n t )
s u b s e t s .
W h e n t h e p r o b l e m is s e t i n t h is i m p e r a t i v e f o r m , a b e g i n n e r m a y p a n i c
a n d n o t k n o w h o w t o p r o c e e d . S u p p o s e , h o w e v e r , t h a t th e p r o b l e m w e r e
c a s t a s a q u e r y , s u c h a s
( i) H o w m a n y s u b s e t s c a n b e f o r m e d f r o m a s e t o f n o b j e c t s ?
( ii ) P r o v e o r d i s p r o v e : A s e t w i t h n e l e m e n t s h a s 2" s u b s e t s .
I n e i t h e r o f t h e s e f o r m s t h e r e is a l r e a d y t h e i m p l i c i t s u g g e s t i o n t h a t o n e
s h o u l d b e g i n b y c h e c k i n g o u t a f e w s p e c i a l c a s e s . T h i s i s h o w e a c h p r o b l e m
s h o u l d b e a p p r o a c h e d : r e m a i n s k e p t ic a l of th e r e s u l t u n t i l c o n v i n c e d .
S o l u t i o n 1 . W e b e g i n b y e x a m i n i n g w h a t h a p p e n s w h e n th e se t c o n t a i n s
0 , 1 , 2 , 3 e l e m e n t s ; t h e r e s u l t s a r e s h o w n i n t h e f o l l o w i n g t a b l e :
E l e m e n t s N u m b e r o f
n
o f
S
S u b s e t s o f
S
s u b s e t s o f
S
0 n o n e 0 1
1 x { 0 , { x , } 2
2 x „ x
2
0 , {AT,}, { x
2
} , { J C „ x
2
} 4
3 * „ x
2
, *
3
0 , { * , } , { * , } , { * „ * , } 8
O u r p u r p o s e i n c o n s t r u c t i n g t h i s t a b l e is not only to verify t h e r e s u l t , b u t
a l s o t o l o o k f o r p a t t e r n s t h a t m i g h t s u g g e s t h o w t o p r o c e e d i n t h e g e n e r a l
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1.1. Search for a Pattern
3
c a s e . T h u s , w e a i m t o b e a s s y s t e m a t i c a s p o s s i b l e . I n t h i s c a s e , n o t i c e w h e n
n = 3 , we ha ve l i s t e d f i r s t t he subse t s o f { JC
( )
) a n d t h e n , i n t h e s e c o n d
l i n e , e a c h o f t h e s e s u b s e t s a u g m e n t e d b y t h e e l e m e n t x
3
. T h i s i s t h e k e y
i d e a t h a t a l l o w s u s t o p r o c e e d t o h i g h e r v a l u e s o f n. F o r e x a m p l e , w h e n
n = 4 , t h e s u b s e t s o f S = {x
]t
x
2
,x
3
,x
4
} a r e t h e e i g h t s u b s e t s o f {x
]
,x
2
,x
i
}
( s h o w n i n t h e t a b l e ) t o g e t h e r w i t h t h e e i g h t f o r m e d b y a d j o i n i n g
x
4
t o e a c h
o f t h e s e . T h e s e s i x t e e n s u b s e t s c o n s t i t u t e t h e e n t i r e c o l l e c t i o n o f p o s s i b i l i -
t i e s ; t h u s , a s e t w i t h 4 e l e m e n t s h a s 2
4
( = 1 6) s u b s e t s .
A p r o o f b a s e d o n th i s i d e a i s a n e a s y a p p l i c a t i o n o f m a t h e m a t i c a l
i n d u c t i o n ( s e e S e c t i o n 2 . 1 ).
S o l u t i o n 2 . A n o t h e r w a y t o p r e s e n t t h e i d e a o f t h e l a s t s o l u t i o n i s t o a r g u e
a s f o l l o w s . F o r e a c h n , l e t A„ d e n o t e t h e n u m b e r o f ( d i f f e r e n t ) s u b s e t s o f a
s e t w i t h n ( d i f f e r e n t ) e l e m e n t s . L e t S b e a se t w i t h n + 1 e l e m e n t s , a n d
d e s i g n a t e o n e of it s e l e m e n t s b y
x.
T h e r e i s a o n e - t o - o n e c o r r e s p o n d e n c e
b e t w e e n t h o s e s u b s e t s o f S w h i c h d o n o t c o n t a i n x a n d t h o s e s u b s e t s t h a t
d o c o n t a i n x ( n a m e l y , a s u b s e t T of t h e f o r m e r ty p e c o r r e s p o n d s t o
T
U { * } ) . T h e f o r m e r t y p e s a r e a l l s u b s e t s o f 5 - { x } , a s e t w i t h
n
e l e m e n t s , a n d t h e r e f o r e , i t m u s t b e t h e c a s e t h a t
T h i s r e c u r r e n c e r e l a t io n , t r u e f o r n = 0 , 1 , 2 , 3 c o m b i n e d w i t h t h e f a c t
t h a t A
0
= 1, i m p l i e s t h a t A„~ 2". (A„ = 2A„^, = 2
2
A„_
2
= ••• = 2 ^ 4
0
= 2" .)
S o l u t i o n 3 . A n o t h e r s y s t e m a t i c e n u m e r a t i o n of s u b s e t s c a n b e c a r r i e d o u t
b y c o n s t r u c t i n g a " t r e e " . F o r t h e c a s e n = 3 a n d S =» th e t re e i s a s
s h o w n b e l o w :
E a c h b r a n c h o f t h e t r e e c o r r e s p o n d s t o a d i s t i n c t s u b s e t o f 5 ( t h e b a r o v e r
t h e n a m e o f t h e e l e m e n t m e a n s t h a t i t i s n o t i n c l u d e d i n t h e s e t c o r r e s p o n d -
i n g to t h a t b r a n c h ) . T h e t r e e i s c o n s t r u c t e d i n t h r e e s t a g e s, c o r r e s p o n d i n g t o
t h e t h r e e e l e m e n t s o f S . E a c h e l e m e n t o f S l e a d s t o t w o p o s s i b i l i t i e s : e i t h e r
i t i s i n t h e s u b s e t o r i t i s n o t , a n d t h e s e c h o i c e s a r e r e p r e s e n t e d b y t w o
b r a n c h e s . A s e a c h e l e m e n t i s c o n s i d e r e d , t h e n u m b e r o f b r a n c h e s d o u b l e s .
A
n+ t
=2A„.
Subset
{a. b, c}
U, bj
{a, c}
{b, c}
{b}
{c}
9
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t.
H euristics
T h u s , f o r a t h r e e - e l e m e n t s e t, t h e n u m b e r of b r a n c h e s is 2 x 2 x 2 = 8 . F o r
a n n - e l e m e n t s e t t h e n u m b e r o f b r a n c h e s i s
2 X 2 X
• • •
X 2. - 2" ;
n
t h u s , a s e t w i t h n e l e m e n t s h a s
2"
s u b s e t s .
S o l u t i o n 4 .
S u p p o s e w e e n u m e r a t e s u b s e t s a c c o r d i n g t o t h e i r s iz e . F o r
e x a m p l e , w h e n S = {a,b,c,d} , t h e s u b s e t s a r e
N u m b e r of N u m b e r o f
e l e m e n t s s u b s e t s
0 0 i
1 { 0 } , { 6 } . { c ) , { r f } 4
2
{a,b}, {a,c}, {a,d}, {b,c}, {b,d},
{
c , d )
6
3
{a,b,c}, {a,b,d}, \a,c,d), {b,c,d}
4
4 {a,b,c,d} 1
T h i s b e g i n n i n g c o u l d p r o m p t t h e f o l l o w i n g a r g u m e n t . L e t
S
b e a s e t w i th
n e l e m e n t s . T h e n
N o . o f s u b s e t s o f S
1
= 2 s u b s e t s o f S w i t h
k
e l e m e n t s )
T h e f i n a l s t e p i n t h i s c h a i n of e q u a l i t i e s f o l l o w s f r o m t h e b i n o m i a l t h e o r e m ,
u p o n s e t t i n g
x
= 1 a n d
y = 1.
S o l u t i o n 5 .
A n o t h e r s y s t e m a t i c b e g i n n i n g i s i l l u s t r a t e d in T a b l e 1
.1 ,
w h i c h
l i s t s t h e s u b s e t s o f
S
= {X,,JC
2
, JC
3
}. T o u n d e r s t a n d t h e p a t t e r n h e r e , n o t i c e
t h e c o r r e s p o n d e n c e o f s u b s c r i p t s i n t h e l e f t m o s t c o l u m n a n d t h e o c c u r r e n c e
Table 1 .1
Subset
Tr ip le
B in a r y n u m b er D ec im a l n u m b e r
0
( 0 , 0 , 0 )
0
0
{*3>
( 0 , 0 , 1 ) 1 1
< *
2
}
(0 ,1 ,0 ) 10
2
( 0 , 1 , 1 )
11 3
{*,)
( 1 , 0 , 0 )
100
4
(1 ,0 ,1 ) 101 5
( 1 , 1 , 0 )
110 6
( 1 . 1 , 1 )
i n 7
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5
o f l ' s i n t h e s e c o n d c o l u m n o f t r i p l e s . S p e c i f i c a l l y , if A i s a su bs e t o f
5 = { x „ x
2
, . . . , x „ l , d e f i n e a , , t o r
i
= 1 , 2 , . . . . n , b y
f l if a , < = A ,
a, = i
' [ O if a,£A.
