101 problems in algebra from the training of the usa imo team
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1 0 1 P R O B L E M S I N A L G E B R A
F R O M T H E T R A I N I N G O F T H E U S A I M O T E A M
T A N D R E E S C U t Z F E N D
A M T P U B L I S H I N G
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1 0 1 P R O B L E M S I N A L G E B R A
R O M T H E T R A I N I N G O F T H E U S A 1 M O T E A M
T A N D R U S C U F t Z F F N G
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P u b l i s h e d b y
A M T P U B L I S H I N G
A u s t r a l i a n M a t h e m a t i c s T r u s t
U n i v e r s i t y o f C a n b e r r a A C T 2 6 0 1
A U S T R A L I A
C o p y r i g h t 2 0 0 1 A M T P u b l i s h i n g
T e l e p h o n e : + 6 1 2 6 2 0 1 5 1 3 7
A M T T L i m i t e d A C N 0 8 3 9 5 0 3 4 1
N a t i o n a l L i b r a r y o f A u s t r a l i a C a r d N u m b e r a n d I S S N
A u s t r a l i a n M a t h e m a t i c s T r u s t E n r i c h m e n t S e r i e s I S S N 1 3 2 6 - 0 1 7 0
1 0 1 P r o b l e m s i n A l g e b r a I S B N 1 8 7 6 4 2 0 1 2 X
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T H E A U S T R A L I A N M A T H E M A T I C S T R U S T
E N R I C H M E N T S E R I E S
E D I T O R I A L C O M M I T T E E
C h a i r m a n
G R A H A M H P O L L A R D , C a n b e r r a A U S T R A L I A
E d i t o r
P E T E R J T A Y L O R , C a n b e r r a A U S T R A L I A
W A R R E N J A T K I N S , C a n b e r r a A U S T R A L I A
E D J B A R B E A U , T o r o n t o C A N A D A
G E O R G E B E R Z S E N Y I , T e r r a H a u t e U S A
R O N D U N K L E Y , W a t e r l o o C A N A D A
W A L T E R E M I E N T K A , L i n c o l n U S A
N I K O L A Y K O N S T A N T 1 N O V , M o s c o w R U S S I A
A N D Y L i u , E d m o n t o n C A N A D A
J O R D A N B T A B O V , S o f i a B U L G A R I A
J O H N W E B B , C a p e T o w n S O U T H A F R I C A
T h e b o o k s i n t h i s s e r i e s a r e s e l e c t e d f o r t h e i r m o t i v a t i n g , i n t e r e s t i n g
a n d s t i m u l a t i n g s e t s o f q u a l i t y p r o b l e m s , w i t h a l u c i d e x p o s i t o r y s t y l e
i n t h e i r s o l u t i o n s . T y p i c a l l y , t h e p r o b l e m s h a v e o c c u r r e d i n e i t h e r
n a t i o n a l o r i n t e r n a t i o n a l c o n t e s t s a t t h e s e c o n d a r y s c h o o l l e v e l .
T h e y a r e i n t e n d e d t o b e s u f f i c i e n t l y d e t a i l e d a t a n e l e m e n t a r y l e v e l
f o r t h e m a t h e m a t i c a l l y i n c l i n e d o r i n t e r e s t e d t o u n d e r s t a n d b u t , a t
t h e s a m e t i m e , b e i n t e r e s t i n g a n d s o m e t i m e s c h a l l e n g i n g t o t h e
u n d e r g r a d u a t e a n d t h e m o r e a d v a n c e d m a t h e m a t i c i a n . I t i s b e l i e v e d
t h a t t h e s e m a t h e m a t i c s c o m p e t i t i o n p r o b l e m s
a r e
a p o s i t i v e
i n f l u e n c e o n t h e l e a r n i n g a n d e n r i c h m e n t o f m a t h e m a t i c s .
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T H E A U S T R A L I A N M A T H E M A T I C S T R U S T
E N R I C H M E N T S E R I E S
B o o K s I N T H E S E R I E S
1 A U S T R A L I A N M A T H E M A T I C S C O M P E T I T I O N B O O K 1 1 9 7 8 - 1 9 8 4
J D E d w a r d s , D J K i n g E t P J O ' H a l l o r a n
2
M A T H E M A T I C A L T O O L C H E S T
A W P l a n k E t N H W i l l i a m s
3
T O U R N A M E N T O F T O W N S Q U E S T I O N S A N D S O L U T I O N S 1 9 8 4 - 1 9 8 9
P J T a y l o r
4
A U S T R A L I A N M A T H E M A T I C S C O M P E T I T I O N B O O K 2 1 9 8 5 - 1 9 9 1
P J O ' H a l l o r a n , G P o l l a r d E t P J T a y l o r
5 P R O B L E M S O L V I N G V I A T H E A M C
W A t k i n s
6
T O U R N A M E N T O F T O W N S Q U E S T I O N S A N D S O L U T I O N S 1 9 8 0 - 1 9 8 4
P J T a y l o r
7
T O U R N A M E N T O F T O W N S Q U E S T I O N S A N D S O L U T I O N S 1 9 8 9 - 1 9 9 3
P J T a y l o r
1 8
A S I A N P A C I F I C M A T H E M A T I C S O L Y M P I A D S 1 9 8 9 - 2 0 0 0
H L a u s c h E t C B o s c h G i r a l
9 M E T H O D S O F P R O B L E M S O L V I N G B O O K 1
J B T a b o v F t P J T a y l o r
1 0 C H A L L E N G E 1 9 9 1 - 1 9 9 5
J B H e n r y , J D o w s e y , A R E d w a r d s , U M o t t e r s h e a d ,
A N a k o s E t G V a r d a r o
i i
U S S R M A T H E M A T I C A L O L Y M P I A D S 1 9 8 9 - 1 9 9 2
A M S l i n k o
1 1 2
A U S T R A L I A N M A T H E M A T I C A L O L Y M P I A D S 1 9 7 9 - 1 9 9 5
H L a u s c h E t P J T a y l o r
1 3
C H I N E S E M A T H E M A T I C S C O M P E T I T I O N S A N D O L Y M P I A D S 1 9 8 1 - 1 9 9 3
A L i u
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1 1 4 P O L I S H E t A U S T R I A N M A T H E M A T I C A L O L Y M P I A D S 1 9 8 1 - 1 9 9 5
M E K u c z m a E t E W i n d i s c h b a c h e r
1 1 5
T O U R N A M E N T O F T O W N S Q U E S 1 1 O N S A N D S O L U T I O N S 1 9 9 3 - 1 9 9 7
P J T a y l o r E t A M S t o r o z h e v
1 1 6 A U S T R A L I A N M A T H E M A T I C S C O M P E T I T I O N B O O K 3 1 9 9 2 - 1 9 9 8
W J A t k i n s , J E M u n r o E t P J T a y l o r
1 7
S E E K I N G S O L U T I O N S
J C B u r n s
1 8
1 0 1 P R O B L E M S I N A L G E B R A
T A n d r e e s c u E t Z F e n g
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P R E F A C E
T h i s b o o k c o n t a i n s o n e h u n d r e d h i g h l y r a t e d p r o b l e m s u s e d i n t h e t r a i n -
i n g a n d t e s t i n g o f t h e U S A I n t e r n a t i o n a l M a t h e m a t i c a l O l y m p i a d ( I M O )
t e a m . I t i s n o t a c o l l e c t i o n o f o n e h u n d r e d v e r y d i f f i c u l t , i m p e n e t r a b l e
q u e s t i o n s . I n s t e a d , t h e b o o k g r a d u a l l y b u i l d s s t u d e n t s ' a l g e b r a i c s k i l l s
a n d t e c h n i q u e s . T h i s w o r k a i m s t o b r o a d e n s t u d e n t s ' v i e w o f m a t h e m a t -
i c s a n d b e t t e r p r e p a r e t h e m f o r p o s s i b l e p a r t i c i p a t i o n i n v a r i o u s m a t h e -
m a t i c a l c o m p e t i t i o n s . I t p r o v i d e s i n - d e p t h e n r i c h m e n t i n i m p o r t a n t a r e a s
o f a l g e b r a b y r e o r g a n i z i n g a n d e n h a n c i n g s t u d e n t s ' p r o b l e m - s o l v i n g t a c -
t i c s a n d s t r a t e g i e s . T h e b o o k f u r t h e r s t i m u l a t e s s t u d e n t s ' i n t e r e s t f o r
f u t u r e s t u d y o f m a t h e m a t i c s .
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I N T R O D U C T I O N
I n t h e U n i t e d S t a t e s o f A m e r i c a , t h e s e l e c t i o n p r o c e s s l e a d i n g t o p a r -
t i c i p a t i o n i n t h e I n t e r n a t i o n a l M a t h e m a t i c a l O l y m p i a d ( I M O ) c o n s i s t s
o f a s e r i e s o f n a t i o n a l c o n t e s t s c a l l e d t h e A m e r i c a n M a t h e m a t i c s C o n -
t e s t 1 0 ( A M C 1 0 ) , t h e A m e r i c a n M a t h e m a t i c s C o n t e s t 1 2 ( A M C 1 2 ) ,
t h e A m e r i c a n I n v i t a t i o n a l M a t h e m a t i c s E x a m i n a t i o n ( A I M E ) , a n d t h e
U n i t e d S t a t e s o f A m e r i c a M a t h e m a t i c a l O l y m p i a d ( U S A M O ) . P a r t i c i -
p a t i o n i n t h e A I M E a n d t h e U S A M O i s b y i n v i t a t i o n o n l y , b a s e d o n
p e r f o r m a n c e i n t h e p r e c e d i n g e x a m s o f t h e s e q u e n c e . T h e M a t h e m a t i -
c a l O l y m p i a d S u m m e r P r o g r a m ( M O S P ) i s a f o u r - w e e k , i n t e n s e t r a i n -
i n g o f 2 4 - 3 0 v e r y p r o m i s i n g s t u d e n t s w h o h a v e r i s e n t o t h e t o p o f t h e
A m e r i c a n M a t h e m a t i c s C o m p e t i t i o n s . T h e s i x s t u d e n t s r e p r e s e n t i n g t h e
U n i t e d S t a t e s o f A m e r i c a i n t h e I M O a r e s e l e c t e d o n t h e b a s i s o f t h e i r
U S A M O s c o r e s a n d f u r t h e r I M O - t y p e t e s t i n g t h a t t a k e s p l a c e d u r i n g
M O S P . T h r o u g h o u t M O S P , f u l l d a y s o f c l a s s e s a n d e x t e n s i v e p r o b l e m
s e t s g i v e s t u d e n t s t h o r o u g h p r e p a r a t i o n i n s e v e r a l i m p o r t a n t a r e a s o f
m a t h e m a t i c s . T h e s e t o p i c s i n c l u d e c o m b i n a t o r i a l a r g u m e n t s a n d i d e n t i -
t i e s , g e n e r a t i n g f u n c t i o n s , g r a p h t h e o r y , r e c u r s i v e r e l a t i o n s , t e l e s c o p i n g
s u m s a n d p r o d u c t s , p r o b a b i l i t y , n u m b e r t h e o r y , p o l y n o m i a l s , t h e o r y o f
e q u a t i o n s , c o m p l e x n u m b e r s i n g e o m e t r y , a l g o r i t h m i c p r o o f s , c o m b i n a t o -
r i a l a n d a d v a n c e d g e o m e t r y , f u n c t i o n a l e q u a t i o n s a n d c l a s s i c a l i n e q u a l i -
t i e s .
O l y m p i a d - s t y l e e x a m s c o n s i s t o f s e v e r a l c h a l l e n g i n g e s s a y p r o b l e m s . C o r -
r e c t s o l u t i o n s o f t e n r e q u i r e d e e p a n a l y s i s a n d c a r e f u l a r g u m e n t . O l y m -
p i a d q u e s t i o n s c a n s e e m i m p e n e t r a b l e t o t h e n o v i c e , y e t m o s t c a n b e
s o l v e d w i t h e l e m e n t a r y h i g h s c h o o l m a t h e m a t i c s t e c h n i q u e s , c l e v e r l y a p -
p l i e d .
H e r e i s s o m e a d v i c e f o r s t u d e n t s w h o a t t e m p t t h e p r o b l e m s t h a t f o l l o w .
