D
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n
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A L A Y S I A A S S E A N S C H O
M
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A T H O L Y M P I A D S 2 0 1 1 C O O N T E S T
S O L U T I O N S A N D A N S W E R S 1
1
II t e m s i n pp a r e n t h e s e s a r e n o o t r e q u i r e d .
9 2
M E T H O D 1 : S t r a t e g y : C o u n t t h e n u m b e r o f t i m e s e a c h d i g i t a p p e a r s . M u l t ip l y e a c h v a lu e b y t h e n u m b e r o f t im e s it a p p e a r s . A d d t h o s e p r o d u c t s . i s 9 2 . 1 0 × 2 + 9 × 4 + 6 × 6 = 9 2 . T h e s u m M E T H O D 2 : S t r a t e g y : S e p a r a t e i n t o a r e c t a n g l e o f 2 s a n d a t r i a n g l e o f 4 s . 2 2 2 6 4 4 4 4 4
T h e s u m
2
M E c v c
2 2 6 6 4 4
2 6 6 6 4 4
2 2 2 2
2 2 2 2
=
2 2 2 2
2 2 2 2
2 2 2 2
+
4 4 4 4 4
4 4 4 4 4 4 4 4 4 4
2
4
i s 1 6 × 2 + 1 5 × 4 = 9 2 .
E T H O D 1 : S t r a t e g a c h n i c k e l is w o r t h e n ts m o re . T h e n e a lu e o f 1 0 c e n t s . T o l l e c t i o n .
( q u a r t e r s )
y : C o m p a r e e a c h c o i n t o t h e a v e r a g e v a l u e .
5 c e n t s le s s th a n th e a v e r a c h q u a r te r , c o m b in e d w ith h e r e a r e 1 2 n i c k e l s , s o 4 q
M E T H O D 2 : S t r a t e g y : L e t Q r e p r e s e n t t h e n u m + 6 0 . T h e t o t a l n u m b e r q u a r t e r s m u s t b e a d d e
a g e 3 n u a rr
; e a c h ic k e ls t e r r s m
q u , h a u ss
a r te r is s a n a v t b e a d
w o r t h 1 5 e r a g e d e d t o t h e
3 9
U s e a l g e b r a . b e r o f q u a r t e r s t o b e a d d e d . T h e t o t a l v a l u e o f t h e c o i n s i s 2 5 Q o f c o i n s si s Q + 1 2 . T h e n 2 5 Q Q + +1 2 2 0 = 1 0 . S o l v i n g , Q = 4 . T h u s , 4
( s u m s )
d .
F O L L O W - U P S : H o w m a n y $ 5 b i l l s m u s t b e a d d e d t o t w e n t y $ 1 0 0 b i l l s s o t h a t t h e a v e r a g e v a l u e o f a l l t h e b i l l s i s $ 1 0 ? [ 3 6 0 ]
3
3
4
5 ( f a c t o r s )
S t r a t e g y : F i n d t h e l e a s t a n d g r e a t e s t p o s s i b l e s u m s .
T p e + T
h o x 1 h
e l e a s t p o s s ib l e s s ib l e s u m i s o b tr e m e s is a ls o a 5 in c lu s iv e , y o u e r e a r e 3 9 d i f f e
s u m ta in p o s s h o re n
is o b t a in e d b y a d d in g – 1 2 a n d e d b y a d d in g 7 a n d 8 . E v e r y in t e s ib le s u m . B y e x a m in in g a n u m b u ld s e e 2 3 n e g a t iv e s u m s , 1 5 p o t s u u m m s t h a t c a n n b e o b b t a i n e d .
– 1 1 . g e r b e r lin s it iv e
T h e e tw e e fr o s u m
g re e n m s ,
a t e s t t h e s e – 2 3 t o a n d z e r o .
F O L L O W - U P S : H o w m a n y d i s t i n c t s u m s c a n b e o b t a i n e d b y a d d i n g t w o d i ffe r e n t i n t e g e r s c h o s e n fr o m t h e c o n s e c u t i v e e v e n i n t e g e r s fr o m - 1 2 t o + 8 , i n c l u s i v e ? [ 1 9 ]
C o p y r ig h t ©
2 0 1 0 b y M a l a y s ia A S E A N
S c h o o ls M a t h O ly m p ia d s .