I t i s c l e a r t h a t w e c a n n o w i d e n t i f y a s u b s e t A o f S w i t h ( o j , a
2
,. . . , a„), a n
n - t u p l e o f O 's a n d l ' s . C o n v e r s e l y , e a c h s u c h n - t u p l e w i l l c o r r e s p o n d t o a
u n i q u e s u b s e t o f S. T h u s , t h e n u m b e r o f s u b s e t s o f 5 i s e q u a l t o t h e
n u m b e r of n - t u p l e s o f O 's a n d l ' s . T h i s l a t t e r s et is o b v i o u s l y i n o n e - t o - o n e
c o r r e s p o n d e n c e w i t h t h e s e t o f n o n n e g a t i v e b i n a r y n u m b e r s le s s t h a n 2 " .
T h u s , e a c h n o n n e g a t i v e i n te g e r l es s t h a n 2" c o r r e s p o n d s t o e x a c t l y o n e
s u b s e t o f S, a n d c o n v e r s e l y . T h e r e f o r e , i t m u s t b e t h e c a s e t h a t S h a s 2"
s u b s e t s .
N o r m a l l y , w e w i ll g i v e o n l y o n e s o l u t i o n t o e a c h e x a m p l e — a s o l u t i o n
w h i c h s e r v e s t o i l l u s t r a t e t h e h e u r i s t i c u n d e r c o n s i d e r a t i o n . I n t h i s f i r s t
e x a m p l e , h o w e v e r , w e s i m p l y w a n t e d t o r e i t e r a t e t h e e a r l i e r c l a i m t h a t a
s i n g l e p r o b l e m c a n u s u a l l y b e w o r k e d i n a v a r i e t y of w a y s . T h e l e s s o n t o b e
l e a r n e d i s t h a t o n e s h o u l d r e m a i n f l e x i b l e i n t h e b e g i n n i n g s t a g e s o f
p r o b l e m e x p l o r a t i o n . If a n a p p r o a c h d o e s n ' t s e e m t o l e a d a n y w h e r e , d o n ' t
d e s p a i r , b u t s e a r c h f o r a n e w i d e a . D o n ' t g e t f i x a t e d o n a s in g l e i d e a u n t i l
y o u ' v e h a d a c h a n c e t o t h i n k b r o a d l y a b o u t a v a r i e t y o f a l t e r n a t i v e
a p p r o a c h e s .
1 . 1 . 2 . L e t S„
0
, 5 „ , i , a n d S „
2
d e n o t e t h e s u m o f e v e r y t h i r d e l e m e n t i n t h e
n t h r o w o f P a s c a l ' s T r i a n g l e , b e g i n n i n g o n t h e l e f t w i t h t h e f i rs t e l e m e n t ,
t h e s e c o n d e l e m e n t , a n d t h e t h i r d e l e m e n t r es p e c t i v e ly . M a k e a c o n j e c t u r e
c o n c e r n i n g t h e v a l u e o f S
1 0 0 J
.
S o l u t i o n . W e b e g i n b y e x a m i n i n g l o w - o r d e r c a se s w i t h t h e h o p e o f f i n d i n g
p a t t e r n s t h a t m i g h t g e n e ra l i z e. I n T a b l e 1 .2 , t h e n o n u n d e r l i n e d t e r m s a r e
t h o s e w h i c h m a k e u p t h e s u m m a n d s o f t h e s i ng l y u n d e r l i n e d a n d
1 2 i 2 I 2 - 1
1 3 1 1 3 2 " 3 3
1 4 6 4 ] 4 5 5 6
+
1 5 1 2 10 5 i 5 11 10 " II
J k £ 1 5 2 0 1 5 £ 1 6 22* 21 21
I 7 21. 35 35 21 7 i 7 4 3 4 3 4 2 "
i ^ i
1
—
1
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6
t.
H euristics
d o u b l y u n d e r l i n e d t e r m s a r e t h o s e o f a n d
S
n2
,
r e s p e c t i v e l y . T h e t h r e e
c o l u m n s o n t h e right s h o w t h a t , i n e a c h c a s e , t w o o f t h e s u m s a r e e q u a l ,
w h e r e a s t h e t h i r d i s e i t h e r o n e l a r g e r ( i n d i c a t e d b y a s u p e r s c r i p t + ) o r o n e
s m a l l e r ( i n d i c a t e d b y a s u p e r s c r i p t — ) . I t a l s o a p p e a r s t h a t t h e u n e q u a l
t e r m i n t h i s s e q u e n c e c h a n g e s w i t h i n a c y c l e o f s ix . T h u s , f r o m t h e p a t t e r n
e s t a b l i s h e d i n t h e f i rs t r o w s , w e e x p e c t t h e a n o m a l y f o r n = 8 t o o c c u r i n
t h e m i d d l e c o l u m n a n d it w il l b e o n e ' ^ ^ t h a n t h e o t h e r t w o .
W e k n o w t h a t S „ # + 5
b J
+ S „
a
= 2 " ( se e 1 .1 .1) . S i nc e 100 = 6 x 16 + 4 ,
w e e x p e c t t h e u n e q u a l t e r m t o o c c u r i n t h e t h i r d c o l u m n (S 10 0.2 ) a n d t o b e
o n e m o r e t h a n t h e o t h e r tw o . T h u s
S
l000
= Sioo.i = ^100,2
-
a n d 5
| 0 0
, +
Sioo,i
+
S |oo. i
+
1
=
2
J 0
° . F r o m t h e s e e q u a t i o n s w e a r e l e d t o c o n j e c t u r e
t h a t '
S
l Q M
= i
-
1
A f o r m a l p r o o f o f t h is c o n j e c t u r e i s a s t r a i g h t f o r w a r d
application of
m a t h e m a t i c a l i n d u c t i o n ( se e C h a p t e r 2 ).
1 . 1 3 .
Lttx
it
x
2
,x
3
,... b e a s e q u e n c e o f n o n z e r o r e a l n u m b e r s s a t i s f y i n g
E s t a b l i s h n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s o n x
t
a n d x
2
f o r x„ t o b e a n
i n t e g e r f o r i n f i n i t e l y m a n y v a l u e s o f n .
S o l u t i o n .
T o g e t a f e e l f o r t h e s e q u e n c e , w e w i ll c o m p u t e t h e f i r s t f e w t e r m s ,
e x p r e s s i n g t h e m i n t e r m s o f x, a n d x
2
. W e h a v e ( o m i t t i n g t h e a l g e b r a )
- '
4
3*i - 2X
2
'
x,x
2
W e a r e f o r t u n a t e in t h i s p a r t i c u l a r i n s t a n c e t h a t t h e c o m p u t a t i o n s a r e
m a n a g e a b l e a n d a p a t t e r n e m e r g e s . A n ea s y i n d u c t i o n a r g u m e n t e s t a b l i s h e s
t h a t
- ( n — I ) * , — ( n — 2 ) x
2
'
w h i c h , o n i s o l a t i n g t h e c o e f f i c i e n t o f n, t a k e s t h e f o r m
* I * J
( * i - J t j ) » + ( 2 * 2 - * i ) '
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7
I n t h i s f o r m , w e s e e t h a t i f
x
,
x
2
,
t h e d e n o m i n a t o r w i ll e v e n t u a l l y e x c e e d
t h e n u m e r a t o r i n m a g n i t u d e , s o x
n
t h e n w i ll n o t b e a n i n t e g e r . H o w e v e r , if
JC, = x
2
, a l l t h e t e r m s o f t h e s e q u e n c e a r e e q u a l . T h u s ,
x„
is a n i n t e g e r f o r
i n f i n i t e l y m a n y v a l u e s o f
n
i f and on ly i f
x, = x
2
.
1 . 1 . 4 .
F i n d p o s i t i v e n u m b e r s
n
a n d
a
y
,a
2
a„
su ch th a t a , + • • • +
a„
= 1 0 0 0 a n d t h e p r o d u c t a , a
2
• • • a „ i s a s l a r ge a s po ss ib l e .
S o l u t i o n . W h e n a p r o b l e m i n v o l v e s a p a r a m e t e r w h i c h m a k e s t h e a n a l y s i s
c o m p l i c a t e d , i t is o f t e n h e l p f u l i n t h e d i s c o v e r y s t a g e t o r e p l a c e it t e m p o r a r -
i ly w i t h s o m e t h i n g m o r e m a n a g e a b l e . I n t h i s p r o b l e m , w e m i g h t b e g i n b y
e x a m i n i n g a s e q u e n c e o f s p e c ia l c a s e s o b t a i n e d b y r e p l a c i n g 1 0 0 0 i n t u r n
w i t h 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 I n t h i s w a y w e a r e l e d t o d i s c o v e r t h a t i n a
m a x i m u m p r o d u c t
( i ) n o a , w i l l b e g r e a t e r t h a n 4 ,
( i i ) no o , wi l l equa l I ,
( ii i( a l l a ' s c a n b e t a k e n t o b e 2 o r 3 ( b e c a u s e 4 = 2 x 2 a n d 4 = 2 + 2 ) ,
( i v ) a t m o s t t w o
a-
s w il l e q u a l 2 ( b e c a us e 2 x 2 x 2 < 3 x 3 a n d 2 + 2 + 2
= 3 + 3).
E a c h o f t h e s e i s e a s y t o e s t a b l i s h . T h u s , w h e n t h e p a r a m e t e r i s 1 0 0 0 a s i n
t he p r o b l e m a t h a n d , t h e m a x i m u m p r o d u c t m u s t b e 3
3 3 2
X 2
2
.