T a k e y o u r t i m e V e r y f e w c o n t e s t a n t s c a n s o l v e a l l t h e g i v e n p r o b -
l e m s .
T r y t o m a k e c o n n e c t i o n s b e t w e e n p r o b l e m s . A v e r y i m p o r t a n t
t h e m e o f t h i s w o r k i s : a l l i m p o r t a n t t e c h n i q u e s a n d i d e a s f e a t u r e d
i n t h e b o o k a p p e a r m o r e t h a n o n c e
O l y m p i a d p r o b l e m s d o n ' t " c r a c k " i m m e d i a t e l y . B e p a t i e n t . T r y
d i f f e r e n t a p p r o a c h e s . E x p e r i m e n t w i t h s i m p l e c a s e s . I n s o m e c a s e s ,
w o r k i n g b a c k w a r d f r o m t h e d e s i r e d r e s u l t i s h e l p f u l .
E v e n i f y o u c a n s o l v e a p r o b l e m , d o r e a d t h e s o l u t i o n s . T h e y m a y
c o n t a i n s o m e i d e a s t h a t d i d n o t o c c u r i n y o u r s o l u t i o n s , a n d t h e y
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v i i i
I n t r o d u c t i o n
m a y d i s c u s s s t r a t e g i c a n d t a c t i c a l a p p r o a c h e s t h a t c a n b e u s e d e l s e -
w h e r e . T h e f o r m a l s o l u t i o n s a r e a l s o m o d e l s o f e l e g a n t p r e s e n t a -
t i o n t h a t y o u s h o u l d e m u l a t e , b u t t h e y o f t e n o b s c u r e t h e t o r t u r o u s
p r o c e s s o f i n v e s t i g a t i o n , f a l s e s t a r t s , i n s p i r a t i o n a n d a t t e n t i o n t o
d e t a i l t h a t l e d t o t h e m . W h e n y o u r e a d t h e s o l u t i o n s , t r y t o r e -
c o n s t r u c t t h e t h i n k i n g t h a t w e n t i n t o t h e m . A s k y o u r s e l f , " W h a t
w e r e t h e k e y i d e a s ? " " H o w c a n I a p p l y t h e s e i d e a s f u r t h e r ? "
G o b a c k t o t h e o r i g i n a l p r o b l e m l a t e r , a n d s e e i f y o u c a n s o l v e i t
i n a d i f f e r e n t w a y . M a n y o f t h e p r o b l e m s h a v e m u l t i p l e s o l u t i o n s ,
b u t n o t a l l a r e o u t l i n e d h e r e .
A l l t e r m s i n b o l d f a c e a r e d e f i n e d i n t h e G l o s s a r y . U s e t h e g l o s s a r y
a n d t h e r e a d i n g l i s t t o f u r t h e r y o u r m a t h e m a t i c a l e d u c a t i o n .
M e a n i n g f u l p r o b l e m s o l v i n g t a k e s p r a c t i c e . D o n ' t g e t d i s c o u r a g e d
i f y o u h a v e t r o u b l e a t f i r s t . F o r a d d i t i o n a l p r a c t i c e , u s e t h e b o o k s
o n t h e r e a d i n g l i s t .
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A C K N O W L E D G E M E N T S
T h a n k s t o T i a n k a i L i u w h o h e l p e d i n p r o o f r e a d i n g a n d p r e p a r i n g s o l u -
t i o n s .
M a n y p r o b l e m s a r e e i t h e r i n s p i r e d b y o r f i x e d f r o m m a t h e m a t i c a l c o n t e s t s
i n d i f f e r e n t c o u n t r i e s a n d f r o m t h e f o l l o w i n g j o u r n a l s :
H i g h - S c h o o l M a t h e m a t i c s , C h i n a
R e v i s t a M a t e m a t i c a T i m i o a r a , R o m a n i a
K v a n t , R u s s i a
W e d i d o u r b e s t t o c i t e a l l t h e o r i g i n a l s o u r c e s o f t h e p r o b l e m s i n t h e s o l u -
t i o n p a r t . W e e x p r e s s o u r d e e p e s t a p p r e c i a t i o n t o t h e o r i g i n a l p r o p o s e r s
o f t h e p r o b l e m s .
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A B B R E V I A T I O N S A N D N O T A T I O N S
A b b r e v i a t i o n s
A H S M E
A m e r i c a n H i g h S c h o o l M a t h e m a t i c s
E x a m i n a t i o n
A I M E A m e r i c a n I n v i t a t i o n a l M a t h e m a t i c s
E x a m i n a t i o n
A M C 1 0 A m e r i c a n M a t h e m a t i c s C o n t e s t 1 0
A M C 1 2
A m e r i c a n M a t h e m a t i c s C o n t e s t 1 2 ,
w h i c h r e p l a c e s A H S M E
A R M L
A m e r i c a n R e g i o n a l M a t h e m a t i c s L e a g u e
I M O
I n t e r n a t i o n a l M a t h e m a t i c a l O l y m p i a d
U S A M O
U n i t e d S t a t e s o f A m e r i c a M a t h e m a t i c a l O l y m p i a d
M O S P
M a t h e m a t i c a l O l y m p i a d S u m m e r P r o g r a m
P u t n a m
T h e W i l l i a m L o w e l l P u t n a m M a t h e m a t i c a l
C o m p e t i t i o n
S t . P e t e r s b u r g
S t . P e t e r s b u r g ( L e n i n g r a d ) M a t h e m a t i c a l
O l y m p i a d
N o t a t i o n s f o r N u m e r i c a l S e t s a n d F i e l d s
Z t h e s e t o f i n t e g e r s
t h e s e t o f i n t e g e r s m o d u l o n
t h e s e t o f p o s i t i v e i n t e g e r s
t h e s e t o f n o n n e g a t i v e i n t e g e r s
t h e s e t o f r a t i o n a l n u m b e r s
t h e s e t o f p o s i t i v e r a t i o n a l n u m b e r s
t h e s e t o f n o n n e g a t i v e r a t i o n a l n u m b e r s
t h e s e t o f n - t u p l e s o f r a t i o n a l n u m b e r s
t h e s e t o f r e a l n u m b e r s
t h e s e t o f p o s i t i v e r e a l n u m b e r s
t h e s e t o f n o n n e g a t i v e r e a l n u m b e r s
t h e s e t o f n - t u p l e s o f r e a l n u m b e r s
t h e s e t o f c o m p l e x n u m b e r s
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C O N T E N T S
P R E F A C E
v i i
I N T R O D U C T I O N
i x
A C K N O W L E D G E M E N T S
x i
A B B R E V I A T I O N S A N D N O T A T I O N S
x i i i
1 . I N T R O D U C T O R Y P R O B L E M S
1
2 . A D V A N C E D P R O B L E M S
1 3
3 . S O L U T I O N S T O I N T R O D U C T O R Y P R O B L E M S 2 7
4 . S O L U T I O N S T O A D V A N C E D P R O B L E M S
6 5
G L O S S A R Y
1 3 1
F U R T H E R R E A D I N G
1 3 7
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I N T R O D U C T O R Y P R O B L E M S
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1 . I N T R O D U C T O R Y P R O B L E M S
P r o b l e m 1
L e t a , b , a n d c b e r e a l a n d p o s i t i v e p a r a m e t e r s . S o l v e t h e e q u a t i o n
a + b x + b + c x + c + a x =
b - a x + c - b x + a - c x .
P r o b l e m 2
F i n d t h e g e n e r a l t e r m o f t h e s e q u e n c e d e f i n e d b y x 0 = 3 , x 1 = 4 a n d
a
x n + 1 = x n _ 1 - n x n
f o r a l l ' n E N .
P r o b l e m 3
L e t x 1 , x 2 i . . . , X . b e a s e q u e n c e o f i n t e g e r s s u c h t h a t
( i ) - 1 < x , < 2 , f o r z = 1 , 2 . . . . , n ;
( i i ) x 1 + x 2 +
+ x n
1 9 ;
( i i i )
D e t e r m i n e t h e m i n i m u m a n d m a x i m u m p o s s i b l e v a l u e s o f
x i + x 2 + + x n .
P r o b l e m 4
T h e f u n c t i o n f , d e f i n e d b y
f ( x ) =
a x + b
c x + d '
w h e r e a , b , c , a n d d a r e n o n z e r o r e a l n u m b e r s , h a s t h e p r o p e r t i e s
f ( 1 9 ) = 1 9 , f ( 9 7 ) = 9 7 ,
a n d f ( f ( x ) ) = x ,
f o r a l l v a l u e s o f x , e x c e p t -
d
c
F i n d t h e r a n g e o f f .
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2
1 . I n t r o d u c t o r y P r o b l e m s
P r o b l e m 5
P r o v e t h a t
( a - b ) 2
b > 0 .
P r o b l e m 6
S e v e r a l ( a t l e a s t t w o ) n o n z e r o n u m b e r s a r e w r i t t e n o n a b o a r d . O n e m a y
e r a s e a n y t w o n u m b e r s , s a y a a n d b , a n d t h e n w r i t e t h e n u m b e r s a + 2
a n d b - a i n s t e a d .
2
P r o v e t h a t t h e s e t o f n u m b e r s o n t h e b o a r d , a f t e r a n y n u m b e r o f t h e
p r e c e d i n g o p e r a t i o n s , c a n n o t c o i n c i d e w i t h t h e i n i t i a l s e t .
P r o b l e m 7
T h e p o l y n o m i a l
1 - x + x 2 - x 3 + . . . + x 1 6 _ x 1 7
m a y b e w r i t t e n i n t h e f o r m
a o + a 1 y + a 2 y 2 + . . . + a 1 6 y 1 6 + a 1 7 y 1 7 ,
w h e r e y = x + 1 a n d a s a r e c o n s t a n t s .
F i n d a 2 .
P r o b l e m 8
L e t a , b , a n d c b e d i s t i n c t n o n z e r o r e a l n u m b e r s s u c h t h a t
1
1 1
= c + - .
+ b = b + -
c
P r o v e t h a t I a b c l = 1 .
P r o b l e m 9
F i n d p o l y n o m i a l s f ( x ) , g ( x ) , a n d h ( x ) , i f t h e y e x i s t , s u c h t h a t f o r a l l x ,
1 - 1
i f x < - 1
I f ( x ) I - I g ( x ) I + h ( x ) =
3 x + 2
i f - 1 < x < 0
- 2 x + 2 i f x > 0 .
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1 . I n t r o d u c t o r y P r o b l e m s 3
P r o b l e m 1 0
F i n d a l l r e a l n u m b e r s x f o r w h i c h
7
x + 2 7 "
1 2 x + 1 8 x
6
P r o b l e m 1 1
F i n d t h e l e a s t p o s i t i v e i n t e g e r m s u c h t h a t
f o r a l l p o s i t i v e i n t e g e r s n .
P r o b l e m 1 2
L e t a , b , c , d , a n d e b 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
a b c d e = a + b + c + d + e .
F i n d t h e m a x i m u m p o s s i b l e v a l u e o f m a x { a , b , c , d , e } .
P r o b l e m 1 3
E v a l u a t e
3 4
2 0 0 1
1 + 2 + 3 + 2 + 3 + 4
1 9 9 9 + 2 0 0 0 + 2 0 0 1
P r o b l e m 1 4
L e t x = v f a 2 + a + 1 - ' 1 a 2 - - a + 1 , a E R .
F i n d a l l p o s s i b l e v a l u e s o f x .
P r o b l e m 1 5
F i n d a l l r e a l n u m b e r s x f o r w h i c h
1 0 x + 1 l x + 1 2 x = 1 3 x + 1 4 x .
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4 1 . I n t r o d u c t o r y P r o b l e m s
P r o b l e m 1 6
L e t f : N x N - N b e a f u n c t i o n s u c h t h a t f ( 1 , 1 )
= 2 ,
f ( m + 1 , n ) = f ( m , n ) + m a n d f ( m , n + 1 ) = f ( m , n ) - n
f o r a l l m , n E N .
F i n d a l l p a i r s ( p , q ) s u c h t h a t f ( p , q ) = 2 0 0 1 .