w w w . m a s m o . in fo
5
3 8 ( o r e q u i v a l e n t )
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A L A Y S I A A S S E A N S C H O
M 4
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A T H O L Y M P I A D S 2 0 1 1 C O
N TT E S T
M T s a s a
E T H O D 1 : S t r a t e g y : U s e t h e d i v i s i b i l i t y r u l e s t o f i n d f a c t o r s . h e s u m o f t h e d ig it s o f 5 6 1 is 1 2 , s o 3 is a f a c t o r a n d 5 6 1 ÷ 3 a t is f ie s t h e t e s t o f d iv is ib i lit y f o r m u lt ip l e s o f 1 1 ( t h a t i s , 1 – 8 + ls o a f a c t o r a n d 1 8 7 ÷ 1 1 = 1 7 . T h u s 5 6 1 f a c t o r s in to 3 × 1 1 × h o w s a ll f a c t o r p a ir s o f 5 6 1 . O f t h e 8 f a c t o r s , 3 a r e p r im e . T h e r e 5 f a c t o r s o f 5 5 6 6 1 1 w h i c h a r e n o t p r r i m e .
M C a 5
E T H O a ll th e ll its fa 6 1 w h F n d ( a
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= 1 7 1 7 re f
8 7 . T h e = 0 ), s o . T h e ta o r e , t h e
n 1 8 7 1 1 i s b le r e
D 2 : S t r a t e g y : F i n d t h e t o t a l n u m b e r o f f a c t o r s w i t h o u t f a c t o r i n g . p r i m e f a c t o r s P , Q , a n d R . T h e i r p r o d u c t i s P Q R a n d t h e t a b l e s h o w s c t o r p a i r s . I n a l l , 3 o f t h e 8 f a c t o r s a r e p r i m e . T h u s , t h e r e a r r e 5 f a c t o r s o f i c h a r e n o t p r i m e .
O L L O W -U P S : (1 o t p r i m e ? [ 1 2 ] ( 2 o e s it h a v e a lto g e 4 ) W h y is 1 n e ith e t l e a s t 3 fa c to r s. 1
) N ) N th e r p h a
is th e is th e r ? [ 3 2 r im e n s o n ly
p ro d u p r o d u ] ( 3 ) W o r c o m o n e fa
c t o f 4 d i ffe r e n t p r i m c t o f 5 d i ffe r e n t p r i m h a t is th e le a s t n u m p o s i t e ? [ A p r i m e h c t o r . ]
e n u m e n u m b e r w a s e x a
b e r s . H o w b e r s . H o w ith e x a c tly c tly 2 fa c to
m a n y m a n y 8 fa c t o rs a n d
u p p o s e fo r e x a m p le th a t B C = 6 a n d C D = 4 . T h e n B P a n d P C e a c h r e 3 , a n d C Q a n d Q D e a c h a r e 2 , a n d t h e a r e a o f r e c t a n g le A B C D is 4 . h e a r e a o f ∆ A B P i s 2 1 ⋅ 4 ⋅ 3 = 6 , o f ∆ P C Q i s 2 1 ⋅ 3 ⋅ 2 = 3 , a n d 1 ⋅ 2
6 ⋅ 2 = 6 . T h e t o t a l u n s h a d e d a r e a i s 6 + 3 + 6 = 1 5 , a n d t h e a r e a o f ∆ A P Q i s 9 . T h e a r r e e a o f Δ A P Q i s 2 9 4 = 38 o f t h e a r e e a o f ∆ Q D A i s
o f r e c c t a n n g l e A B C D . M E T H O D 2 : S t r a t e g y : S p l i t t h e f i g u r e i n t o m o r e c o n v e n i e n t s h a p e s .
D r a w P R a n d Q S p a r a l l e l t o t h e s i d e s o f t h e r e c t a n g l e a s s h o w n . T h e a r e a o f ∆ A B P i s 2 1 t h e a r e a o f r e c t a n g l e A B P R , w h i c h i s 2 1 t h e a r e a o f A B C D . T h e n ∆ A B P i s
1 4
t h e t h e a r e a o f A B C D .
T h e a r e a o f ∆ Q D A i s 2 1 t h e a r e a o f r e c t a n g l e S Q D A w h i c h i s 2 1 t h e a r e a o f A B C D . S o ∆ Q D A i s 4 1 t h e a r e a o f A B C D . T h e a r e a o f ∆ P C Q i s 2 1 t h e a r e a o f r e c t a n g l e P C Q T w h i c h i s 4 1 t h e t h e a r e a o f A B C D . S o ∆ P C Q i s 8 1 t h e a r e a o f A B C D . T h e u n s h a d e d a r e a t h e n i s 4 1 +
5 1 1 4 + 8 = 8
o f t h e a r e a o f A B C D .