1 . 1 3 . L e t S b e a s e t a n d * b e b i n a r y o p e r a t i o n o n
S
s a t i s f y i n g t h e t w o
l a w s
x * x = x f o r a l l x in S,
(x*y)*z e (y*z)*x fo r a l l x,y,z in S.
S h o w t h a t x *y = y * x f o r a l l x, y in S.
S o l u t i o n . T h e s o l u t i o n , w h i c h a p p e a r s s o n e a t l y b e l o w , is a c t u a l l y t h e e n d
r e s u l t o f c o n s i d e r a b l e s c r a t c h w o r k ; t h e p r o c e d u r e c a n o n l y b e d e s c r i b e d a s
a s e a r c h f o r p a t t e r n ( t h e p r i n c i p l e p a t t e r n i s t h e c y c l i c n a t u r e of t h e f a c t o r s
i n t h e s e c o n d c o n d i t i o n ) . W e h a v e , f o r a l l x ,_ y i n S ,
x*y ~{x y)
= [y*(x*y)]*x - [(x*y)*x]*y'« {(y * x) * x) * y - [(x*x)*y]*y
~[(y*y)]*ix*x)=y*x.
P r o b l e m s
D e v e l o p a f e el f o r t h e f o l l o w i n g p r o b l e m s b y s e a r c h i n g f o r p a t t e r n s . M a k e
a p p r o p r i a t e c o n j e c t u r e s , a n d t h i n k a b o u t h o w t h e p r o o f s m i g h t b e c a r r i e d
o u t .
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18 . Heuristics
1 . 1 . 6 . B e g i n n i ng w i th 2 a n d 7 , t h e s e q u e n c e 7 , 1 , 4 , 7 , 4 , 2 , 8 - , . . . i s c o n -
s tructed by mult ip ly ing success ive pairs of i ts members and adjo in ing the
resul t as the next one or two members of the sequence , depending on
whether the product is a one- or a two-digi t number . Prove tha t the d ig i t 6
a p p e a r s a n infinite num ber of limes in th e sequence.
1 .1 .7 . L e t S, deno te the sequence o f pos i t ive in tege rs 1 ,2 ,3 ,4 ,5 ,6 ,
and de f ine the sequence S
n +
, in terms of S„ by ad din g 1 to those in tegers in
S„ which are d iv is ib le by n . Thus, for example , S
2
is 2 , 3 , 4 ,5 , 6 , 7 , . . . , S
3
i s 3 , 3 ,5 ,5 ,7 ,7 D e te rm ine those in tege rs n wi th the p rope r ty tha t the
first n - 1 integers in S„ a re n.
1 .1 .8 . Pro ve tha t a l ist can be m a d e of all th e sub sets of a f ini te set in su ch
a way that
(i) th e em pty set is f irst in th e l ist ,
( i i ) each subset occurs exact ly once , and
(ii i) each subset in the l ist is obtained either by adding one element to the
preceding subset or by dele t ing one e lement of the preceding subset .
1 .1 .9 . De te rm ine the nu m be r of odd b inom ia l coe f f ic ien ts in the expans ion
of (x + 7 ) '
0 0 0
. (See 4.3.5.)
1 . 1 . 1 0 . A we l l -known theorem asse r t s tha t a p r ime p > 2 can be wri t ten as
a sum of two perfec t squares (
p
=
m
2
+ n
2
, with
m
a n d
n
integers) if and
only if p i s one more than a mul t ip le o f 4 . Make a con jec tu re concern ing
which p r im es p > 2 can be wr i t ten in each o f the fo llowing fo rm s , us ing
(not necessarily positive) integers x a n d y: (a) x
2
+ 1 6y
2
, (b ) 4x
2
+ 4 x y +
5y
2
. (See 1.5.10.)
1 .1 .11 . I f is a seq uen ce such th a t for n > I , (2 - a„)a„ = 1 , w ha t
h a p p e n s t o a
n
as n ten ds tow ard inf inity ? (See 7.6.4.)
1 .1 .12 .
Le t 5 be a se t , an d let » be a b ina ry ope ra t io n on S sa t is fy ing the
laws
X
x * ( x * y) = y fo r all x, y in S , '
fo r all x, y in S.
=
V '
J
£ y ° ( y " y)
Show tha t x »y = y * x for a l l x, y in S. s>fr~ / a -'to/1
= X [ y ° ( y
A d d i t i o n a l E x a m p l e s "-
Mos t induc t ion p rob lems a re based on the discovery of a pa t tern . Thus, the
problems in Sect ions 2 .1 , 2 .2 , 2 .3 , 2 .4 offer addi t ional prac t ice in th is
heuristic. Also see 1.7.2, 1.7.7, 1.7.8, 2.5.6, 3.1.1, 3.4.6, 4.3.1, 4.4.1, 4.4.3,
4.4.15, 4.4.16, 4.4.17.
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1.2. Draw a Figure
II
1.2. Draw a Figure
W h e n e v e r p o s s i b l e it is h e l p f u l t o d e s c r i b e a p r o b l e m p i c t o r ia l l y , b y m e a n s
o f a f i g u r e , a d i a g r a m , o r a g r a p h . A d i a g r a m m a t i c r e p r e s e n t a t i o n u s u a l l y
m a k e s it e a s i e r t o a s s i m i l a t e t h e r e l e v a n t d a t a a n d t o n o t i c e r e l a t i o n s h i p s
a n d d e p e n d e n c e s .
1 . 2 . 1 .
A c h o r d o f c o n s t a n t l e n g t h s li d e s a r o u n d i n a s e m i c i r cl e . T h e
m i d p o i n t of t h e c h o r d a n d t h e p r o j e c t i o n s o f i ts e n d s u p o n t h e b a s e f o r m
t h e v e r t i c e s o f a t r i a n g l e . P r o v e t h a t t h e t r i a n g l e is i s o s c e l e s a n d n e v e r
c h a n g e s i t s s h a p e .
S o l u t i o n . L e t AB d e n o t e t h e b a s e o f t h e s e m i c i r c l e , l e t XY b e t h e c h o r d , M
t h e m i d p o i n t o f XY, C a n d D t h e p r o j e c t i o n s o f X a n d Y o n AB ( F i g u r e
1 . 1 ) . L e t t h e p r o j e c t i o n o f
M
o n t o
AB
b e d e n o t e d b y
N.
T h e n
N is
t h e
m i d p o i n t o f
CD
a n d it f o l l o w s t h a t A
CMD
i s i so s c e l e s .
T o s h o w t h a t t h e s h a p e o f t h e t r i a n g l e i s i n d e p e n d e n t of t h e p o s i t i o n o f
t h e c h o r d , i t s u f f i c e s t o s h o w t h a t Z
MCD
r e m a i n s u n c h a n g e d , o r e q u i v a -
l e n c y , t h a t L XCM i s c o n s t a n t , f o r a l l p o s i t i o n s o f XY. T o s e e t h a t t h i s i s
t h e c a s e , e x t e n d XC t o c u t t h e c o m p l e t e d c i r c le a t Z ( F i g u r e 1 .2 ) . T h e n CM
i s p a r a l l e l t o
ZY
( C a n d
M
a r e t h e m i d p o i n t s o f
XZ
a n d
XY,
r e s p e c t i v e l y ) ,
X
A
A C . N D B
Figure 1.1.
Z
Figure 1.2.
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10 t. Heuristics
Figure 1.3.
a n d c o n s e q u e n t l y LXCM = LXZY. B u t LXZY e q u a l s o n e - h a l f t h e a r c
XY, a n d t h is a r c d e p e n d s o n l y o n t h e l e n g t h o f t h e c h o r d XY. T h i s
c o m p l e t e s t h e p r o o f .
O n e m i g h t a s k : H o w i n t h e w o r l d d i d a n y o n e e v e r t h i n k t o e x t e n d
XC
in
t h i s w a y ? T h i s i s p r e c i s e ly t h e s t e p t h a t m a k e s t h e a r g u m e n t s o p r e t t y , a n d
i t is i n d e e d a v e r y d i f f i c u l t s t e p t o m o t i v a t e . A b o u t a l l t h a t c a n b e s a i d i s
t h a t t h e u s e o f a u x i l i a r y l i n e s a n d a r c s ( o f t e n f o u n d b y r e f l e c t i o n , e x t e n s i o n ,
o r r o t a t i o n ) i s a c o m m o n p r a c t i c e i n g e o m e t r y . J u s t t h e a w a r e n e s s o f t h i s
f a c t w i ll a d d t o t h e p o s s i b l e a p p r o a c h e s i n a g i v e n p r o b l e m .
A n o t h e r i n t e r e s t i n g a p p r o a c h t o th i s p r o b l e m i s t o c o o r d i n a t i z e t h e
p o i n t s a n d t o p r o c e e d a n a l y t i c a l l y . T o s h o w t h a t t h e s h a p e o f t h e t r i a n g l e i s
i n d e p e n d e n t o f t h e p o s i t i o n o f t h e c h o r d , it s u f f i c e s t o s h o w t h a t t h e
h e i g h t - t o - b a s e r a t i o ,
MN /CD
, i s c o n s t a n t .
L e t
O
d e n o t e t h e m i d p o i n t o f
AB,
a n d l et
$
=
L YOB.
I t i s c l e a r t h a t t h e
e n t i r e c o n f i g u r a t i o n is c o m p l e t e l y d e t e r m i n e d b y (F i g u re 1 .3 ) .