P r o b l e m 1 7
L e t f b e a f u n c t i o n d e f i n e d o n [ 0 , 1 ] s u c h t h a t
f ( O ) = f ( l ) = 1 a n d I f ( a ) - f ( b ) I < I a - b I ,
f o r a l l a
b i n t h e i n t e r v a l [ 0 , 1 ] .
P r o v e t h a t
I f ( a ) - f ( b ) I 6 t h e e q u a t i o n
1 1
1
+
2
+ . . . +
2
= 1
x 1
x 2
x
h a s i n t e g e r s o l u t i o n s .
P r o b l e m 2 1
F i n d a l l p a i r s o f i n t e g e r s ( a , b ) s u c h t h a t t h e p o l y n o m i a l a x 1 7 + b x l s + 1
i s d i v i s i b l e b y x 2 - x - 1 .
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1 . I n t r o d u c t o r y P r o b l e m s 5
P r o b l e m 2 2
G i v e n a p o s i t i v e i n t e g e r n , l e t p ( n ) b e t h e p r o d u c t o f t h e n o n - z e r o d i g i t s
o f n . ( I f n h a s o n l y o n e d i g i t , t h e n p ( n ) i s e q u a l t o t h a t d i g i t . ) L e t
S = p ( 1 ) + p ( 2 ) +
+ p ( 9 9 9 ) .
W h a t i s t h e l a r g e s t p r i m e f a c t o r o f S ?
P r o b l e m 2 3
L e t x n 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 u c h t h a t
x i _ 2 x n _ 1
x n
2 x n _ 2 - x n _ 1
f o r n = 3 , 4 , . . . .
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 1 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 .
P r o b l e m 2 4
S o l v e t h e e q u a t i o n
x 3 - 3 x = x + 2 .
P r o b l e m 2 5
F o r a n y s e q u e n c e o f r e a l n u m b e r s A = { a 1 , a 2 , a 3 ,
} , d e f i n e D A t o b e
t h e s e q u e n c e { a 2 - a 1 , a 3 - a 2 , a 4 - a 3 , . . . } . S u p p o s e t h a t a l l o f t h e t e r m s
o f t h e s e q u e n c e A ( A A ) a r e 1 , a n d t h a t a 1 9 = a 9 2 = 0 .
F i n d a 1 .
P r o b l e m 2 6
F i n d a l l r e a l n u m b e r s x s a t i s f y i n g t h e e q u a t i o n
2 x + 3 x - 4 x + 6 x - 9 x = 1 .
P r o b l e m 2 7
P r o v e t h a t
8 0
1 6 < E v 1 < 1 7 .
k = 1
k
P r o b l e m 2 8
D e t e r m i n e t h e n u m b e r o f o r d e r e d p a i r s o f i n t e g e r s ( m , n ) f o r w h i c h m n >
0 a n d
m 3 + n 3 + 9 9 m n = 3 3 3 .
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6
1 . I n t r o d u c t o r y P r o b l e m s
P r o b l e m 2 9
L e t a , b , a n d c b e p o s i t i v e r e a l n u m b e r s s u c h t h a t a + b + c < 4 a n d
a b + b c + c a > 4 .
P r o v e t h a t a t l e a s t t w o o f t h e i n e q u a l i t i e s
l a - b i < 2 ,
l b - c l < 2 ,
I c - a l < 2
a r e t r u e .
P r o b l e m 3 0
E v a l u a t e
n
1
E
( n - k ) ( n + k )
k = O
P r o b l e m 3 1
L e t 0 < a < 1 . S o l v e
f o r p o s i t i v e n u m b e r s x .
P r o b l e m 3 2
W h a t i s t h e c o e f f i c i e n t o f x 2 w h e n
( 1 + x ) ( 1 + 2 x ) ( 1 + 4 x ) . . . ( 1 + 2 n X )
i s e x p a n d e d ?
P r o b l e m 3 3
L e t m a n d n b e d i s t i n c t p o s i t i v e i n t e g e r s .
F i n d t h e m a x i m u m v a l u e o f I x ' - x n l w h e r e x i s a r e a l n u m b e r i n t h e
i n t e r v a l ( 0 , 1 ) .
P r o b l e m 3 4
P r o v e t h a t t h e p o l y n o m i a l
( x - a l ) ( x - a 2 ) . . . ( x - a n ) - 1 ,
w h e r e a l , a 2 , , a n a r e d i s t i n c t i n t e g e r s , c a n n o t b e w r i t t e n a s t h e p r o d -
u c t o f t w o n o n - c o n s t a n t p o l y n o m i a l s w i t h i n t e g e r c o e f f i c i e n t s , i . e . , i t i s
i r r e d u c i b l e .
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1 . I n t r o d u c t o r y P r o b l e m s 7
P r o b l e m 3 5
F i n d a l l o r d e r e d p a i r s o f r e a l n u m b e r s ( x , y ) f o r w h i c h :
( 1 + x ) ( 1 + x 2 ) ( 1 + x 4 ) = l + y 7
a n d
( 1 + y ) ( 1 + y 2 ) ( 1 + y 4 )
=
1 + x 7 .
P r o b l e m 3 6
S o l v e t h e e q u a t i o n
2 ( 2 x - 1 ) x 2 + ( 2 x - 2 - 2 ) x = 2 x + 1
- 2
f o r r e a l n u m b e r s x .
P r o b l e m 3 7
L e t a b e a n i r r a t i o n a l n u m b e r a n d l e t n b e a n i n t e g e r g r e a t e r t h a n 1 .
P r o v e t h a t
( a +
a 2 - 1 ) + ( a
-
a 2 - 1 )
i s a n i r r a t i o n a l n u m b e r .
P r o b l e m 3 8
S o l v e t h e s y s t e m o f e q u a t i o n s
( x 1 - x 2 + x 3 ) 2
= x 2 ( x 4 + x 5 - x 2 )
( x 2 - x 3 + x 4 ) 2
= x 3 ( x 5 + x 1 - x 3 )
( x 3 - x 4 + x 5 ) 2
= x 4 ( x 1 + x 2 - x 4 )
( x 4 - x 5 + x 1 ) 2
= x 5 ( x 2 + x 3 - x 5 )
( x 5 - x 1 + x 2 ) 2
= x l ( x 3 + x 4 - x l )
f o r r e a l n u m b e r s x 1 , x 2 , x 3 , x 4 , x 5 .
P r o b l e m 3 9
L e t x , y , a n d z b e c o m p l e x n u m b e r s s u c h t h a t
x + y + z = 2 ,
x 2 + y 2 + z 2 = 3
a n d
E v a l u a t e
x y z = 4 .
1
1
1
x y + z - l + y z + x - l + z x + y - 1
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8
1 . I n t r o d u c t o r y P r o b l e m s
P r o b l e m 4 0
M r . F a t i s g o i n g t o p i c k t h r e e n o n - z e r o r e a l n u m b e r s a n d M r . T a f i s g o i n g
t o a r r a n g e t h e t h r e e n u m b e r s a s t h e c o e f f i c i e n t s o f a q u a d r a t i c e q u a t i o n
x 2 + x + = 0 .
M r . F a t w i n s t h e g a m e i f a n d o n l y i f t h e r e s u l t i n g e q u a t i o n h a s t w o
d i s t i n c t r a t i o n a l s o l u t i o n s .
W h o h a s a w i n n i n g s t r a t e g y ?
P r o b l e m 4 1
G i v e n t h a t t h e r e a l n u m b e r s a , b , c , d , a n d e s a t i s f y s i m u l t a n e o u s l y t h e
r e l a t i o n s
a + b + c + d + e = 8 a n d a 2 + b 2 + c 2 + d 2 + e 2 = 1 6 ,
d e t e r m i n e t h e m a x i m u m a n d t h e m i n i m u m v a l u e o f a .
P r o b l e m 4 2
F i n d t h e r e a l z e r o s o f t h e p o l y n o m i a l
P a ( x ) = ( x 2 + 1 ) ( x - 1 ) 2 - a x 2 ,
w h e r e a i s a g i v e n r e a l n u m b e r .
P r o b l e m 4 3
P r o v e t h a t
1 3
2 n - 1
1
2 4
2 n
7 3 7
f o r a l l p o s i t i v e i n t e g e r s n .
P r o b l e m 4 4
L e t
P ( x ) = a o x n + a l
x n - 1 +
. . . + a , ,
b e a n o n z e r o p o l y n o m i a l w i t h i n t e g e r c o e f f i c i e n t s s u c h t h a t P ( r ) _
P ( s ) = 0 f o r s o m e i n t e g e r s r a n d s , w i t h 0 < r < s .
P r o v e t h a t a k < - s f o r s o m e k .
P r o b l e m 4 5
L e t m b e a g i v e n r e a l n u m b e r .
F i n d a l l c o m p l e x n u m b e r s x s u c h t h a t
( X ) 2
x 2
2
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1 . I n t r o d u c t o r y P r o b l e m s
P r o b l e m 4 6
T h e s e q u e n c e g i v e n b y x o = a , x 1 = b , a n d
x n + 1 -
1 ( X n - 1
+ 1 I .
2
i s p e r i o d i c .
P r o v e t h a t a b = 1 .
P r o b l e m 4 7
L e t a , b , c , a n d d b e r e a l n u m b e r s s u c h t h a t
P r o v e t h a t
( a 2 + b 2 - 1 ) ( c 2 + d 2 - 1 ) > ( a c + b d - 1 ) 2 .
a 2 + b 2 > 1 a n d c 2 + d 2 > 1 .
P r o b l e m 4 8
F i n d a l l c o m p l e x n u m b e r s z s u c h t h a t
( 3 z + 1 ) ( 4 z + 1 ) ( 6 z + 1 ) ( 1 2 z + 1 ) = 2 .
9
P r o b l e m 4 9
L e t x 1 i x 2 ,
- , x n _ 1 i b e t h e z e r o s d i f f e r e n t f r o m 1 o f t h e p o l y n o m i a l
P ( x ) = x n - 1 , n > 2 .
P r o v e t h a t
1
1 1
1 - x 1
+
1 - x 2
+ . . . +
1 - x n _ 1
n - 1
2
P r o b l e m 5 0
L e t a a n d b b e g i v e n r e a l n u m b e r s . S o l v e t h e s y s t e m o f e q u a t i o n s
x - y x 2 - y z
a ,
1 - x 2 + y 2
y - x x 2 - y 2
= b
1 - x 2 + y 2
f o r r e a l n u m b e r s x a n d y .
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A D V A N C E D P R O B L E M S
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2 . A D V A N C E D P R O B L E M S
P r o b l e m 5 1
E v a l u a t e
0 2 0 0
2
5
0 ) +
( 2 0 0 0 )
+ ( 2 0 8 0 0 )
+ . . . +
( 2 0 0 0 )
P r o b l e m 5 2
L e t x , y , z b e p o s i t i v e r e a l n u m b e r s s u c h t h a t x 4 + y 4 + z 4 = 1 .
D e t e r m i n e w i t h p r o o f t h e m i n i m u m v a l u e o f
x 3
y
3
z 3
1 - x 8 + 1 y 8 + 1 - z 8 .
P r o b l e m 5 3
F i n d a l l r e a l s o l u t i o n s t o t h e e q u a t i o n
2 X + 3 2 : + 6 X = x 2 .
P r o b l e m 5 4
L e t { a n } n > 1 b e a s e q u e n c e s u c h t h a t a l = 2 a n d
a n 1
a n + i =
2 +
a
n
f o r a l l n e N .
F i n d a n e x p l i c i t f o r m u l a f o r a n .
P r o b l e m 5 5
L e t x , y , a n d z b e p o s i t i v e r e a l n u m b e r s . P r o v e t h a t
x + y
x +
( x + y ) ( x + z )
y +
( y + z ) ( y + x )
z
+
f ( n ) a n d
f ( f ( n ) ) = 3 n
f o r a l l n .
E v a l u a t e f ( 2 0 0 1 ) .
P r o b l e m 5 8
L e t F b e t h e s e t o f a l l p o l y n o m i a l s f ( x ) w i t h i n t e g e r s c o e f f i c i e n t s s u c h
t h a t f ( x ) = 1 h a s a t l e a s t o n e i n t e g e r r o o t .