P Q R 1 × P Q R 3 × Q R 1 1 × P R 1 7 × P Q
f a c t o r s o f N a r e fa c t o r s r s ? [ 3 0 ] a c o m p o s i t e h a s
S t r a t e g y : F i n d t h e u n s h a d e d a r e a . M E T H O D 1 : S t r a t e g y : A s s i g n c o n v e n i e n t n u m e r i c a l l e n g t h s t o t h e s i d e s .
S a 2 T
5 6 1 1 × 5 6 1 3 × 1 8 7 1 1 × 5 1 1 7 × 3 3
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D
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A L A Y S I A A S S E A N S C H O
M 6
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M E T H O D 1 : S t r a t e g y : A d d d e c i m a l s i n a c o n v e n i e n t o r d e r . R e w r i t e e a c h t e r m a s a d e c i m a l a n d t h e n c o m b i n e t e r m s o f t h e s a m e s ig n .
1 − 1 2 0 + 1
3 4 0 0 − 1 0 0 0 =
1 − 0 . 2 + 0 . 0 3 − 0 . 0 4
=
6
I t e m s i n pp a r e n t h e s e s a r e n o t r e q u i r e d .
0 . 8 2 6
( 1 + 0 . 0 3 ) − ( 0 . 2 + 0 . 0 0 4 )
= 1
. 0 3 − 0 . 2 0 4 = 0 . 8 2 6 M E T H O D D 2 : S t r a t e g y : E l i m i n a t e t h e d e n o m i n a t o r s ( t e m p o r a r i l y ) . M u lt ip ly e a c h t e r m b y 1 0 0 0 a n d r e d u c e . T h e n d iv id e b y 1 0 0 0 t o u n d o s te p 1 . 7
1 0 0 0 − 2 10 00 0 + 3 1 0 0 0 0 0 − 14 00 00 00 = 1 0 0 0 − 2 0 0 + 3 0 − 4 = 8 2 6 N o w
7
5
d i v i d e b y 1 0 0 0 t o u n d o t h e f i r s t s t e p . T h e r e s u l t i s 0 . . 8 2 6 .
S t r a t e g y : W o r k b a c k w a r d s .
8
4 7
A m y p i c k e d a w h o l e n u m b e r , s o s h e s t a r t t e d w i t h 5 .
8
M E T H 2 7 , 3 5 T h e n is tw ic
O D 1 : S t r a t e , a n d 3 2 a re 2 7 + 3 5 + 3 2 e t h e p e r im e
M E T H O L e t a , b a re : (1 ) O n e w a (3 ) fro m
g y : U s e t h e s y m m e t r y o f t h e g i v e n i n fo r m a t i o n .
( 3 )
a + c = 3 5 2 a + b + c = 6 2 b + c = 3 2
)
9
e a c h t h e s u m o f a d i f f e r e n t p a i r o f s i d e s o f t h e t r ia n g le . is t h e s u m o f th e t h r e e s id e s , e a c h c o u n t e d t w ic e . T h u s 9 4 . t e r a n d t h e p e r i m e t e r o f t t h e t r i i a n g l e i s 4 7 c m
D D 2 : S t r a t e g y : U s e a l g e b r , a n d c r e p r e s e n t t h e s id a + b = 2 7 , (2 ) a + c = 3 y t o s o lv e t h is s y s te m is t h e r e s u l t . ( 1 ) a + b = 2 7
( 2 )
( c m
3
a t o fi n d t h e l e n g t h o f e a c h s i d e .
e s o f t h e t r i a n g l e . T h e n t h e t h r e e e q u a t i o n s 5 , a n d ( 3 ) b + c = 3 2 . t o a d d e q u a t i o n s ( 1 ) a n d ( 2 ) , a n d t h e n s u b t r a c t 1 0
N o w a d d ( 4 ) a n d ( 3 ) : ( 4 ) a = 1 5
b + c = 3 2
2 5
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D
i v i s i o
n
M
A L A Y S I A A S S E A N S C H O
M
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C o n t i n u e e d d
F o o v
9
O f f o
L L O W a p a r a 3 d i ffe lu m e o
- U P S : ( 1 ) S u lle lo g r a m . W r e n t fa c e s o f f t h e b o x ? [ 1
p p o h a a b 4 0
s e t h e 3 n u m b e r s g i v e n i n t h e p r o b l e m a r e e a c h t h e s u m o f 2 s i d e s t i s t h e p e r i m e t e r o f t h e p a r a l l e l o g r a m ? [ 6 2 2 / 3 c m ] ( 2 ) T h e a r e a o x ( r e c t a n g u l a r s o l i d ) a r e 2 0 , 2 8 , a n d 3 5 s q c m . W h a t i s t h e c m 3 ]
S t r a t e g y : C o n t i n u e t h e s e q u e n c e a n d l o o k fo r a p a t t e r n .