L e t a - L XOY. U s i n g t h i s n o t a t i o n ,
CD = c o s ® - c o s ( 0 + a ) ,
s i n 9 + s i n ( B + a )
M N ^ i ,
a n d t h e h e i g h t - b a s e r a t i o i s
s in t f + s in (0 + a )
F
W = 2 ( c o s 9 - c o s ( « + , , ) r
I t i s n o t i m m e d i a t e l y c l e a r t h a t t h i s q u a n t i t y is i n d e p e n d e n t o f 0 ; t h i s i s t h e
co n t e n t o f 1 .8 .1 a n d 6 .6 .7 .
1 . 2 . 2 . A p a r t i c l e m o v i n g o n a s t r a i g h t l i n e s t a r t s f r o m r e s t a n d a t t a i n s a
v e l o c i t y t >
0
a f t e r t r a v e r s i n g a d i s t a n c e j
0
. If t h e m o t i o n is s u ch t h a t t h e
a c c e l e r a t i o n w a s n e v e r i n c r e a s in g , f i n d t h e m a x i m u m t i m e f o r t h e t r a n s -
v e r s e .
S o l u t i o n .
F o c u s a t t e n t i o n o n t h e g r a p h o f t h e v e l o c i t y u = o ( / ) ( F i g u r e
1.4).
W e a r e g i v e n t h a t o ( 0 ) = 0 , a n d t h e g r a p h o f v i s n e v e r c o n c a v e u p w a r d
( b e c a u s e t h e a c c e l e r a t i o n , do/dt, i s n e v e r i n c r e a s i n g ) . T h e a r e a u n d e r t h e
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1.2. Draw a Figure
I I
c u r v e is e q u a l t o J
0
( d i s t a n c e t r a v e r s e d = $'f,v(t)dt). F r o m t h i s r e p r e s e n t a -
t i o n , i t i s c l e a r t h a t w e w i l l m a x i m i z e t h e t i m e o f t r a v e r s e w h e n t h e c u r v e
D ( / ) f r o m 0 t o
P
i s a s t r a i g h t l i n e ( F i g u r e 1 .5 ). A t t h e m a x i m u m t i m e /
0
,
| /
0
u
0
= j
0
, o r e q u i v a l e n t l y , i
0
= 2 s
0
/ v
0
.
1 . 2 3 . I f
a
a n d
b
a r e p o s i t iv e i n t e g e r s w i t h n o c o m m o n f a c t o r , s h o w t h a t
S o l u t i o n . W h e n
b
= 1, w e w i l l u n d e r s t a n d t h a t t h e s u m o n t h e l e f t i s 0 s o
t h e r e s u l t h o l d s .
I t i s n o t c l e a r h o w a f i g u r e c o u l d b e u s e f u l i n e s t a b l i s h i n g t h i s p u r e l y
a r i t h m e t i c i d e n ti t y . Y e t , t h e s t a t e m e n t i n v o l v e s t w o i n d e p e n d e n t v a r i a b l e s ,
a
a n d
b,
a n d
a/b, 2a/b,
3 a /
b ,
. . . a r e t h e v a l u e s o f t h e f u n c t i o n
f ( x ) = a x/b w h e n x = 1 , 2 , 3 , . . . , r e s p e c t i v e l y . I s it p o s s i b l e t o i n t e r p r e t
J a / 6 ] , \ 2 a / b \ , . . . g e o m e t r i c a l l y ?
T o m a k e t h i n g s c o n c r e t e , c o n s i d e r t h e c a s e
a = 5
a n d
b
= 7 . T h e p o i n t s
P
k
= {k,5k/1), k =
1 , 2 , . . . , 6 , e a c h l i e o n th e l in e
y
= 5 x / 7 , a n d
1
5 k / I
] e q u a l s t h e n u m b e r o f l a t t i c e p o i n t s o n t h e v e r t i c a l l i n e t h r o u g h
P
k
w h i c h l i e a b o v e t h e x - a x i s a n d b e l o w P
k
. T h u s , 2 A - i i ^k/1 J e q u a l s t h e
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12
t.
H euristics
A
2
4
3
S
2 3 4 5 6 7
Figure 1 .6 .
8
n u m b e r o f l a t t i c e p o i n t s i n t e r i o r t o A
ABC
( s e e F i g u r e 1 .6 ) . B y s y m m e t r y ,
t h i s n u m b e r i s o n e - h a l f t h e n u m b e r o f l a t t i c e p o i n t s in t h e i n t e r i o r o f
r e c t a n g l e ABCD. T h e r e a r e 4 x 6 = 2 4 l a t t i c e p o i n t s i n A BCD, w h i c h
m e a n s t h a t t r i a n g l e ABC c o n t a i n s 12 i n t e r i o r l a t t i c e p o i n t s .
T h e s a m e a r g u m e n t g o e s t h r o u g h i n t h e g e n e r a l c a s e . T h e c o n d i t i o n t h a t
a a n d b h a v e n o c o m m o n f a c t o r a s s u r e s u s t h a t n o n e o f t h e l a t t i c e p o i n t s i n
t h e i n t e r i o r o f ABCD w i l l f a l l o n t h e l i n e / = ax/b. T h u s ,
= j ( N o . of l a t t i c e p o i n t s i n t h e i n t e r i o r o f ABCD)
1 . 2 . 4 ( T h e h a n d s h a k e p r o b l e m ) . M r . a n d M r s . A d a m s r e c e n t ly a t t e n d e d
a p a r t y a t w h i c h t h e r e w e r e t h r e e o t h e r c o u p l e s . V a r i o u s h a n d s h a k e s t o o k
p l a c e . N o o n e s h o o k h a n d s w i t h h i s / h e r o w n s p o u s e , n o o n e s h o o k h a n d s
w i t h t h e s a m e p e r s o n t w i c e , a n d of c o u r s e , n o o n e s h o o k h i s / h e r o w n h a n d .
A f t e r a ll t h e h a n d s h a k i n g w a s f i n i s h e d , M r . A d a m s a s k e d e a c h p e r s o n ,
i n c l u d i n g h i s w i f e , h o w m a n y h a n d s h e o r s h e h a d s h a k e n . T o h i s s u r p r i s e ,
e a c h g a v e a d i f f e r e n t a n s w e r . H o w m a n y h a n d s d i d M r s . A d a m s s h a k e ?
S o l u t i o n .
A l t h o u g h a d i a g r a m is n o t e s s e n t i a l t o t h e s o l u t i o n , it i s h e l p f u l t o
v i e w t h e d a t a g r a p h i c a l l y i n t h e f o l l o w i n g f a s h i o n . R e p r e s e n t t h e e i g h t
i n d i v i d u a l s b y t h e e i g h t d o t s a s s h o w n i n F i g u r e 1 . 7 .
N o w t h e a n s w e r s t o M r . A d a m s ' q u e r y m u s t h a v e b e e n t h e n u m b e r s
0 , 1 , 2 , 3 , 4 , 5 , 6 . T h e r e f o r e , o n e o f t h e i n d i v i d u a l s , s a y A, h a s s h a k e n h a n d s
w i t h s i x o t h e r s , s a y B,C,D
S
E,F,G. I n d i c a t e t h i s o n t h e g r a p h b y d r a w i n g
l i n e s e g m e n t s f r o m
A
t o t h e s e p o i n t s , a s i n F i g u r e 1 .8 .
F r o m t h i s d i a g r a m , w e s e e t h a t H m u s t b e t h a t p e r s o n w h o h a s s h a k e n
n o o n e ' s h a n d . F u r t h e r m o r e , A a n d H m u s t b e s p o u s e s , b e c a u s e A h a s
( a - 1 ) (6 - 1 )
2
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1.2. Draw a Figure II
Figure 1.9.
B y s u p p o s i t i o n , o n e o f B,C,D,E,F,G, h a s s h a k e n f i v e h a n d s . B y
r e l a b e l i n g if n e c e s s a r y w e m a y a s s u m e t h i s p e r s o n i s 8 . A l s o , w e m a y
a s s u m e w i t h o u t l o s s o f g e n e r a l i t y t h a t t h e f i v e w i t h w h o m B h a s s h a k e n
h a n d s a r e l a b e l e d
A,C,D,E,F.
T h i s i s s h o w n i n F i g u r e 1 .9 . F r o m t h i s
s k e t c h w e e a s i l y s e e t h a t G is t h e o n l y p e r s o n w h o c o u l d h a v e a n s w e r e d
"one",
a n d
B
a n d
G
m u s t b e s p o u s e s .
A g a i n , a s b e f o r e , b y r e l a b e l i n g t h e p o i n t s C , D, E i f n e c e s s a r y , w e m a y
a s s u m e t h a t C s h o o k f o u r h a n d s a n d t h a t t h e y b e l o n g e d t o A,B,D,E. T h e
c o r r e s p o n d i n g d i a g r a m i s g i v e n i n F i g u r e 1 .1 0. U s i n g t h e s a m e r e a s o n i n g a s
a b o v e , F a n d C a r e s p o u s e s , a n d c o n s e q u e n t l y , D a n d E a r e s p o u s e s .
E a c h o f
D
a n d
E
h a s s h a k e n h a n d s w i t h t h r e e o t h e r s . S i n c e M r . A d a m s
d i d n o t r e c e i v e t w o " t h r e e " a n s w e r s , D a n d E m u s t c o r r e s p o n d t o M r . a n d
M r s .
Adam s; that is to
s a y , M r s . A d a m s s h o o k h a n d s w i t h t h r e e o t h e r s .