F o r e a c h i n t e g e r k > 1 , f i n d m k , t h e l e a s t i n t e g e r g r e a t e r t h a n 1 f o r
w h i c h t h e r e e x i s t s f E F s u c h t h a t t h e e q u a t i o n f ( x ) = M k h a s e x a c t l y
k d i s t i n c t i n t e g e r r o o t s .
P r o b l e m 5 9
L e t x 1 = 2 a n d
2
x ' + 1 = x n - x , + 1 ,
f o r n > 1 .
P r o v e t h a t
1
0 .
S o l u t i o n 5 , A l t e r n a t i v e 1
N o t e t h a t
+
` < 1 0 ( w h i c h i m p l i e s a >
a b > b ) .
P r o b l e m 6 [ S t . P e t e r s b u r g 1 9 8 9 ]
S e v e r a l ( a t l e a s t t w o ) n o n z e r o n u m b e r s a r e w r i t t e n o n a b o a r d . O n e m a y
e r a s e a n y t w o n u m b e r s , s a y a a n d b , a n d t h e n w r i t e t h e n u m b e r s a + 2
a n d b - 2 i n s t e a d .
P r o v e t h a t t h e s e t o f n u m b e r s o n t h e b o a r d , a f t e r a n y n u m b e r o f t h e
p r e c e d i n g o p e r a t i o n s , c a n n o t c o i n c i d e w i t h t h e i n i t i a l s e t .
S o l u t i o n 6
L e t S b e t h e s u m o f t h e s q u a r e s o f t h e n u m b e r s o n t h e b o a r d . N o t e t h a t
S i n c r e a s e s i n t h e f i r s t o p e r a t i o n a n d d o e s n o t d e c r e a s e i n a n y s u c c e s s i v e
o p e r a t i o n , a s
\ 2
C a + 2
J + ( b - 2 ) 2 = 4 ( a 2 + b 2 ) > a 2 + b 2
w i t h e q u a l i t y o n l y i f a = b = 0 .
T h i s c o m p l e t e s t h e p r o o f .
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
3 1
P r o b l e m 7 [ A I M E 1 9 8 6 ]
T h e p o l y n o m i a l
1 - x + x 2 - x 3 + + x 1 6 - x 1 7
m a y b e w r i t t e n i n t h e f o r m
a o + a l y + a 2 y 2 + . . . + a 1 6 y 1 6 + a 1 7 y 1 7 ,
w h e r e y = x + 1 a n d a s a r e c o n s t a n t s . F i n d a 2 .
S o l u t i o n 7 , A l t e r n a t i v e 1
L e t f ( x ) d e n o t e t h e g i v e n e x p r e s s i o n . T h e n
x f ( x ) = x - x 2 + x 3 - . . . - x 1 8
a n d
H e n c e
( 1 + x ) f ( x ) = 1 - x 1 8 .
f ( x ) = f ( y - 1 ) =
1 - ( y -
1 ) 1 8
-
1 - ( y - 1 ) 1 8
1 + ( y - 1 )
y
T h e r e f o r e a 2 i s e q u a l t o t h e c o e f f i c i e n t o f y 3 i n t h e e x p a n s i o n o f
1 - ( y - 1 ) 1 s
a 2 =
( 1 8 )
= 8 1 6 .
S o l u t i o n 7 , A l t e r n a t i v e 2
L e t f ( x ) d e n o t e t h e g i v e n e x p r e s s i o n . T h e n
f ( x ) = f ( y - 1 ) = 1
( y - 1 ) + ( y - 1 ) 2 ( y - 1 )
1 7
= 1 + (
T h u s
2
) +
( 1 ) , + . . .
+
( 1 7 )
=
( 1 8 ) .
a 2 = ( 2
H e r e w e u s e d t h e f o r m u l a
( n )
k + ( k + 1 ) -
( k
+ 1
a n d t h e f a c t t h a t
( 2 2 )
_
( 3 3 )
= 1 .
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3 2
3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
P r o b l e m 8
L e t a , b , a n d c b e d i s t i n c t n o n z e r o r e a l n u m b e r s s u c h t h a t
a + - = b + - = c + - .
P r o v e t h a t I a b c I = 1 .
S o l u t i o n 8
F r o m t h e g i v e n c o n d i t i o n s i t f o l l o w s t h a t
a
.
b =
b
b c c
,
b - c = c c a a , a n d c - a =
a a b b
M u l t i p l y i n g t h e a b o v e e q u a t i o n s g i v e s ( a b c ) 2 = 1 , f r o m w h i c h t h e d e s i r e d
r e s u l t f o l l o w s .
P r o b l e m 9 [ P u t n a m 1 9 9 9 ]
F i n d p o l y n o m i a l s f ( x ) , g ( x ) , a n d h ( x ) , i f t h e y e x i s t , s u c h t h a t f o r a l l x ,
- 1
i f x < - 1
I f ( X ) I - I g ( x ) I + h ( x ) =
3 x + 2
i f - 1 < x < 0
- 2 x + 2 i f x > 0 .
S o l u t i o n 9 , A l t e r n a t i v e 1
S i n c e x = - 1 a n d x = 0 a r e t h e t w o c r i t i c a l v a l u e s o f t h e a b s o l u t e
f u n c t i o n s , o n e c a n s u p p o s e t h a t
F ( x ) = a l x + 1 l + b l x l + c x + d
( c - a - b ) x + d - a i f x < - 1
( a + c - b ) x + a + d i f - 1 < x < 0
( a + b + c ) x + a + d i f x > 0 ,
w h i c h i m p l i e s t h a t a = 3 / 2 , b = - 5 / 2 , c = - 1 , a n d d = 1 / 2 .
H e n c e f ( x ) = ( 3 x + 3 ) / 2 , g ( x ) = 5 x / 2 , a n d h ( x ) = - x + z .
S o l u t i o n 9 , A l t e r n a t i v e 2
N o t e t h a t i f r ( x ) a n d s ( x ) a r e a n y t w o f u n c t i o n s , t h e n
r + s + I r - s l
m a x ( r , s ) =
2
T h e r e f o r e , i f F ( x ) i s t h e g i v e n f u n c t i o n , w e h a v e
F ( x ) = m a x { - 3 x - 3 , 0 } - m a x { 5 x , 0 } + 3 x + 2
= ( - 3 x - 3 + 1 3 x + 3 1 ) / 2 - ( 5 x + I 5 x l ) / 2 + 3 x + 2
.
1 ( 3 x + 3 ) / 2 1 - 1 5 x / 2 1 - x +
2 1
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
3 3
P r o b l e m 1 0
F i n d a l l r e a l n u m b e r s x f o r w h i c h
8 x + 2 7 x
7
1 2 x + 1 8 x 6
S o l u t i o n 1 0
B y s e t t i n g 2 x = a a n d 3 x = b , t h e e q u a t i o n b e c o m e s
a 3 + b 3
7
a 2 b + b 2 a
6
a 2 - a b + b 2
_ 7
a b
6 '
6 a 2
- 1 3 a b + 6 b 2 = 0 ,
( 2 a - 3 b ) ( 3 a - 2 b ) = 0 .
T h e r e f o r e 2 x + 1 = 3 x + 1 o r 2 x - 1 = 3 x - 1 , w h i c h i m p l i e s t h a t x = - 1 a n d
x = 1 .
I t i s e a s y t o c h e c k t h a t b o t h x = - 1 a n d x = 1 s a t i s f y t h e g i v e n e q u a t i o n .
P r o b l e m 1 1 [ R o m a n i a 1 9 9 0 ]
F i n d t h e l e a s t p o s i t i v e i n t e g e r m s u c h t h a t
C 2 n l n
n
< m
f o r a l l p o s i t i v e i n t e g e r s n .
S o l u t i o n 1 1
N o t e t h a t
C 2 n )
3 5
= ( 1 + 1 ) 2 n = 4 n
T h u s m = 4 .
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3 4
3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
P r o b l e m 1 2
L e t a , b , c , d , a n d e b 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
a b c d e = a + b + c + d + e .
F i n d t h e m a x i m u m p o s s i b l e v a l u e o f m a x { a , b , c , d , e } .
S o l u t i o n 1 2 , A l t e r n a t i v e 1
S u p p o s e t h a t a < b < c < d < e . W e n e e d t o f i n d t h e m a x i m u m v a l u e o f
e . S i n c e
e < a + b + c + d + e < 5 e ,
t h e n e < a b c d e < 5 e , i . e .
1 < a b c d < 5 .
H e n c e ( a , b , c , d ) = ( 1 , 1 , 1 , 2 ) , ( 1 , 1 , 1 , 3 ) , ( 1 , 1 , 1 , 4 ) , ( 1 , 1 , 2 , 2 ) , o r
( 1 , 1 , 1 , 5 ) , w h i c h l e a d s t o m a x { e } = 5 .
S o l u t i o n 1 2 , A l t e r n a t i v e 2
A s b e f o r e , s u p p o s e t h a t a < b < c < d < e . N o t e t h a t
1 _
1 + 1 + 1 + 1 + 1
b c d e c d e a
d e a b
e a b c a b c d
l a n d e - 1 < 4 o r e < 5 .
I t i s e a s y t o s e e t h a t ( 1 , 1 , 1 , 2 , 5 ) i s a s o l u t i o n .
T h e r e f o r e m a x { e } = 5 .
C o m m e n t : T h e s e c o n d s o l u t i o n c a n b e u s e d t o d e t e r m i n e t h e m a x i -
m u m v a l u e o f { x 1 i X 2 . . . . , x . n } , w h e n x i , x 2 , . . . , x n 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
1 1 1 2 . . . 1 n = X 1 + X 2 + . . . + 1 n .
P r o b l e m 1 3
E v a l u a t e
3 4 2 0 0 1
1 + 2 + 3 + 2 - 3 + 4 + + 1 9 9 9
+ 2 0 0 0 + 2 0 0 1
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
S o l u t i o n 1 3
N o t e t h a t
k + 2
_
k + 2
k + ( k + 1 ) + ( k + 2 )
k [ 1 + k + 1 + ( k + 1 ) ( k + 2 ) ]
1
k ( k + 2 )
k + 1
( k + 2 )
( k + 2 ) - 1
( k + 2 )
( k + 1 )
( k + 2 )
B y t e l e s c o p i n g s u m , t h e d e s i r e d v a l u e i s e q u a l t o
1
1
2 2 0 0 1
P r o b l e m 1 4
L e t x = a 2 + a + 1 - a 2 - a + 1 , a E R .
F i n d a l l p o s s i b l e v a l u e s o f x .
S o l u t i o n 1 4 , A l t e r n a t i v e 1
S i n c e
a n d
x =
w e h a v e
S q u a r i n g b o t h s i d e s o f
a 2 + I a l + 1 > I a l
2 a
a
2 +
a + 1 + a 2 - a + 1 '
j x j < 1 2 a / a l = 2 .
x +
v l - a - 2
a 2 + a + 1
3 5
y i e l d s
2 x a 2 - a + 1 = 2 a - x 2 .
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3 6 3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
S q u a r i n g b o t h s i d e s o f t h e a b o v e e q u a t i o n g i v e s
x 2 ( 2 2 - 4 )
2 2
2
2
2
=
( x
- 1 ) a
= x
- 4 ) o r ax
4 ( x 2 - 1 )
S i n c e a 2 > 0 , w e m u s t h a v e
2 2 ( x 2 - 4 ) ( X 2
- 1 ) > 0 ,
S i n c e I x I < 2 , x 2 - 4 < 0 w h i c h f o r c e s x 2 - 1 < 0 . T h e r e f o r e , - 1 < x < 1 .
C o n v e r s e l y , f o r e v e r y x E ( - 1 , 1 ) t h e r e e x i s t s a r e a l n u m b e r a s u c h t h a t
x = a 2 + a + 1 - a 2 - a + 1 .
S o l u t i o n 1 4 , A l t e r n a t i v e 2
L e t A = ( - 1 / 2 , / / 2 ) , B = ( 1 / 2 , / / 2 ) , a n d P = ( a , 0 ) .
T h e n P
i s a p o i n t o n t h e x - a x i s a n d w e a r e l o o k i n g f o r a l l p o s s i b l e v a l u e s o f
d = P A - P B .