T h e s i x t h t e r m i s – ( 3 s o o n . T h e te rm s re p t e r m i s - 6 , a n d s o t h
+ – 6 + 1 ) = – ( – 2 ) = 2 . T h e s e q u e n c e i s 1 , 2 , 3 , - 6 , , 1 , 2 , 3 , - 6 , 1 , 2 , 3 , - 6 , , a n d e a t i n g r o u p s o f f o u r . T h u s , e v e r y f o u r t h t e r m i s - 6 . T h e r e f o r e t h e 1 0 0 t h e 9 9 t h t e r m i s 3 .
F O L L O W - U P S : ( 1 ) A s e q u e n c e b e g i n s 1 , 2 , 3 , … T h e f o u r t h t e r m i s t h e t h i r d l e s s t h e s e c o n d ; t h e f i f t h t e r m i s t h e f o u r t h l e s s t h e t h i r d , a n d s o o n . W h a t i s t h e 4 9 t h t e r m ? [ - 1 ]
1 0
M E T H O D D 1 : S t r a In th e s e rie s 1 + 2 5 t e r m s . P a ir th a n d s o o n . T h e s T h e n u m b e r w ith
te g y : F in d th e s u m
o f t h e n u m b e r s .
3 + e s e u m o u t
g e t w s , w 5 0 a id d le
5 … + te r m s a o f e a c h a p a ir is
4 9 , w e s f o llo p a ir i s th e m
fr o n o
o m r k in d th n e ,
1 to 4 9 g fro m e re a re 2 5 . T h
b y th e 1 2 e s u
a d o u p a m
d in g 2 t w e n t y f o u r t i m e s , s o t h e s e r i e s h a s t s id e i n w a r d ( 1 + 4 9 ) + ( 3 + 4 7 ) + ( 5 + 4 5 ) , ir s . is t h e n 1 2 × 5 0 + 2 5 = 6 2 5 . T h e n ,
6 2 5 i s 2 5 . M E T H O D 2 : S t r a t e g y : L T h e ta b le b e lo w e x a m s q u a re r o o t is e q u a l to N u m b e r o f T e 1 2 3 4 5
o o k fo r a p a t t e r n i n t h e p a r t i a l s u m s .
in e s t h e s q u a r e r o o t o f t h e s u m s o f t h e f ir s t f e w t e r m s . I n e a c h c a s e , t h e t h e n u m b e r o f t e r m s a d d e d . r m s S u m o f T e r m s S q u a r e R o o t t o f t t h e S u u m m o f T e r m s 1 1 (1 + 3 = ) 4 2 (1 + 3 + 5 = )9 3 1 6 4 2 5 5
T h e r e a r e 2 5 t e r m s in t h e g iv e n s e q u e n c e ( 2 5 o d d n u m b e r s f r o m s q u a r e r o o t o f t h e s u m o f t h e s e r ie s is 2 5 .
1 t o 4 9 ) . T h e r e f o r e t h e
( N o t e : O t h e r F O L L O W - U P p r o b l e m s r e l a t e d t o s o m e o f t h e a b o v e c a n b e fo u n d i n o u r b o o k s “ M a t h O l y m p i a d C o n t e s t P r o b l e m s fo r E l e m e n t a r y a n d M i d d l e S c h o o l s ” a n d “ C r e a t i v e P r o b l e m S o l v i n g i n S c h o o l M a t h e m a t i c s . ” )
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D
i v i s i o
n
M
A L A Y S I A A S S E A N S C H O
M 1 1
M E TT H T h e p 2 0 2 = n u m b
M
O
L S
A T H O L Y M P I A D S 2 0 1 1 C O
N T T E S T
O D 1 : S t r a t e g y : L o o k f o r a p e r f e c t s q u a r e n e a r 4 2 0 . a g e n u m b e r s d if f e r b y 1 , s o t h e f a c to r s o f 4 2 0 m u s t b e n e a r ly e q u a l. 4 0 0 , s o t r y 2 0 a s a f a c t o r . 4 2 0 = 2 0 × 2 1 , a n d t h e s u m o f t t h e t w o p a g e e r s i s 4 1 .