P r o b l e m s
1 . 2 . 5 . T w o p o l e s , w i t h h e i g h t s a a n d
b,
a r e a d i s t a n c e
d
a p a r t ( a l o n g l e v e l
g r o u n d ) . A g u y w i r e s t r e t c h e s f r o m t h e t o p o f e a c h o f t h e m t o s o m e p o i n t
P
o n t h e g r o u n d b e t w e e n t h e m . W h e r e s h o u l d P b e l o c a t e d to minimize t h e
t o t a l l e n g t h o f t h e w i r e ? ( H i n t : L e t t h e p o l e s b e e r e c t e d a t p o i n t s C a n d
D,
a n d t h e i r t o p s b e l a b e l e d A a n d B, r e s p e c t i v e l y . W e w i s h t o m i n i m i z e
AP + PB. A u g m e n t t h i s d i a g r a m b y r e f l e c t i n g i t i n t h e b a s e l i n e CD.
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14
t. Heuristics
S u p p o s e
B
r e f l e c t s t o
B' (PB = PB).
N o w t h e p r o b l e m i s: W h e r e s h o u l d
P
b e l o c a t e d t o m i n i m i z e
AP + PB" )
1 . 2 . 6 .
L e t
ABC
b e a n a c u t e - a n g l e d t r i a n g l e , a n d l e t
D
b e o n t h e i n t e r i o r o f
t h e s e g m e n t
AB.
L o c a t e p o i n t s
E
o n
AC
a n d
F
o n
CB
s u c h t h a t t h e
i n s c r i b e d t r i a n g l e
DEF
w il l h a v e m i n i m u m p e r i m e t e r . ( H i n t : R e f l e c t
D
i n
l i n e
AC
t o a p o i n t
D'\
r e f l e c t
D
i n
CB
t o a p o i n t
D"
a n d c o n s i d e r t h e l i ne
s e g m e n t D'D".)
1 . 2 . 7 . A r e c t a n g u l a r r o o m m e a s u r e s 3 0 f e e t i n l e n g t h a n d 1 2 f e e t i n h e i g h t ,
a n d t h e e n d s a r e 1 2 f e e t i n w i d t h . A f ly , w i t h a b r o k e n w i n g , r e s t s a t a p o i n t
o n e f o o t d o w n f r o m t h e c e i l i n g a t t h e m i d d l e o f o n e e n d . A s m u d g e o f f o o d
i s l o c a t e d o n e f o o t u p f r o m t h e f l o o r a t t h e m i d d l e of t h e o t h e r e n d . T h e f l y
h a s j u s t e n o u g h e n e r g y t o
walk
4 0 f e e t . S h o w t h a t t h e r e i s a p a t h a l o n g
w h i c h t h e f ly c a n w a l k t h a t w i l l e n a b l e it t o g e t t o t h e f o o d .
1 . 2 . 8 . E q u i l a t e r a l t r i a n g l e s
A BP
a n d
ACQ
a r e c o n s t r u c t e d e x t e r n a l l y o n
t h e s i d e s AB a n d AC o f t r i a n g l e ABC. P r o v e t h a t CP = BQ. ( H i n t : F o r a
n i c e s o l u t i o n , r o t a t e t h e p l a n e o f t h e t r i a n g l e 6 0 ° a b o u t t h e p o i n t A, in a
d i r e c t i o n w h i c h t a k e s
B
i n t h e d i r e c t i o n of C . W h a t h a p p e n s t o t h e l i n e
s e g m e n t C P ? )
1 . 2 . 9 .
L e t
a
a n d
b
b e g i v e n p o s i t i v e r e a l n u m b e r s w i t h
a < b.
I f tw o
points
a r e s e l e c t e d a t r a n d o m f r o m a s tr a i g h t li n e s e g m e n t o f l e n g t h
b,
w h a t i s t h e
p r o b a b i l i t y t h a t t h e d i s t a n c e b e t w e e n t h e m i s a t le a s t a ? ( H i n t : L e t
x
a n d /
d e n o t e t h e r a n d o m l y c h o s e n n u m b e r s f r o m t h e i n t e r v a l [ 0 ,6 ] , a n d c o n s i d e r
t h e s e i n d e p e n d e n t r a n d o m v a r i a b l e s o n t w o s e p a r a t e a x e s . W h a t a r e a
c o r r e s p o n d s t o | J C
- y\ >
a ? )
1 . 2 . 1 0 . G i v e a g e o m e t r i c i n t e r p r e t a t i o n t o t h e f o l l o w i n g p r o b l e m . L e t / b e
d i f f e r e n t i a b l e w i th / ' c o n t i n u o u s o n [a, b]. S h o w t h a t if t h e r e i s a n u m b e r c
i n ( a ,
b\
s u c h t h a t
f'(c) =
0 , t h e n w e c a n f i n d a n u m b e r
d
i n
(a, b)
s u c h t h a t
1 J . 1 1 . L e t
a
a n d
b
b e r e a l n u m b e r s ,
a < b.
I n d i c a t e g e o m e t r i c a l l y t h e
p r e c i s e l o c a t i o n o f e a c h o f t h e f o l l o w i n g n u m b e r s : (a + b)/2 (= + {b)\
}a + Lb; } a + i 6 ; [m/(m + n)]a + \n/{m + rt)]b, w h e r e m> 0 a n d
n > 0 . ( T h e l a t t e r n u m b e r c o r r e s p o n d s t o t h e c e n t e r o f g r a v i t y o f a s y s t e m
of
t w o
masses
— o ne , o f m a s s
m,
l o c a t e d a t
a,
a n d t h e o t h e r , of m a s s
n,
l o c a t e d a t
b.)
1 . 2 . 1 2 .
U s e t h e g r a p h o f
y
= s in
x
t o s h o w t h e f o l l o w i n g . G i v e n t r i a n g l e
ABC,
g + C
2 '
( b ) — S _ s i n
B
+ — 2 — s i n C < s in ( —
—
—
B
+ — c l m > 0 , « > 0 .
v
' m + n m + n
V m + n
m
+
n
)
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1.3. Formulate an Equivalent Problem
15
1 . 2 . 1 3 . U s e a d i a g r a m ( a r e c t a n g u l a r a r r a y (a , a , ) ) t o s h o w t h a t
i - 0 y - 0 y = 0/
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1.3. Form ulate an Equ ivalent Problem 17
a l e n t p r o b l e m i s t o f i n d a l l x ( o t h e r t h a n x = 1) w h i c h s a t i s f y x
5
= 1 . T h e s e
a r e t h e f i v e f i f t h r o o t s o f u n i t y , g i v e n b y
x , = c o s ^ i r + • i s i n $ i r ,
x, = c o s f f f + / ' si n J i :
A s a b y - p r o d u c t o f h a v i n g w o r k e d t h is p r o b l e m t w o d i f f e r e n t w a y s ,
1
s ee t h a t
, ^ . . , - 1 + V 5 . ' / l o " + ~ 2 V 5
r
i n + < s in f i r = + 1 - —
E q u a t i n g r ea l a n d i m a g i n a r y p a r t s y i e l d s
- 1 + V 5 V l O + 2V 5
co s 7 2 ° = J
v
, s in 7 2 ° = - r - ^ — .
4 4
( S i m i l a r f o r m u l a s c a n b e f o u n d f o r x
2
, x
3
, a n d x
4
. )
1 3 J .
P
i s a p o i n t i n s i d e a g i v e n t r i a n g l e
ABC; D, £, F
a r e t h e f ee t o f t h e
p e r p e n d i c u l a r s f r o m P t o t h e l i n e s BC, CA, AB, r e s p e c t i v e l y . F i n d a l l P f o r
w h i c h
B C . C A . A B
PD PE PF
S o l u t i o n . D e n o t e t h e l e n g t h s o f BC, AC, AB b y a,b,c, r e s p e c t i v e l y , a n d
PD, PE, PF b y p, q, r , r e s p e c t i v e l y ( s e e F i g u r e 1 .1 1 ). W e w i s h t o m i n i m i z e
a/p + b/q + c/r.
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18 t. Heuristics
N o t i c e t h a t
A r e a
ABC
= A r e a A
BCP
+ A r e a
A CAP +
A r e a
{\ ABP
\ap
+
{b q
+ j c r
ap +• bq + cr
2
T h u s ,
ap + bq + cr
i s a c o n s t a n t , i n d e p e n d e n t o f t h e p l a c e m e n t o f
P.
T h e r e f o r e , i n s t e a d o f m i n i m i z i n g a f p + b /q + c/r , we wil l minimize
(ap + bq + c r X a f p + b/q + c/r). ( T h i s s t e p w i ll a p p e a r m o r e n a t u r a l
a f t e r a s t u d y of i n e q u a l i t i e s w i t h c o n s t r a i n t s t a k e n u p i n S e c t i o n 7 . 3 . ) W e
h a v e
> a
2
+ b
2
+• c
2
+• lab
+ 2
be
+ 2 a c
= ( a + b + c)
2
.
T h e i n e q u a l i t y i n t h e s e c o n d s t e p fo l l o w s f r o m t h e f a c t t h a t f o r a n y t w o
p o s i t i v e n u m b e r s
x
a n d
y
w e h a v e
x / y + y / x >
2 , w i t h e q u a l i t y i f a n d
on ly i f
x = y.
A s a r e s u l t o f t h i s f a c t ,
(ap + bq + cr)(a/p + b/q + c/r)
wil l
a t t a i n i t s m i n i m u m v a l u e (a + b + c ) w h e n , a n d o n l y w h e n , p - q - r .