B y t h e T r i a n g l e I n e q u a l i t y , S P A - P B j < I A B I = 1 . A n d i t i s c l e a r
t h a t a l l t h e v a l u e s - 1 < d < 1 a r e i n d e e d o b t a i n a b l e . I n f a c t , f o r s u c h
a d , a h a l f h y p e r b o l a o f a l l p o i n t s Q s u c h t h a t Q A - Q B = d i s w e l l
d e f i n e d . ( P o i n t s A a n d B a r e f o c i o f t h e h y p e r b o l a . )
S i n c e l i n e A B i s p a r a l l e l t o t h e x - a x i s , t h i s h a l f h y p e r b o l a i n t e r s e c t s t h e
x - a x i s , i . e . , P i s w e l l d e f i n e d .
P r o b l e m 1 5
F i n d a l l r e a l n u m b e r s x f o r w h i c h
l O x + 1 1 x + 1 2 x = 1 3 x + W .
S o l u t i o n 1 5
I t i s e a s y t o c h e c k t h a t x = 2 i s a s o l u t i o n . W e c l a i m t h a t i t i s t h e o n l y
o n e . I n f a c t , d i v i d i n g b y 1 3 2 o n b o t h s i d e s g i v e s
( 1 0 )
x
+ ( 1 1 ) x +
/
( 1 3 ) 2
= 1 +
( 1 3 ) 2 .
T h e l e f t h a n d s i d e i s a d e c r e a s i n g
f u n c t i o n
o f x a n d t h e r i g h t h a n d s i d e
i s a n i n c r e a s i n g f u n c t i o n o f x .
T h e r e f o r e t h e i r g r a p h s c a n h a v e a t m o s t o n e p o i n t o f i n t e r s e c t i o n .
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
3 7
C o m m e n t : M o r e g e n e r a l l y ,
a 2 + ( a + 1 ) 2 + .
.
. + ( a + k ) 2
= ( a + k +
1 ) 2
+ ( a + k + 2 ) 2 +
+ ( a + 2 k ) 2
f o r a = k ( 2 k + 1 ) , k E N .
P r o b l e m 1 6 [ K o r e a n M a t h e m a t i c s C o m p e t i t i o n 2 0 0 1 ]
L e t f : N x N - N b e a f u n c t i o n s u c h t h a t f ( 1 , 1 ) = 2 ,
f ( m + 1 , n ) = f ( m , n ) + m a n d f ( m , n + 1 ) = f ( m , n ) - n
f o r a l l m , n E N .
F i n d a l l p a i r s ( p , q ) s u c h t h a t f ( p , q ) = 2 0 0 1 .
S o l u t i o n 1 6
W e h a v e
f ( p , q )
= f ( p - 1 , q ) + p - 1
= f ( p - 2 , q ) + ( p - 2 ) + ( p - 1 )
f ( l , q ) +
p ( p - 1 )
2
f ( 1 , q - 1 ) - ( q - 1 ) + p ( p - 1 )
2
f ( 1 , 1 ) -
q ( q - 1 )
+
p ( p - 1 )
2 2
2 0 0 1 .
T h e r e f o r e
p ( p - 1 )
-
q ( q - 1 )
2 2
=
1 9 9 9 ,
( p - q ) ( p + q - 1 ) = 2 . 1 9 9 9 .
N o t e t h a t 1 9 9 9 i s a p r i m e n u m b e r a n d t h a t p - q < p + q - 1 f o r p , q E N .
W e h a v e t h e f o l l o w i n g t w o c a s e s :
1 . p - q = 1 a n d p + q - 1 = 3 9 9 8 . H e n c e p = 2 0 0 0 a n d q = 1 9 9 9 .
2 . p - q = 2 a n d p + q - 1 = 1 9 9 9 . H e n c e p = 1 0 0 1 a n d q = 9 9 9 .
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3 8 3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
T h e r e f o r e ( p , q ) = ( 2 0 0 Q , 1 9 9 9 ) o r ( 1 0 0 1 , 9 9 9 ) .
P r o b l e m 1 7 [ C h i n a 1 9 8 3 ]
L e t f b e a f u n c t i o n d e f i n e d o n [ 0 , 1 ] s u c h t h a t
f ( 0 ) = f ( 1 ) = 1 a n d l f ( a ) - f ( b ) I < I a - b I ,
f o r a l l a 5 4 b i n t h e i n t e r v a l [ 0 , 1 ] .
P r o v e t h a t
I f ( a ) - f ( b ) I 1 / 2 . B y s y m m e t r y , w e m a y a s s u m e t h a t a > b . T h e n
I f ( a ) - f ( b ) I
= I f ( a ) - f ( 1 ) + f ( 0 ) - f ( b ) I
I f ( a ) - f ( 1 ) I + I f ( 0 ) - f ( b ) I
< I a - 1 I + I 0 - b I
1 - a + b - 0
1 - ( a - b )
1
0 y i e l d s
3 - 2 y 3 + 2 y
3
- y -
3
T h u s t h e p o s s i b l e v a l u e s f o r y a r e 0 , 1 , a n d 2 , w h i c h l e a d t o t h e s o l u t i o n s
( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , a n d ( 2 , 2 ) .
T h e r e f o r e , t h e i n t e g e r s o l u t i o n s o f t h e e q u a t i o n a r e ( x , y ) = ( 1 , 0 ) , ( 0 , 1 ) ,
( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , a n d ( n , - n ) , f o r a l l n E Z .
P r o b l e m 1 9 [ K o r e a n M a t h e m a t i c s C o m p e t i t i o n 2 0 0 1 ]
L e t
f ( x ) =
2
4 x + 2
f o r r e a l n u m b e r s x . E v a l u a t e
f ( 2 0 0 1 ) + f ( 2 0 0 1 ) +
+ f ( 2 0 0 0 1 )
S o l u t i o n 1 9
N o t e t h a t f h a s a h a l f - t u r n s y m m e t r y a b o u t p o i n t ( 1 / 2 , 1 / 2 ) . I n d e e d ,
2 _
2 - 4 x
_ 4 x
f ( 1 - x ) =
4 1 - x + 2
4 + 2 . 4 x
4 x + 2 '
f r o m w h i c h i t f o l l o w s t h a t f ( x ) + f ( 1 - x ) = 1 .
T h u s t h e d e s i r e d s u m i s e q u a l t o 1 0 0 0 .
P r o b l e m 2 0
P r o v e t h a t f o r n > 6 t h e e q u a t i o n
1 1
1
2
+ 2 +
. + 2 = 1
x 1 x 2
x n
h a s i n t e g e r s o l u t i o n s .
S o l u t i o n 2 0
N o t e t h a t
1
1 1
1
1
a 2
( 2 a ) 2 + ( 2 a ) 2 + ( 2 a ) 2
+ ( 2 a ) 2 '
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4 0 3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
f r o m w h i c h i t f o l l o w s t h a t i f ( x , i x 2 , , x n ) _ ( a , , a 2 ,
g e r s o l u t i o n t o
1 1
1
2
+ . . . +
- 2
+
2
2
1 2
n
t h e n
( x 1 , x 2 , . . .
, x n - 1 , x n , x n + l , X n + 2 ,
x n + 3 )
= ( a 1 a 2 . . . a n - 1 , 2 a n , 2 a n , 2 a n , 2 a n , )
i s a n i n t e g e r s o l u t i o n t o
1 1
2
+ 2 +
+
2
x l x 2
x n + 3
= 1 .
) i s a n i n t e -
T h e r e f o r e w e c a n c o n s t r u c t t h e s o l u t i o n s i n d u c t i v e l y i f t h e r e a r e s o l u t i o n s
f o r n = 6 , 7 , a n d 8 .
S i n c e x 1 = 1 i s a s o l u t i o n f o r n = 1 , ( 2 , 2 , 2 , 2 ) i s a s o l u t i o n f o r n = 4 ,
a n d ( 2 , 2 , 2 , 4 , 4 , 4 , 4 ) i s a s o l u t i o n f o r n = 7 .
I t i s e a s y t o c h e c k t h a t ( 2 , 2 , 2 , 3 , 3 , 6 ) a n d ( 2 , 2 , 2 , 3 , 4 , 4 , 1 2 , 1 2 ) a r e s o l u -
t i o n s f o r n = 6 a n d n = 8 , r e s p e c t i v e l y . T h i s c o m p l e t e s t h e p r o o f .
P r o b l e m 2 1 [ A I M E 1 9 8 8 ]
F i n d a l l p a i r s o f i n t e g e r s ( a , b ) s u c h t h a t t h e p o l y n o m i a l
a x 1 7 + b x 1 6 + 1
i s d i v i s i b l e b y x 2 - x - 1 .
S o l u t i o n 2 1 , A l t e r n a t i v e 1
L e t p a n d q b e t h e r o o t s o f x 2 - x - 1 = 0 .
B y V i e t a ' s t h e o r e m ,
p + q = 1 a n d p q = - 1 . N o t e t h a t p a n d q m u s t a l s o b e t h e r o o t s o f
a x 1 7 + b x 1 6 + 1 = 0 . T h u s
a p 1 7 + b p 1 6 =
- 1 a n d a q
1 7 + b g 1 6
= - 1 .
M u l t i p l y i n g t h e f i r s t o f t h e s e e q u a t i o n s b y q 1 6 , t h e s e c o n d o n e b y p 1 6
a n d u s i n g t h e f a c t t h a t p q = - 1 , w e f i n d
a p + b = - q 1 6 a n d a q + b =
- p ' s
( 1 )
T h u s
1 s 1 6
a =
p
- q
= ( p 8 + g 8 ) ( p 4 + g 4 ) ( p 2 + q 2 ) ( p + q )
p - q
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
S i n c e
p + q =
1 ,
p 2 + q 2 =
( p + q ) 2 - 2 p q = 1 + 2 = 3 ,
p 4 + q 4
=
( p 2 + q 2 ) 2 - 2 p 2 g 2 = 9 - 2 = 7 ,
p 8 + q 8
=
( p 4 + q 4 ) 2 - 2 p 4 g 4 = 4 9 - 2 = 4 7 ,
i t f o l l o w s t h a t a = 1 - 3 . 7 - 4 7 = 9 8 7 .
L i k e w i s e , e l i m i n a t i n g a i n ( 1 ) g i v e s
- b =
p 1 7 - q 1 7
p - q
p 1 6 + p 1 5 q + p 1 4 g 2 +
. . . +
q ' 6
( p 1 6 + q ' 6 ) + p q ( p 1 4 + q 1 4 ) + p 2 g 2 ( p 1 2 + q 1 2 )
+ . . . +
p 7 g 7 ( p 2 + q 2 ) + p 8 g 8
( p 1 6 + q 1 6 )
_
( p 1 4
+
q 1 4 ) +
. . . - ( p 2 + q 2 ) + 1 .
4 1
F o r n > 1 , l e t k 2 n = p 2 n + q 2 n . T h e n k 2 = 3 a n d k 4 = 7 , a n d
k 2 n + 4 = p 2 n + 4 +
q 2 n + 4
= ( p 2 n + 2 + q 2 n + 2 ) ( p 2 + q 2 ) - p 2 g 2 ( p 2 n + q 2 n )
= 3 k 2 n + 2 - k 2 n
f o r n > 3 . T h e n k 6 = 1 8 , k 8 = 4 7 , k l o = 1 2 3 , k 1 2 = 3 2 2 , k 1 4 = 8 4 3 ,
k 1 6 = 2 2 0 7 .
H e n c e
- b = 2 2 0 7 - 8 4 3 + 3 2 2 - 1 2 3 + 4 7 - 1 8 + 7 - 3 + 1 = 1 5 9 7
o r
( a , b ) = ( 9 8 7 , - 1 5 9 7 ) .
S o l u t i o n 2 1 , A l t e r n a t i v e 2
T h e o t h e r f a c t o r i s o f d e g r e e 1 5 a n d w e w r i t e
( c 1 5 x 1 5 - C 1 4 X 1 4 + . . . + c 1 x - c o ) ( x 2 - x - 1 ) = a x 1 7 + b x 1 6 + 1 .
C o m p a r i n g c o e f f i c i e n t s :
x
0
c o = 1 ,
x 1 :
c o - c , = 0 , c 1 = 1
x 2 : - c o - c 1 + c 2 = 0 , c 2 = 2 ,
a n d f o r 3 < k < 1 5 , x k :
- C k - 2 - C k - 1 + C k = O .
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4 2 3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
I t f o l l o w s t h a t f o r k < 1 5 , c k = F k + 1 ( t h e F i b o n a c c i n u m b e r ) .
T h u s a = c l 5 = F 1 6 = 9 8 7 a n d b = - c 1 4 - c 1 5 = - F 1 7 = - 1 5 9 7 o r
( a , b ) = ( 9 8 7 , - 1 5 9 7 ) .
C o m m e n t :
C o m b i n i n g t h e t w o m e t h o d s , w e o b t a i n s o m e i n t e r e s t i n g
f a c t s a b o u t s e q u e n c e s k 2 , , , a n d F 2 , , , _ 1 . S i n c e
3 F 2 n + 3 - F 2 n + 5 = 2 F 2 n + 3 - F 2 n + 4 = F 2 n + 3 - F 2 n + 2 = F 2 . + 1 ,
i t f o l l o w s t h a t F 2 , , , _ 1 a n d k 2 n s a t i s f y t h e s a m e r e c u r s i v e r e l a t i o n . I t i s
e a s y t o c h e c k t h a t k 2 = F 1 + F 3 a n d k 4 = F 3 + F 5 .
T h e r e f o r e k e n = F 2 n _ 1 + F 2 n + i a n d
F 2 n + 1 = k e n - k e n - 2 + k 2 n _ 4 - . . . +
( _ 1 ) n - 1 k 2
+ ( - 1 ) n .
P r o b l e m 2 2 [ A I M E 1 9 9 4 ]
G i v e n a p o s i t i v e i n t e g e r n , l e t p ( n ) b e t h e p r o d u c t o f t h e n o n - z e r o d i g i t s
o f n . ( I f n h a s o n l y o n e d i g i t , t h e n p ( n ) i s e q u a l t o t h a t d i g i t . ) L e t
S = p ( 1 ) + p ( 2 ) + . . . + p ( 9 9 9 ) .
W h a t i s t h e l a r g e s t p r i m e f a c t o r o f S ?
S o l u t i o n 2 2
C o n s i d e r e a c h p o s i t i v e i n t e g e r l e s s t h a n 1 0 0 0 t o b e a t h r e e - d i g i t n u m b e r
b y p r e f i x i n g O s t o n u m b e r s w i t h f e w e r t h a n t h r e e d i g i t s . T h e s u m o f t h e
p r o d u c t s o f t h e d i g i t s o f a l l s u c h p o s i t i v e n u m b e r s i s
( 0 . 0 . 0 + 0 . 0 .
= ( 0 + 1 + . . . + 9 ) 3 - 0 .
H o w e v e r , p ( n ) i s t h e p r o d u c t o f n o n - z e r o d i g i t s o f n . T h e s u m o f t h e s e
p r o d u c t s c a n b e f o u n d b y r e p l a c i n g 0 b y 1 i n t h e a b o v e e x p r e s s i o n , s i n c e
i g n o r i n g 0 ' s i s e q u i v a l e n t t o t h i n k i n g o f t h e m a s 1 ' s i n t h e p r o d u c t s . ( N o t e
t h a t t h e f i n a l 0 i n t h e a b o v e e x p r e s s i o n b e c o m e s a 1 a n d c o m p e n s a t e s
f o r t h e c o n t r i b u t i o n o f 0 0 0 a f t e r i t i s c h a n g e d t o 1 1 1 . )
H e n c e
S = 4 6 3 - 1 = ( 4 6 - 1 ) ( 4 6 2 + 4 6 + 1 ) = 3 3 . 5 . 7 . 1 0 3 ,
a n d t h e l a r g e s t p r i m e f a c t o r i s 1 0 3 .
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
4 3
P r o b l e m 2 3 [ P u t n a m 1 9 7 9 ]
L e t x n 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 u c h t h a t
1 n - 2 x n - 1
x n =
2 x n - 2 - 1 n - 1
f o r n = 3 , 4 , . . . .
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 1 a n d x 2 f o r x n 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 2 3 , A l t e r n a t i v e 1
W e h a v e
1 2 1 n - 2 - x n _ 1 2
1
x n
1 n - 2 x n - 1
1 n - 1
X n - 2
L e t y n = 1 / x n . T h e n Y n - Y n - 1 = Y n - 1 - Y n - 2 , i . e . , y n i s a n a r i t h m e t i c
s e q u e n c e . I f x n i s a n o n z e r o i n t e g e r w h e n n i s i n a n i n f i n i t e s e t S , t h e
y n ' s f o r n E S s a t i s f y - 1 < y n < 1 .
S i n c e a n a r i t h m e t i c s e q u e n c e i s u n b o u n d e d u n l e s s t h e c o m m o n d i f f e r e n c e
i s 0 , Y n - Y n - 1 = 0 f o r a l l n , w h i c h i n t u r n i m p l i e s t h a t x 1 = x 2 = m , a
n o n z e r o i n t e g e r .
C l e a r l y , t h i s c o n d i t i o n i s a l s o s u f f i c i e n t .
S o l u t i o n 2 3 , A l t e r n a t i v e 2
A n e a s y i n d u c t i o n s h o w s t h a t
1 1 x 2
1 1 x 2
1 n
( n - 1 ) x 1 - ( n - 2 ) x 2
( x 1 - x 2 ) n + ( 2 x 2 - x 1 ) '
f o r n = 3 , 4 , . . . .
I n t h i s f o r m w e s e e t h a t x n w i l l 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 i f a n d o n l y i f x l = x 2 = m f o r s o m e n o n z e r o i n t e g e r m .
P r o b l e m 2 4
S o l v e t h e e q u a t i o n
x 3 - 3 x = x + 2 .
S o l u t i o n 2 4 , A l t e r n a t i v e 1
I t i s c l e a r t h a t x > - 2 . W e c o n s i d e r t h e f o l l o w i n g c a s e s .
1 . - 2 < x < 2 . S e t t i n g x = 2 c o s a , 0 < a < 7 r , t h e e q u a t i o n b e c o m e s
8 c o s 3 a - 6 c o s a = 2 ( c o s a + 1 ) .
o r
2 c o s 3 a = V 4 c o s t
2 ,
a
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4 4
3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
f r o m w h i c h i t f o l l o w s t h a t c o s 3 a = c o s 2 2 '
T h e n 3 a - 2 = 2 m 7 r , m E Z , o r 3 a + 2 = 2 n 7 r , n E Z .
S i n c e 0 < a < 7 r , t h e s o l u t i o n i n t h i s c a s e i s
x = 2 c o s O = 2 ,
x = 2 c o s 4 5 , a n d x = 2 c o s 4 7
.
2 . x > 2 . T h e n x 3 - 4 x = x ( x 2 - 4 ) > 0 a n d
x 2 - x - 2 = ( x - 2 ) ( x + 1 ) > 0
o r
I t f o l l o w s t h a t
x 3 - 3 x > x > x + 2 .
H e n c e t h e r e a r e n o s o l u t i o n s i n t h i s c a s e .
T h e r e f o r e , x = 2 , x = 2 c o s 4 7 r / 5 , a n d x = 2 c o s 4 7 r / 7 .
S o l u t i o n 2 4 , A l t e r n a t i v e 2
F o r x > 2 , t h e r e i s a r e a l n u m b e r t > 1 s u c h t h a t
x = t
2
1
+ t 2 .
T h e e q u a t i o n b e c o m e s
( t 2 + ) 3
- 3 I t 2 +
2 ) =
1 t 2 +
2 + 2 ,
i s +
1
t 6
= t +
1
t ,
( t 7 - 1 ) ( t 5 - 1 ) = 0 ,
w h i c h h a s n o s o l u t i o n s f o r t > 1 .
H e n c e t h e r e a r e n o s o l u t i o n s f o r x > 2 .
F o r - 2 < x < 2 , p l e a s e s e e t h e f i r s t s o l u t i o n .
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3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
4 5
P r o b l e m 2 5 [ A I M E 1 9 9 2 ]
F o r a n y s e q u e n c e o f r e a l n u m b e r s A = { a l , a 2 , a 3 ,
} , d e f i n e A A t o b e
t h e s e q u e n c e { a 2 - a l , a 3 - a 2 , a 4 - a 3 , . . . } .
S u p p o s e t h a t a l l o f t h e t e r m s o f t h e s e q u e n c e A ( A A ) a r e 1 , a n d t h a t
a i 9 = a 9 2 = 0 .
F i n d a l .
S o l u t i o n 2 5
S u p p o s e t h a t t h e f i r s t t e r m o f t h e s e q u e n c e A A i s d .
T h e n
A A = { d , d + l , d + 2 , . . . }
w i t h t h e n t h t e r m g i v e n b y d + ( n - 1 ) .
H e n c e
A = { a l , a , + d , a l + d + ( d + 1 ) , a l + d + ( d + 1 ) + ( d + 2 ) , . . . }
w i t h t h e n t h t e r m g i v e n b y
a n = a l + ( n - 1 ) d + 2 ( n - 1 ) ( n - 2 ) .
T h i s s h o w s t h a t a n i s a q u a d r a t i c p o l y n o m i a l i n n w i t h l e a d i n g c o e f f i c i e n t
1 / 2 .
S i n c e a l g = a 9 2 = 0 , w e m u s t h a v e
a n = 2 ( n - 1 9 ) ( n - 9 2 ) ,
s o a l = ( 1 - 1 9 ) ( 1 - 9 2 ) / 2 = 8 1 9 .
P r o b l e m 2 6 [ K o r e a n M a t h e m a t i c s C o m p e t i t i o n 2 0 0 0 ]
F i n d a l l r e a l n u m b e r s x s a t i s f y i n g t h e e q u a t i o n
2 x + 3 x - 4 X + 6 X - 9 X = 1 .
S o l u t i o n 2 6
S e t t i n g 2 x = a a n d 3 x = b , t h e e q u a t i o n b e c o m e s
1 + a
2
+ b
2
- a - b - a b = 0 .
M u l t i p l y i n g b o t h s i d e s o f t h e l a s t e q u a t i o n b y 2 a n d c o m p l e t i n g t h e
s q u a r e s g i v e s
( 1 - a ) 2 + ( a - b ) 2 + ( b - 1 ) 2 = 0 .
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4 6
3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
T h e r e f o r e 1 = 2 ' = 3 ' , a n d x = 0 i s t h e o n l y s o l u t i o n .
P r o b l e m 2 7 [ C h i n a 1 9 9 2 ]
P r o v e t h a t
1
8 0
1 6 < ) ' - < 1 7 .
k =
S o l u t i o n 2 7
N o t e t h a t
T h e r e f o r e
v k
2 ( k + 1 - m i )
=
k + 1 + f a 2 + 4 x 2 = a 2 + ( 8
_
4
a )
( 1 )
o r
0 > 5 a 2 - 1 6 a = a ( 5 a - 1 6 ) .
T h e r e f o r e 0 < a < 1 6 / 5 , w h e r e a = 0 i f a n d o n l y i f b = c = d = e = 2
a n d a = 1 6 / 5 i f a n d o n l y i f b = c = d = e = 6 / 5 .
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5 6
3 . S o l u t i o n s t o I n t r o d u c t o r y P r o b l e m s
S o l u t i o n 4 1 , A l t e r n a t i v e 2
B y t h e R M S - A M i n e q u a l i t y , ( 1 ) f o l l o w s f r o m
b 2 + c 2 + d 2 + e 2 >
( b + c + d + e ) 2
_
( 8 - a ) 2
4 4
a n d t h e r e s t o f t h e s o l u t i o n i s t h e s a m e .
P r o b l e m 4 2
F i n d t h e r e a l z e r o s o f t h e p o l y n o m i a l
P a ( x ) = ( x 2 + 1 ) ( x - 1 ) 2 - a x 2 ,
w h e r e a i s a g i v e n r e a l n u m b e r .
S o l u t i o n 4 2
W e h a v e
( x 2 + 1 ) ( x 2
- 2 x + 1 ) - a x e = 0 .
D i v i d i n g b y x 2 y i e l d s
( + )
( x _ 2 + . )
B y s e t t i n g y = x + 1 / x , t h e l a s t e q u a t i o n b e c o m e s
y 2 - 2 y - a = 0 .