E q u i v a l e n l l y , a / p + b/q + c /r a t t a i n s a m i n i m u m v a l u e w h e n P i s l o c a t e d
a t t h e i n c e n t e r o f t h e t r i a n g l e .
U
. 4 .
P r o v e t h a t i f t n a n d n a r e p o s i t i v e i n t e g e r s a n d 1 < k < n , t h e n
S o l u t i o n . T h e s t a t e m e n t of t h e p r o b l e m c o n s t i t u t e s o n e of t h e f u n d a m e n t a l
i d e n t i t i e s i n v o l v i n g b i n o m i a l c o e f f i c i e n t s . O n t h e l e f t s i d e i s a s u m o f
p r o d u c t s of b i n o m i a l c o e f f i c i e n t s . O b v i o u s l y , a d i r e c t s u b s t i t u t i o n o f f a c t o -
r i a ls f o r b i n o m i a l c o e f f i c i e n t s p r o v i d e s n o i n s i g h t .
Q u i t e o f t e n , f i n i t e s e r i e s ( e s p e c i a l ly t h o s e w h i c h i n v o l v e b i n o m i a l c o e f f i -
c i e n t s ) c a n b e s u m m e d c o m b i n a t o r i a l l y . T o u n d e r s t a n d w h a t is m e a n t h e r e ,
t r a n s f o r m t h e s e r ie s p r o b l e m i n t o a c o u n t i n g p r o b l e m i n t h e f o l l o w i n g
m a n n e r . L e t S = A U B, w h e r e A i s a se t w i th n e l e m e n t s a n d B is a se t,
d i s j o i n t f r o m
A,
w i t h
m
e l e m e n t s . W e w il l c o u n t , i n t w o d i f f e r e n t w a y s , t h e
n u m b e r o f (d i s t i nc t ) f c -subse ts o f
S.
O n t h e o n e h a n d , t h i s n u m b e r i s ( " J " ) .
O n t h e o t h e r h a n d , t h e n u m b e r o f A > s u b se t s o f S w i t h e x a c t l y / e l e m e n t s
( a p + l
+ +
^
+
£
' 1
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1.3. Formulate an Equivalent Problem
19
f r o m A ( a n d k - i e l e m e n t s f r o m 8) i s ( " X * - / ) - I* f o l l o w s t h a t
+ = N o . of Ar-subsets of S
k
= 2 ( N o . o f ^ - s u b s e t s o f S w i t h i e l e m e n t s f r o m A)
/ = o
( A n o t h e r s o l u t i o n t o t h i s p r o b l e m , b a s e d o n t h e p r o p e r t i e s o f p o l y n o m i a l s ,
i s g i ve n in 4 .3 .2 . )
C o u n t i n g p r o b l e m s c a n o f t e n b e s i m p l if i e d b y " i d e n t i f y i n g " ( b y m e a n s
o f a o n e - t o - o n e c o r r e s p o n d e n c e ) t h e e l e m e n t s o f o n e s e t w i t h t h o s e o f
a n o t h e r s e t w h o s e e l e m e n t s c a n m o r e e a s il y b e c o u n t e d . T h e n e x t t h r e e
e x a m p l e s i l l u s t r a t e t h e i d e a .
1 3 . 5 . O n a c i r c l e n p o i n t s a r e s e l e c t e d a n d t h e c h o r d s j o i n i n g t h e m i n p a i r s
a r e d r a w n . A s s u m i n g t h a t n o t h r e e of t h e s e c h o r d s a r e c o n c u r r e n t ( e x c e p t
a t t h e e n d p o m t s ) , h o w m a n y p o i n t s of i n t e r s e c t i o n a r e t h e r e ?
S o l u t i o n .
T h e c a s e s f o r « = 4 , 5 , 6 a r e s h o w n i n F i g u r e 1 . 1 2 . N o t i c e t h a t
e a c h ( i n te r i o r ) i n t e r s e c t i o n p o i n t d e t e r m i n e s , a n d i s d e t e r m i n e d b y , f o u r o f
t h e g i v e n n p o i n t s a l o n g t h e c i r c l e ( t h e s e f o u r p o i n t s w i l l u n i q u e l y p r o d u c e
t w o c h o r d s w h i c h i n t e r s e c t i n t h e i n t e r i o r o f t h e c i r c l e ) . T h u s , t h e n u m b e r
o f i n t e r s e c t i o n p o i n t s i s ( J ) .
1 3 . 6 . G i v e n a p o s i t i v e i n t e g e r n , f i n d t h e n u m b e r o f q u a d r u p l e s o f i n t e g e r s
(a,b,c,d)
s u c h t h a t 0
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2 0
t. Heuristics
a n d t h e s u b s e t s of f o u r o b j e c t s t a k e n f r o m { 0 , 1 , . . . , n + 3 } . S p e c i f i c a l ly ,
l e t (a,b,c,d), 0
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21
1 3 . 1 2 .
G i v e n n o b j e c t s a r r a n g e d i n a r o w . A s u b s e t o f t h e s e o b j e c t s i s
c a l l e d unfriendly if n o t w o o f i t s e l e m e n t s a r e c o n s e c u t i v e . S h o w t h a t t h e
n u m b e r of u n f r i e n d l y s u b s e t s e a c h h a v i n g k e l e m e n t s is ( "
+
' ) . ( H i n t :
A d o p t a n i d e a s i m i l a r t o t h a t u s e d i n 1 . 3 . 6 . )
1 3
. 1 3 .
L e t a ( n ) b e t h e n u m b e r of r e p r e s e n t a t i o n s of t h e p o s i t i v e i n t e g e r
n
a s a s u m of 1 's a n d 2 ' s t a k i n g o r d e r i n t o a c c o u n t . L e t b(n) b e t h e n u m b e r
o f r e p r e s e n t a t i o n s o f
n
a s a s u m o f i n t e g e r s g r e a t e r t h a n 1 , a g a i n t a k i n g
o r d e r i n t o a c c o u n t a n d c o u n t i n g t h e s u m m a n d n . T h e t a b l e b e l o w s h o w s
t h a t a ( 4 ) = 5 a n d
b(6)
= 5:
a - s u m s _ ft -sums
1 + 1 + 2 - 4 + 2
1 + 2 + 1 3 + 3
2 + 1 + 1 2 + 4
2 + 2 2 + 2 + 2
1 + 1 + 1 + 1 6
( a ) S h o w t h a t
a(n)
=
b(n
+ 2 ) f o r e a c h « , b y d e s c r i b i n g a o n e - t o - o n e
c o r r e s p o n d e n c e b e t w e e n t h e a - s u m s a n d 6 - s u m s .
( b ) S h o w t h a t a ( l ) = l , o ( 2 ) - 2 , a n d f o r n > 2 , a ( « ) = a ( n - l ) +
a(n - 2 .
1 3 . 1 4 . B y f i n d i n g t h e a r e a of a t r i a n g l e i n t w o d i f f e r e n t w a y s , p r o v e t h a t if
p i , p
2
. p i a r e t h e a l t i t u d e s o f a tr i a n g l e a n d r i s t h e r a d i u s o f i t s i n s c r i b e d
c i r c le , t h e n \ / p
{
+
1
/ p
2
+
1
/ / > , =
1
/ r .
1 3 . 1 5 . U s e a c o u n t i n g a r g u m e n t t o p r o v e t h a t f o r i n t e g e r s r , n , 0
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2 2
t.
Heuristics
1.4. Modify the Problem
I n t h e c o u r s e of w o r k o n p r o b l e m
A
w e m a y b e l e d t o c o n s i d e r p r o b l e m .
B
.
C h a r a c t e r i s t i c a l l y , t h is c h a n g e i n p r o b l e m s i s a n n o u n c e d b y s u c h p h r a s e s a s
" i t s u f f ic e s t o s h ow t h a t . . . " o r " w e m a y a s s u m e t h a t . . . " o r " w i th o u t
l o s s o f g e n e r a l i t y . . . " . I n t h e l a s t s e c t i o n w e l o o k e d a t e x a m p l e s i n w h i c h
A a n d B w e r e e q u i v a l e n t p r o b l e m s , t h a t is , t h e s o l u t i o n o f e i t h e r o n e o f
t h e m i m p l i e d t h e s o l u t i o n o f t h e o t h e r . I n t h i s s e c t i o n w e l o o k a t c a s e s
w h e r e t h e s o l u t i o n o f t h e m o d i f i e d ( o r a u x i l i a r y ) p r o b l e m , p r o b l e m B,
i m p l i e s t h e s o l u t i o n o f
A,
b u t n o t n e c e s s a r i l y v i c e v e r s a .
1 . 4 . 1 . G i v e n p o s i t iv e n u m b e r s a,b,c,d, p r o v e t h a t
a
3
+ b
3
+
c
3
+
b
3
+ c
3
+ dl
+
c
3
+ d
3
+
a
3
+
d
3
+ a
3
+
b
3
a + 6 + c b + c + d c + d+a d+a+b
> a
2
+ b
2
+ c
2
+ d
2
.
S o l u t i o n .
B e c a u s e o f t h e s y m m e t r y i n t h e p r o b l e m , it is sufficient to prove
t h a t f o r a l l p o s i t i v e n u m b e r s x, y, a n d z
x
3
+ y
3
+ zl_ x
2
+y
2
+ z
2
x+y+z 3
Fo r i f t h i s w er e t h e ca s e , t h e l e f t s i d e o f t h e o r i g in a l i n eq u a l i t y i s a t l e a s t
a
2
+ b
2
+ c
2
, b
2
+ c
2
+ dj_ , c
2
+ d
2
+ a
2
, d
2
+ a
2
+ b
2
3 3 3
+
3
= a
2
+ b
2
+ c
2
+ d
2
.