I t f o l l o w s t h a t
x + 1 = 1 f
l + a ,
x
w h i c h i n t u r n i m p l i e s t h a t , i f a > 0 , t h e n t h e p o l y n o m i a l P a ( x ) h a s t h e
r e a l z e r o s
1 + 1 + a f a - 2 , / - l - - + a - 2
X 1 , 2 =
2
I n a d d i t i o n , i f a > 8 , t h e n P a ( x ) a l s o h a s t h e r e a l z e r o s
1 - l + a f a - 2 I + a - 2
X 3 , 4 =
2
P r o b l e m 4 3
P r o v e t h a t
1
3
2 n - 1 1
2 4 . . 2 n
0 . T h e n
s 2
= 2 8 + 3 S + 6 8 > 3 ,
s o s > f , a n d h e n c e [ s ] > 1 .
B u t t h e n s > l s j y i e l d s
2 S > 2 i s J = ( 1 + 1 ) L , , ] > 1 + [ s ] > 8 ,
w h i c h i n t u r n i m p l i e s t h a t
6 S > 4 S = ( 2 8 ) 2 > s 2 .
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4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
S o 2 S + 3 3 + 6 S > s 2 , a c o n t r a d i c t i o n .
T h e r e f o r e x = - 1 i s t h e o n l y s o l u t i o n t o t h e e q u a t i o n .
P r o b l e m 5 4
L e t { a n } n > 1 b e a s e q u e n c e s u c h t h a t a l = 2 a n d
a n 1
a n + 1 =
+
2 a
n
f o r a l l n E N .
F i n d a n e x p l i c i t f o r m u l a f o r a n .
S o l u t i o n 5 4
S o l v i n g t h e e q u a t i o n
l e a d s t o x = f V 1 2 - . N o t e t h a t
2
a n + 1 + v / ' 2 -
a 2 n + 2 v f 2 a n + 2
a n +
a n + 1 - v f 2 -
a 2 - 2 V 1 2 a n + 2
a n - v 2
T h e r e f o r e ,
2 ^ - 1
a n + v f 2 -
( a , +
a n - v f 2 -
a n d
( V r - 2 + 1 ) t r y
v / 2 - [ ( v f 2 - + 1 ) 2 n + 1 ]
a n =
( V / 2 - +
1 ) 2 n - 1
P r o b l e m 5 5
L e t x , y , a n d z b e p o s i t i v e r e a l n u m b e r s . P r o v e t h a t
x y
x +
( x + y ) ( x + z )
+ y +
( y + z ) ( y + x )
z
+
z +
( z + x ) ( z + y )
-
S o l u t i o n 5 5
N o t e t h a t
( x + y ) ( x + z ) > x y +
x z .
6 7
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6 8
4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
I n f a c t , s q u a r i n g b o t h s i d e s o f t h e a b o v e i n e q u a l i t y y i e l d s
x 2 + y z > 2 x
y z ,
w h i c h i s e v i d e n t b y t h e A M - G M i n e q u a l i t y . T h u s
x
O ,
w h i c h i m p l i e s t h a t
T h e r e f o r e ,
a i + b i > a , b l ( a , + b 1 ) .
1 _ 1
a + b + 1
a i + b i + a l b l c l
1
a l b l ( a , + b i ) + a l b l c l
a l b l c l
a , b l ( a , + b l + c l )
C l
L i k e w i s e ,
a n d
a l + b l + c l
1 < a l
b + c + 1 - a l + b l + c l
7 7
1
0 .
N o t e t h a t
n f ( m , n - m ) = [ m + ( n - m ) ] f ( m , n - m )
= ( n - m ) f ( m , m + ( n - m ) )
( n - m ) f ( m , n )
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4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
7 9
o r
L i k e w i s e ,
f ( m , n - m ) =
n
n m f ( m , n ) .
g ( m , n - m ) = n n m g ( m , n ) .
S i n c e f ( m , n ) g ( m , n ) , f ( m , n - m ) g ( m , n - m ) .
T h u s ( m , n - m ) E S .
B u t ( m , n - m ) h a s a s m a l l e r s u m m + ( n - m ) = n , a c o n t r a d i c t i o n .
T h e r e f o r e o u r a s s u m p t i o n i s w r o n g a n d f ( x , y ) = l c m ( x , y ) i s t h e o n l y
s o l u t i o n .
P r o b l e m 6 5 [ R o m a n i a 1 9 9 0 ]
C o n s i d e r n c o m p l e x n u m b e r s z k , s u c h t h a t Z k I < 1 , k = 1 , 2 , . . . , n .
P r o v e t h a t t h e r e e x i s t e 1 , .
2 ,
. . , e n E
{ - 1 , 1 } s u c h t h a t , f o r a n y m < n ,
l e l z l + e 2 z 2 + ' ' ' + e m z m l < 2 .
S o l u t i o n 6 5
C a l l a f i n i t e s e q u e n c e o f c o m p l e x n u m b e r s e a c h w i t h a b s o l u t e v a l u e n o t
e x c e e d i n g 1 a g r e e n s e q u e n c e .
C a l l a g r e e n s e q u e n c e { z k } ' 1 h a p p y i f i t h a s a f r i e n d s e q u e n c e { e k } n
k _ 1
o f i s a n d - i s , s a t i s f y i n g t h e c o n d i t i o n o f t h e p r o b l e m .
W e w i l l p r o v e b y i n d u c t i o n o n n t h a t a l l g r e e n s e q u e n c e s a r e h a p p y .
F o r n = 2 , t h i s c l a i m i s o b v i o u s l y t r u e .
S u p p o s e t h i s c l a i m i s t r u e w h e n n e q u a l s s o m e n u m b e r m . F o r t h e c a s e
o f n = m + 1 , t h i n k o f t h e Z k a s p o i n t s i n t h e c o m p l e x p l a n e .
F o r e a c h k , l e t B k b e t h e l i n e t h r o u g h t h e o r i g i n a n d t h e p o i n t c o r r e -
s p o n d i n g t o z k . A m o n g t h e l i n e s 2 1 , L 2 , 6 , s o m e t w o a r e w i t h i n 6 0 o f
e a c h o t h e r ; s u p p o s e t h e y a r e f , a n d f , 3 , w i t h t h e l e f t o v e r o n e b e i n g f , .
T h e f a c t t h a t Q a a n d Q p a r e w i t h i n 6 0 o f e a c h o t h e r i m p l i e s t h a t t h e r e
e x i s t s s o m e n u m b e r e , 3 E { - 1 , 1 } s u c h t h a t z ' = z a + e , 3 z , 3 h a s a b s o l u t e
v a l u e a t m o s t 1 .
N o w t h e s e q u e n c e z ' , z . y , z 4 , z 5 , . . .
, z k + 1
i s a k - t e r m g r e e n s e q u e n c e , s o ,
b y t h e i n d u c t i o n h y p o t h e s i s , i t m u s t b e h a p p y ; l e t e ' , e 7 , e 4 , e 5 ,
. . . , e k + 1
b e i t s f r i e n d .
L e t e a = 1 .
T h e n t h e s e q u e n c e { e i } k + 1 i s t h e f r i e n d o f { z i } k + l . I n d u c t i o n i s n o w
c o m p l e t e .
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8 0
4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
P r o b l e m 6 6 [ A R M L 1 9 9 7 ]
F i n d a t r i p l e o f r a t i o n a l n u m b e r s ( a , b , c ) s u c h t h a t
V 3 2 - 1 =
3 a + 3 b + y c - .
S o l u t i o n 6 6
L e t x a n d y = 3 2 . T h e n y 3 = 2 a n d x = 3 y - 1 . N o t e t h a t
1 = y 3 - 1 = ( y - 1 ) ( y 2 + y + 1 ) ,
a n d
y 2 + y + 1 =
3 y 2 + 3 y + 3
_ y 3 + 3 y 2 + 3 y + 1
-
( y + 1 ) 3
3 3
3
w h i c h i m p l i e s t h a t
o r
O n t h e o t h e r h a n d ,
x 3 = y - 1 =
1
3
y 2 + y + 1
( y + l ) 3
- ' r 3
x
y +
3 = y 3 + 1 = ( y + 1 ) ( y 2 - y + 1 )
f r o m w h i c h i t f o l l o w s t h a t
1 _ y 2 - y + 1
y + l
3
C o m b i n i n g ( 1 ) a n d ( 2 ) , w e o b t a i n
C o n s e q u e n t l y ,
x
V 9
( 3 4 - 3 2 + 1 )
4
2
1
( a , b , c ) =
9 '
9 ' 9
( 1 )
( 2 )
i s a d e s i r e d t r i p l e .
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4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
8 1
P r o b l e m 6 7 [ R o m a n i a 1 9 8 4 ]
F i n d t h e m i n i m u m o f
1
1
1
l o g x l
x 2 -
4
+ l o g x 2
x 3 -
4
+ . . . + l o g x
( i i
-
4
w h e r e x 1 i x 2 , . . . , x n a r e r e a l n u m b e r s i n t h e i n t e r v a l ( 4 , 1 ) .
S o l u t i o n 6 7
S i n c e l o g , , x i s a d e c r e a s i n g f u n c t i o n o f x w h e n 0 < a < 1 a n d , s i n c e
( x - 1 / 2 ) 2 > 0 i m p l i e s x 2 > x - 1 / 4 , w e h a v e
l o g . " x k + i - 1 > l o g x k x k + 1 = 2 l o g x , , x k + 1 =
2 1 o g x k + l
4 l o g x k
I t f o l l o w s t h a t
l o g x l I x 2 -
4
I + l o g , ( X 3 -
4 )
+ . . . + l o g x n
( 1
- 4
)
>
2
l o g x 2
+
l o g x 3
+ . . . +
l o g x n +
l o g X I
-
l o g x i l o g x 2
1 0 9 X , - 1
l o g x n
> 2 n
b y t h e A M - G M i n e q u a l i t y .
E q u a l i t i e s h o l d i f a n d o n l y i f
x 1 = x 2 = . . . = x n = 1 / 2 .
P r o b l e m 6 8 [ A I M E 1 9 8 4 ]
D e t e r m i n e x 2 + y 2 + z 2 + w 2 i f
x 2
y
2 z 2
w 2
2 2 - 1 2 + 2 2 - 3 2 + 2 2 - 5 2 + 2 2 - 7 2
= 1 ,
x 2
2 z 2 W 2
4 2 - 1 2 + 4 2
y
- 3 2
+ 4 2 - 5 2 + 4 2 - 7 2 = 1 ,
x 2
y 2
z 2
, w 2
6 2 - 1 2 + 6 2 - 3 2 + 6 2 - 5 2 + 6 2 - 7 2
x 2 2 z 2
w 2
8 2 - 1 2 + 8 2 - 3 2 + 8 2 - 5 2 + 8 2 - 7 2
= 1
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8 2
4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
S o l u t i o n 6 8
T h e c l a i m t h a t t h e g i v e n s y s t e m o f e q u a t i o n s i s s a t i s f i e d b y x 2 ,
y 2 , z 2 ,
a n d w 2 i s e q u i v a l e n t t o c l a i m i n g t h a t
x 2
y
2 z 2 w 2
t - 1 2 + t - 3 2 + t - 5 2 + t - 7 2
= 1
( 1 )
i s s a t i s f i e d b y t = 4 , 1 6 , 3 6 , a n d 6 4 .
M u l t i p l y i n g t o c l e a r f r a c t i o n s , w e f i n d t h a t f o r a l l v a l u e s o f t f o r w h i c h i t
i s d e f i n e d ( i . e . , t 0 1 , 9 , 2 5 , a n d 4 9 ) , ( 1 ) i s e q u i v a l e n t t o t h e p o l y n o m i a l
e q u a t i o n
P ( t ) = 0 ,
w h e r e
P ( t ) = ( t - 1 ) ( t - 9 ) ( t - 2 5 ) ( t - 4 9 )
- x 2 ( t
- 9 ) ( t - 2 5 ) ( t - 4 9 ) - y 2 ( t - 1 ) ( t - 2 5 ) ( t - 4 9 )
- z 2 ( t - 1 ) ( t - 9 ) ( t - 4 9 ) - w 2 ( t - 1 ) ( t - 9 ) ( t - 2 5 ) .
S i n c e d e g P ( t ) = 4 , P ( t ) = 0 h a s e x a c t l y f o u r z e r o s t = 4 , 1 6 , 3 6 , a n d 6 4 ,
i . e . ,
P ( t ) = ( t - 4 ) ( t - 1 6 ) ( t - 3 6 ) ( t - 6 4 ) .
C o m p a r i n g t h e c o e f f i c i e n t s o f t 3 i n t h e t w o e x p r e s s i o n s o f P ( t ) y i e l d s
1 + 9 + 2 5 + 4 9 + x 2 + y 2 + z 2 + w 2 = 4 + 1 6 + 3 6 + 6 4 ,
f r o m w h i c h i t f o l l o w s t h a t
x 2 + y 2 + z 2 + w 2 = 3 6 .
P r o b l e m 6 9 [ B a l k a n 1 9 9 7 ]
F i n d a l l f u n c t i o n s f : R - + R s u c h t h a t
f ( x f ( x ) + f ( y ) ) = ( f ( x ) ) 2 + y
f o r a l l x , y E J R .
S o l u t i o n 6 9
L e t f ( 0 ) = a . S e t t i n g x = 0 i n t h e g i v e n c o n d i t i o n y i e l d s
f ( f ( y ) ) = a 2 + y ,
f o r a l l y c R .
S i n c e t h e r a n g e o f a 2 + y c o n s i s t s o f a l l r e a l n u m b e r s , f m u s t b e s u r j e c t i v e .
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4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
8 3
T h u s t h e r e e x i s t s b E R s u c h t h a t f ( b ) = 0 .
S e t t i n g x = b i n t h e g i v e n c o n d i t i o n y i e l d s
f ( f ( y ) ) = f ( b f ( b ) + f ( y ) ) = ( f ( b ) ) ' + y = y ,
f o r a l l y E R . I t f o l l o w s t h a t , f o r a l l x , y E R ,
( f ( x ) ) 2 + y = f ( x f ( x ) + f ( y ) )
= f [ f ( f ( x ) ) f ( x ) + f ( n ) ] = f [ f ( x ) f ( f ( x ) ) + y ]
= f ( f ( x ) ) 2 + y = x 2 + y ,
t h a t i s ,
( f ( x ) ) 2 = x 2
( 1 )
I t i s c l e a r t h a t f ( x ) = x i s a f u n c t i o n s a t i s f y i n g t h e g i v e n c o n d i t i o n .
S u p p o s e t h a t f ( x ) 5 4 x . T h e n t h e r e e x i s t s s o m e n o n z e r o r e a l n u m b e r c
s u c h t h a t f ( c ) = - c . S e t t i n g x = c f ( c ) + f ( y ) i n ( 1 ) y i e l d s
I f ( c f ( c ) + f ( y ) ) ] 2 = [ c f ( c ) + f ( y ) ] 2 = [ - c 2 + f ( y ) ] 2 ,
f o r a l l y E R , a n d , s e t t i n g x = c i n t h e g i v e n c o n d i t i o n y i e l d s
f ( c f ( c ) + f ( y ) ) = ( f ( c ) ) 2 + y = c 2 + y ,
f o r a l l y e R .
N o t e t h a t ( f ( y ) ) 2 = y 2 .
I t f o l l o w s t h a t
[ - c 2 + f
( y ) ] 2
= ( c 2 + y ) 2 ,
o r
f ( y ) = - y ,
f o r a l l y E R , a f u n c t i o n w h i c h s a t i s f i e s t h e g i v e n c o n d i t i o n .
T h e r e f o r e t h e o n l y f u n c t i o n s t o s a t i s f y t h e g i v e n c o n d i t i o n a r e f ( x ) = x
o r f ( x ) = - x , f o r x E R .
P r o b l e m 7 0
T h e n u m b e r s 1 0 0 0 , 1 0 0 1 ,
, 2 9 9 9 h a v e
b e e n w r i t t e n o n a b o a r d .
E a c h t i m e , o n e i s a l l o w e d t o e r a s e t w o n u m b e r s , s a y , a a n d b , a n d r e p l a c e
t h e m b y t h e n u m b e r 2 m i n ( a , b ) .
A f t e r 1 9 9 9 s u c h o p e r a t i o n s , o n e o b t a i n s e x a c t l y o n e n u m b e r c o n t h e
b o a r d .
P r o v e t h a t c < 1 .
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8 4
4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
S o l u t i o n 7 0
B y s y m m e t r y , w e m a y a s s u m e a < b . T h e n
2
m i n ( a , b ) =
a
W e h a v e
1 1
1
a
+ b ) '
2
f r o m w h i c h i t f o l l o w s t h a t t h e s u m o f t h e r e c i p r o c a l s o f a l l t h e n u m b e r s
o n t h e b o a r d i s n o n d e c r e a s i n g ( i . e . , t h e s u m i s a m o n o v a r i a n t ) .
A t t h e b e g i n n i n g t h i s s u m i s
_ 1 1 1 1
S
1 0 0 0 + 1 0 0 1 +
+ 2 9 9 9 < c '
w h e r e 1 / c i s t h e s u m a t t h e e n d . N o t e t h a t , f o r 1 < k < 9 9 9 ,
1
1
_
4 0 0 0 4 0 0 0
1
+ >
0 0 0 - k 2 0 0 0 + k 2 0 0 0 2 - k 2
2 0 0 0 2
1 0 0
R e a r r a n g i n g t e r m s i n S y i e l d s
C - 1 0 0 0
( 1 0 0 1
2 9 9 9 )
( 1 0 0 2
2 9 9 8 )
1 1 1
( 1 9 9 9
2 0 0 1
2 0 0 0
1 0 0 0
x
1 0 0 0 +
2 0 0 0
1 ,
o r c < 1 , a s d e s i r e d .
P r o b l e m 7 1 [ B u l g a r i a 1 9 9 8 ]
L e t a l , a 2 , . . . , a , , b e r e a l n u m b e r s , n o t a l l z e r o .
P r o v e t h a t t h e e q u a t i o n
1 + a 1 x + l + a 2 x +
+
l + a , , x = n
h a s a t m o s t o n e n o n z e r o r e a l r o o t .
S o l u t i o n 7 1
N o t i c e t h a t f i ( x ) =
1 - + a x i s c o n c a v e . H e n c e
f ( x ) =
1 + a , x + + l + a , - x
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4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
8 5
i s c o n c a v e .
S i n c e f ' ( x ) e x i s t s , t h e r e c a n b e a t m o s t o n e p o i n t o n t h e c u r v e y = f ( x )
w i t h d e r i v a t i v e 0 .
S u p p o s e t h e r e i s m o r e t h a n o n e n o n z e r o r o o t .
S i n c e x = 0 i s a l s o a r o o t , w e h a v e t h r e e r e a l r o o t s x 1 < x 2 < X 3 . A p -
p l y i n g t h e M e a n - V a l u e t h e o r e m t o f ( x ) o n i n t e r v a l s [ x 1 i x 2 ] a n d [ x 2 i x 3 ] ,
w e c a n f i n d t w o d i s t i n c t p o i n t s o n t h e c u r v e w i t h d e r i v a t i v e 0 , a c o n t r a -
d i c t i o n .
T h e r e f o r e , o u r a s s u m p t i o n i s w r o n g a n d t h e r e c a n b e a t m o s t o n e n o n z e r o
r e a l r o o t f o r t h e e q u a t i o n f ( x ) = n .
P r o b l e m 7 2 [ T u r k e y 1 9 9 8 ]
L e t { a n } b e t h e s e q u e n c e o f r e a l n u m b e r s d e f i n e d b y a 1 = t a n d
a n + 1 = 4 a n ( 1 - a n )
f o r n > 1 .
F o r h o w m a n y d i s t i n c t v a l u e s o f t d o w e h a v e a 1 9 9 8 = 0 ?
S o l u t i o n 7 2 , A l t e r n a t i v e 1
L e t f ( x ) = 4 x ( 1 - x ) . O b s e r v e t h a t
f - 1 ( 0 ) = 1 0 , 1 } ,
f - 1 ( 1 ) = { 1 / 2 } ,
a n d I { y : f ( y ) = x } j = 2 f o r a l l x E [ 0 , 1 ) .
L e t A n = { x E R : f n ( X ) = 0 } ; t h e n
A n + 1 =
{ x E 1 1 8 : f n + 1 ( x ) = 0 }
f - 1 ( [ 0 , 1 ] ) = [ 0 , 1 ] ,
= { x E R : f n ( f ( x ) ) = 0 } = { x E R : f ( x ) E A n } .
W e c l a i m t h a t f o r a l l n > 1 , A n C [ 0 , 1 ] , 1 E A n , a n d
I A n I = 2 n - 1 + 1 .
F o r n = 1 , w e h a v e
A 1 = I x G R I f ( x ) = 0 } = [ 0 , 1 1 ,
a n d t h e c l a i m s h o l d .
N o w s u p p o s e n > 1 a n d A n C [ 0 , 1 ] , 1 E A n , a n d A n I = 2 n - 1 + 1 . T h e n
x c A n + 1 = f ( x ) E A n C [ 0 , 1 ] = x c [ 0 , 1 ] ,
s o A n + 1 C [ 0 , 1 ] .
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8 6
4 . S o l u t i o n s t o A d v a n c e d P r o b l e m s
S i n c e f ( 0 ) = f ( 1 ) = 0 , w e h a v e f ' + 1 ( 1 ) = 0 f o r a l l n > 1 , s o 1 E A , , , + 1
N o w w e h a v e
j A n + 1 j
l { x : f ( x ) E A , , } l
I { x : f ( x ) = a l l
a E A ,
l { x : f ( x ) = 1 } I +
I f x : f ( x ) = a l l
a E A
a E [ 0 , 1 )
T h u s t h e c l a i m h o l d s b y i n d u c t i o n .
F i n a l l y , a 1 9 9 8 = 0 i f a n d o n l y i f f 1 9 9 7 ( t ) = 0 , s o t h e r e a r e 2 1 9 9 6 + 1 s u c h
v a l u e s o f t .
S o l u t i o n 7 2 , A l t e r n a t i v e 2
A s i n t h e p r e v i o u s s o l u t i o n , o b s e r v e t h a t i f f ( x ) E [ 0 , 1 ] t h e n x E [ 0 , 1 ] ,
s o i f a 1 9 9 8 = 0 w e m u s t h a v e t E [ 0 , 1 ] .
N o w c h o o s e 0 E [ 0 , 7 r / 2 ] s u c h t h a t s i n 0 = V .
O b s e r v e t h a t f o r a n y 0 E R ,
f ( s i n e ) = 4 s i n 2 0 ( l - s i n e 0 ) = 4 s i n 2 O c o s t 0 = s i n 2 2 0 ;
s i n c e a 1 = s i n e 0 , i t f o l l o w s t h a t
a 2 = s i n
2 2 0 ,
a 3 = s i n
2
4 0 ,
. . . , a 1 9 9 8 = s i n
2 2 1 9 9 7 0 .
T h e r e f o r e
a 1 9 9 8 = 0
s i n 2
1 9 9 7 0 =
0
0 =
k 7 r
2 1 9 9 7
f o r s o m e k E Z .
T h u s t h e v a l u e s o f t w h i c h g i v e a 1 9 9 8 = 0 a r e
s i n 2 ( k i r / 2 1 9 9 7 )
k E Z , g i v i n g 2 1 9 9 6 + 1 s u c h v a l u e s o f t .
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8 7
P r o b l e m 7 3 [ I M O 1 9 9 7 s h o r t l i s t ]
( a ) D o t h e r e e x i s t f u n c t i o n s f : R - * ] I 8 a n d g : R - > R s u c h t h a t
f ( g ( x ) ) = x 2
a n d
g ( f ( x ) ) = x 3
f o r a l l x E R ?
( b ) D o t h e r e e x i s t f u n c t i o n s f : R - - > R a n d g : R - * R s u c h t h a t
f ( g ( x ) ) = x 2
a n d
g ( f ( x ) ) = x 4
f o r a l l x E l l ?
S o l u t i o n 7 3
( a )
top related