N o w , t o p r o v e t h i s l a t t e r i n e q u a l i t y ,
there is no loss of generality
i n
s u p p o s i n g t h a t
x + y + z =
1 . F o r if n o t , s i m p l y d i v i d e e a c h s i d e o f t h e
i n e q u a l i t y b y (x +y + z)
2
, a n d l e t X = x/(x +y + z ) , Y = y/{x + y + z),
a n d Z - z / { x + y + z).
T h u s , t h e o r i g i n a l p r o b l e m r e d u c e s t o t h e f o l l o w i n g modified p r o b l e m :
G i v e n p o s i t i v e n u m b e r s X, Y, Z s u c h t h a t X + Y + Z = 1 , p r o v e t h a t
A "
3
+ y
3
+Z
3
> y
+z2
.
( F o r a p r o o f o f t h i s i n e q u a l i t y , s e e 7 . 3 . 5 . )
1 . 4 . 2 .
L e t
C
b e a n y p o i n t o n t h e l i n e s e g m e n t
AB
b e t w e e n
A
a n d
B,
a n d
l e t s e m i c i r c l e s b e d r a w n o n t h e s a m e s i d e o f AB w i t h AB, AC, a n d CB a s
d i a m e t e r s ( F i g u r e 1 .1 3 ). A l s o l e t
D
b e a p o i n t o n t h e s e m i c i r c l e h a v i n g
d i a m e t e r AB s u c h t h a t CD is p e r p e n d i c u l a r t o AB, a n d l e t E a n d F b e
p o i n t s o n t h e s e m i c i r c l e s h a v i n g d i a m e t e r s AC a n d CB, r e s p e c t i v e l y , s u c h
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2 3
Figure 1.13.
t h a t
EF
i s a s e g m e n t o f t h e ir c o m m o n t a n g e n t . S h o w t h a t
ECFD
is a
r e c t a n g l e .
S o l u t i o n . N o t e th a t it is sufficient to show t h a t , 4 , E, a n d D a r e c o l l i n e a r (the
same argument
w o u l d s h o w t h a t
B, F,
a n d
D
a r e c o l l i ne a r ) . F o r if t h i s w e r e
t h e c a s e . Z A £ C = 9 0 ° ( £ is o n c ir c l e AEC), L CFB = 9 0 ° ,
a n d t h e r e s u l t h o l d s . It t u r n s o u t , h o w e v e r , t h a t w i t h o u t s o m e i n s i g h t , t h e r e
a r e m a n y w a y s o f g o i n g w r o n g w i t h t h i s a p p r o a c h ; i t 's d i f f i c u l t t o a v o i d
a s s u m i n g t h e c o n c l u s i o n .
O n e w a y o f g a i n i n g i n s i g h t i n t o t h e r e l a t i o n s h i p s a m o n g t h e p a r a m e t e r s
i n a p r o b l e m i s t o n o t i c e t h e e f f e c t w h e n o n e o f t h e m i s a l l o w e d t o v a r y
( p r o b l e m m o d i f i c a t i o n ) . I n t h i s p r o b l e m , l et
D
v a r y a l o n g t h e c i r c u m f e r -
e n c e . L e t G a n d H ( F i g u r e 1 . 14 ) d e n o t e t h e i n t e r s e c t i o n s o f t h e s e g m e n t s
AD a n d BD w i t h t h e c i r c l e s w i t h d i a m e t e r s AC a n d C f l ( a n d c e n t e r s O a n d
O
') r e s p e c t i v e l y . T h e n
LAGC = LADB = L CHB
= 9 0 ° , s o t h a t
GDHC
i s a r e ct a n g l e . F u r t h e r m o r e , L OGC = Z.OCG ( A OGC i s i so sc e l e s ) , a n d
L CGH = L GCD b e c a u s e GH a n d CD a r e d i a g o n a l s o f a r e c t a n g l e .
T h e r e f o r e ,
L OGH L OCD.
N o w , a s
D
m o v e s t o m a k e
CD
p e r p e n d i c u -
l a r t o
AB, L OGH
w i l l a l s o m o v e t o 9 0 " , s o t h a t
GH
i s t a n g e n t t o c i r c l e
0,
a n d G c o i n c i d e s w i t h E. A similar argument s h o w s GH i s t a ng e t t o c i r c l e
0 \
s o
H = F.
T h i s c o m p l e t e s t h e p r o o f . ( N o t e t h e p h r a s e " a s i m i l a r
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2 4 t. Heuristics
a r g u m e n t , " a n o t h e r s i m p l i f y i n g t e c h n i q u e , h a s t h e s a m e e f f e c t w h e n p l a c e d
a f t e r a n a r g u m e n t a s " i t s u f f i c e s t o s h o w t h a t " h a s w h e n p l a c e d b e f o r e t h e
a r g u m e n t . )
N o t e t h a t w e h a v e s o lv e d t h e p r o b l e m b y s o l v i n g a m o r e g e n e r a l
p r o b l e m . T h i s i s a c o m m o n p r o b l e m - s o l v i n g t e c h n i q u e ; w e w i ll s e e m o r e
e x a m p l e s o f i t i n S e c t i o n 1 .1 2 .
1 . 4 . 3 . P r o v e t h a t t h e r e d o n o t e x i s t p o s i t i v e i n t e g e r s x, y, z s u c h t h a t
S o l u t i o n .
S u p p o s e x, y, a n d z a r e p o s i t i v e i n t e g e r s s u c h t h a t x
2
+y
2
+ z
1
=
2
x y z .
S i n c e
x
2
-h y
2
+ z
2
i s ev en ( = 2
x y z ) ,
e i t h e r t w o o f
x, y,
a n d
z
a r e
o d d a n d t h e o t h e r e v e n , o r a l l t h r e e a r e e v e n . S u p p o s e x, y,z a r e e v e n .
T h e n t h e r e a r e p o s i t i v e i n t e g e r s x^,y
v
Z \ s u c h t h a t x = 2x
u
y = 2y
u
z
= 2z,.
F r o m t h e f a c t t h a t
(2x,)
2
+ (2y,f +
( 2 z , )
2
= 2 ( 2 x
1
K 2 ^
1
) ( 2 z , ) i t f o l -
l o w s t h a t
y
l
,z
l
s a t i s f y
x j + yf + z
2
= 2
2
x,y,z,.
A g a i n , f r o m t h i s e q u a -
t ion , if a r e e v e n , a similar argum ent s h o w s t h e r e w i ll b e p o s i t i v e
i n t e g e r s x
2
,y
2
,z
2
s u c h t h a t x\ + y
2
+ z\ = 2
3
x
2
y
2
z
2
.
C o n t i n u e i n t h i s w a y . E v e n t u a l l y w e m u s t a r r i v e a t a n e q u a t i o n o f t h e
f o r m a
2
+ b
2
+ c
2
= 2"abc w he r e n o t a lJ o f a, b, c a r e ev e n ( a n d h e n c e t w o
of
a, b,c
a r e e v e n a n d o n e is o d d ) .
T h u s , w e a r e l e d t o c o n s i d e r t h e f o l l o w i n g m o d i f i e d p r o b l e m : P r o v e
t h e r e d o n o t e x i s t p o s i t i v e i n t e g e r s
x, y, z
a n d
n,
w i t h
x, y
o d d , s u c h t h a t
( T h i s i s P r o b l e m 1 . 9 .3 . )
1 . 4 . 4 . E v a l u a t e dx.
S o l u t i o n .
T h e u s u a l i n t e g r a t i o n t e c h n i q u e s s t u d i e d i n f i r s t - y e a r c a l c u l u s w i l l
n o t w o r k o n t h i s i n t e g r a l . T o e v a l u a t e t h e i n t e g r a l w e w i l l t r a n s f o r m t h e
s i n g l e i n t e g r a l i n t o a d o u b l e i n t e g r a l .
L e t I = f™e~* dx. T h e n
x
2
+ y
2
+ z
2
= Ixyz.
x
2
+ y
2
+ z
2
= 2"xyz.
°° L
e
e y l
dxdy
- r r - " ^
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1.5. Choose Effective Notation
2 5
N o w c h a n g e to a n e q u i v a l e n t in t e g r a l b y s w i t c h i n g t o p o l a r c o o r d i n a t e s .
W e t h e n h a v e
I t f o l l o w s t h a t 1 = y f n / 2 .
A m o d i f i e d ( a u x i l i a r y ) p r o b l e m c a n a r i s e i n m a n y w a y s . I t m a y c o m e
a b o u t w i t h a c h a n g e i n n o t a t i o n ( a s i n 1 . 4 . 4 ; s e e S e c t i o n 1 .5 ) o r b e c a u s e of
s y m m e t r y ( a s in 1 . 4 . 1 ; s e e S e c t i o n 1 .6 ). O f t e n it i s t h e r e s u l t o f " w o r k i n g
b a c k w a r d " ( s e e S e c t i o n 1 .8 ) o r a r g u i n g b y c o n t r a d i c t i o n ( a s i n 1 . 4 .3 ; se e
S e c t i o n 1 .9 ). I t i s n o t u n c o m m o n t o c o n s i d e r a m o r e g e n e r a l p r o b l e m a t t h e
o u t s e t ( a s i n 1 .4 .2 ; s e e S e c t i o n 1 .1 2 ). T h u s w e s e e t h a t p r o b l e m m o d i f i c a t i o n
i s a v e r y g e n e r a l h e u r i s t i c . B e c a u s e o f t hi s, w e w il l d e f e r a d d i n g m o r e
e x a m p l e s a n d p r o b l e m s , p u t t i n g t h e m m o r e a p p r o p r i a t e l y i n t h e m o r e
s p e c i a l i z e d s e c t i o n s w h i c h f o l l o w .
1.5. Choose Effective Notation
O n e o f t h e f i r s t s t e p s i n w o r k i n g a m a t h e m a t i c s p r o b l e m i s t o t r a n s l a t e t h e
p r o b l e m i n t o s y m b o l i c t e r m s . A t t h e o u t s e t , a l l k e y c o n c e p t s s h o u l d b e
i d e n t i f i e d a n d l a b e l e d ; r e d u n d a n c i e s i n n o t a t i o n c a n b e e l i m i n a t e d a s
r e l a t i o n s h i p s a r e d i s c o v e r e d .
1 . 5 . 1 .
O n e m o r n i n g i t s t a r t e d s n o w i n g a t a h e a v y a n d c o n s t a n t r a t e . A
s n o w p l o w s t a r t e d o u t a t 8 : 0 0 A . M . A t 9 : 0 0 A . M . i t h a d g o n e 2 m i l e s . B y
1 0 :0 0 A . M . it h a d g o n e 3 m i l e s . A s s u m i n g t h a t t h e s n o w p l o w r e m o v e s a
c o n s t a n t v o l u m e o f s n o w p e r h o u r , d e t e r m i n e t h e t i m e a t w h i c h i t s t a r t e d
s n o w i n g .
S o l u t i o n . I t i s d i f f i c u l t t o i m a g i n e t h e r e is e n o u g h i n f o r m a t i o n i n t h e
p r o b l e m t o a n s w e r t h e q u e s t i o n . H o w e v e r , if t h e r e is a w a y , w e m u s t
p r o c e e d s y s t e m a t i c a l l y b y f i rs t i d e n t i f y i n g t h o s e q u a n t i t i e s t h a t a r e u n -
k n o w n . W e i n t r o d u c e t h e f o l l o w i n g n o t a t i o n : L e t
t
d e n o t e t h e t i m e t h a t h a s
e l a p s e d s i n c e it s t a r t e d s n o w i n g , a n d l e t T b e t h e t i m e a t w h i c h t h e p l o w
g o e s o u t ( m e a s u r e d f r o m I ~ 0 ) . L e t x(t) b e t h e d i s t a n c e th e p l o w h a s g o n e
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2 6 I. Hfurjslics
a t t i m e
t
( w e a r e o n l y i n t e r e s t e d i n x ( / ) f o r
t > T).
F i n a l l y , l e t
h(i)
d e n o t e
t h e d e p t h o f t h e s n o w a t t i m e t.
W e a r e n o w r e a d y t o t r a n s l a t e t h e p r o b l e m i n t o s y m b o l i c t e r m s . T h e f a c t
t h a t t h e s n o w i s f a l l i n g a t a c o n s t a n t r a t e m e a n s t h a t t h e d e p t h i s i n c r e a s i n g
a t a c o n s t a n t r a t e ; t h a t i s ,
« c, c c o n s t a n t .
dt
I n t e g r a t i n g e a c h s i d e y i e l d s
h(t) = ct + d, Cyd c o n s t a n t s .
S i nc e A( 0 ) = 0 , w e ge t d = 0 . T h u s h{t) = ct.
T h e f a c t t h a t t h e p l o w r e m o v e s s n o w a t a c o n s t a n t r a t e m e a n s t h a t t h e
s p e e d o f t h e p l o w is i n v e r s e l y p r o p o r t i o n a l t o t h e d e p t h a t a n y t i m e
t
( f o r
e x a m p l e , twic$ t h e d e p t h corresponds to h a l f th e s p e e d ) . S y m b o l i c a l l y , for
t > T,
dx
=
_k_
dt h(t) '
k
c o n s t a n t
K
— c o n s t a n t .
I n t e g r a t i n g e a c h s i d e y i e l d s
j c ( f ) = K l o g f + C , C c o n s t a n t .
W e a r e g i v e n th r e e c o n d i t i o n s :
x
= 0 w h e n
t = T, x = 2
w h e n
t ~ T +
1,
a n d x = 3 w h e n t = T + 2. W i t h t w o o f t h e s e c o n d i t i o n s w e c a n e v a l u a t e
t h e c o n s t a n t s
K and C,
a n d w i t h t h e t h i r d , w e c a n s o l v e
for T.
I t
turns out
( t h e d e t a i l s a r e n o t o f i n t e r e s t h e r e ) t h a t
T = ~ 0 - 6 1 8 h o u r s » 3 7 m i n u t e s , 5 s e c o n d s .
T h u s , it s t a r t e d s n o w i n g a t 7 : 2 2 : 5 5 A . M .
1 . 5 . 2 .
( a ) I f n i s a p o s i t i v e i n t e g e r s u c h t h a t 2n + 1 i s a p e r f e c t s q u a r e , s h o w t h a t
n + 1 i s t h e s u m o f t w o s u c c e s s i v e p e r f e c t s q u a r e s .
( b ) I f 3 n + 1 i s a p e r f e c t s q u a r e , s h o w t h a t n + 1 i s t h e s u m o f t h r e e p e r f e c t
s q u a r e s .
S o l u t i o n .
B y i n t r o d u c i n g p r o p e r n o t a t i o n , t h i s r e d u c e s t o a s i m p l e a l g e b r a
p r o b l e m . F o r p a r t ( a ) , s u p p o s e t h a t 2n + 1 = s
2
, s a n i n t e g e r . S i n c e s
2
is an
o d d n u m b e r , s o a l s o i s s. L e t t b e a n i n t e g e r s u c h t h a t s = 2 / + 1 . T h e n
2n + 1 = (2t + l )
2
, a n d s o l v i n g f o r n w e f i n d
1 ) ' ~ 1 . 4 f
2
+ 4f , , 3 , , ,
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1.5. Choose Effective Notation
2 7
A
B C
D
Figure 1.15.
C o n s e q u e n t l y ,
it + 1 = 2 l
2
+ 2 / +
1
=
t
1
+ (t +
I )
2
,
( b ) S u p p o s e
3n +
1
= s
2
, s
a n i n t e g e r . E v i d e n t l y , 5 i s n o t a m u l t i p l e of 3 ,
so s = 3 f ± 1 f o r s o m e i n t e g e r t. T h e n 3 n + 1 » ( 3 ? ± l )
2
, a n d t h e r e f o r e
n + 1 = 3t
2
± 2t + 1 = 2t
2
+ (t ± 1)
2
= i
2
+ t
2
+ (t ± l)
2
.
1 . 5 J . I n t r i a ng l e
ABC, AB = AC, D
is t h e m i d p o i n t of
BC. E
i s t h e f o o t
o f t h e p e r p e n d i c u l a r d r a w n
D
t o
AC,
a n d
F
i s t h e m i d p o i n t o f
DE
( F i g u r e
1 . 1 5 ) . P r o v e t h a t
AF
is p e r p e n d i c u l a r t o
BE.
S o l u t i o n .
W e c a n t r a n s f o r m th e p r o b l e m i n t o a l g e b r a i c t e r m s b y c o o r d i n a -
t i z in g t h e r e l e v a n t p o i n t s a n d b y s h o w i n g t h a t t h e s l o p e s
m
B£
a n d
m
AF
a r e
n e g a t i v e r e c i p r o c a l s .
O n e w a y t o p r o c e e d i s t o t a k e t h e t r i a n g l e a s i t a p p e a r s i n F i g u r e . 1 . 15 :
t a k e D a s t h e o r i g i n ( 0 , 0 ) , A = ( 0 , a ) , B = ( - 6 , 0 ) , a n d C = ( 6 , 0 ) . T h i s i s a
n a t u r a l la b e l i n g o f t h e f i g u r e b e c a u s e i t t a k e s a d v a n t a g e o f t h e b i l a t e r a l
s y m m e t r y o f t h e i s o s c e l e s t r i a n g l e ( s e e t h e e x a m p l e s i n S e c t i o n 1 . 6 ) . H o w -
e v e r , i n t h i s p a r t i c u l a r i n s t a n c e , t h i s n o t a t i o n l e a d s t o s o m e m i n o r c o m p l i c a -
t i o n s w h e n w e l o o k f o r t h e c o o r d i n a t e s o f
E
a n d
F.
A b e t t e r c o o r d i n a t i z a t i o n i s t o t a k e A = ( 0 , 0 ) , B = (4a, 4b), C = (4c , 0 ) ,
a s i n F i g u r e 1 . 1 6. T h e n
a
2
+ b
2
= c
2
,
D = (2a + 2c,2b), E = (2a + 2c,0),
a n d
F
= ( 2 a + 2 c ,
b) .
( A l m o s t n o c o m p u t a t i o n h e r e ; a l l r e l e v a n t p o i n t s a r e
c o o r d i n a t i z e d . ) I t f o l l o w s t h a t
- 3
1
2
± It.
3
3
H e n c e ,
a n d t h e p r o o f i s c o m p l e t e .
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2 8 t. Heuristics
Figure 1.16.
1 . 5 . 4 . L e t
—
1 < 0 ) , a n d t h e r e f o r e ,
A
n
converges to 9
2