1 1
4 1
M E T H O D 2 : S t r a t e g y : C o m b i n e t h e p r i m e f a c t o r s o f 4 2 0 . T h e T h e p r i m e f a c t o r i z a t i o n o f 4 2 0 i s 2 × 2 × 3 × 5 × 7 . C o m b in in g t h e f a c t o r s i n t o t w o p r o d u c t s w h o s e d i f f e r e n c e i s 1 , w e g e t 2 0 × 2 1 . T h e i r s u m i s s 4 1 . F O L L O W - U P S : ( 1 ) T h e s u m o f t h e 6 p a g e n u m b e r s i n a c h a p t e r o f t h e b o o k i s 1 5 3 . W h a t i s t h e n u m b e r o n t h e f i r s t p a g e o f t h e c h a p t e r ? [ 2 3 ]
1 2
1 2
1 2
S t r a t e g y : M i n i m i z e t h e n u m b e r o f a c o r n s i n e a c h h o l e .
E a c h h o le r e q u ir e s a th e s e c o n d , a n d s o o 1 + 2 + 3 + … + 1 1 + a t le a s t 9 1 a c o r n s . T in d ic a t e d . T h e s q u i r r m a k in g 1 4 a c o r n s in F d ( h h
O L L O W i ffe r e n t 2 ) I n h o o le h a s o l e s ? [ 2
d iffe r e n t n u m b e r o f a c o r n s n . C o n t in u e u n t il th e t o t a l n 1 2 = 7 8 , a n d 1 + 2 + 3 + … h e s q u ir r e l p u t s th e fir s t 7 8 e l th e n c a n p u t th e o th e r tw t h a t h o l e . T h e g r e a t e s t p o
- U P S : ( 1 ) A s i d e f r o m 1 , 2 c o u n tin g n u m b e rs h a v e a s w m a n y d i ffe r e n t w a y s c a n a d i ffe r e n t n u m b e r o f a c o r 4 ]
,3 , . . u m o f th e s q n s a n d
, s o u m b + 1 a c o r o a c s ss i b
.,1 1 ,1 4 a 8 0 ? [ 1 : u ir r e l b u n o h o le
I t e m s i n pp a r e n t t h e s e s a r e n o t r e q u u i i r e d . .
p u t 1 i n t h e f i r s t h o l e , 2 i n e r o f a c o r n s is n e a r 8 0 . 3 = 9 1 . 1 3 h o le s r e q u ir e n s i n t o 1 2 h o l e s a s o r n s i n t o t h e 1 2 t h h o l e , l e v a a l l u e o f f N i s 1 2 .
b o v e, th e la r y 2 0 is e m
h o w m a n s t 2 n u m b a c o rn s in p t y ? [ 9 ]
y o th e rs a tw o ( 3 ) 2
e r s e re 1 2 h o le s 0 a c o
ts o a n d if e rn s
( h o l e s )
1 3
f 1 2 1 3 ] a c h i n 3
3 5 1 4
1 3
S t r a t e g y : L i s t t h e p o s s i b l e n u m e r a t o r s a n d d e n o m i n a t o r s .
D e n o t e t h e f r a c t i o n a n d i t s r e c i p r o c a l b y A B a n d AB . T h e i r s u m
h a s a d e n o m in a to r
8 2
o f 1 5 , s o A a n d B m u s t b e c h o s e n f r o m { 1 , 3 , 5 , 1 5 } . N e it h e r A n o r B c a n b e 1 5 , f o r 1 1 5 , 1 3 5 , a n d 1 5 5 a r e e a c h l a r g e r t h a n 2 1 4 5 . L i k e w i s e , n e i t h e r A n o r B c a n b e 1 , f o r 5 1 a n d 3 1 a r e a l s o l a r g e r t h a n 2 1 4 5 . T h e o n l y p o s s i b i l i t y i s t h a t t h e f r a c t i o n s a r e 3 5
a n d
5 3
. I n f a c t, t h e ir s u m
i s 13 5 4
4
= 2 1 5
( u n i t s )
. O f t h e s e t w o , t h e p r o p e r f r a c t i o n i s 35 .
F O L L O W - U P S : W h a t i s t h e l e a s t p o s s i b l e n e c e s s a r i l y p r o p e r ) a n d i t s r e c i p r o c a l ? [ 2 ]
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S t r a t e g y : L o o k fo r a p a t t e r n .
C o n s id e r th e p o in f ir s t q u a d r a n t , f in d C o o r d in a t e s o f (1 , 0 (2 , 1 (3 , 2
ts w h e r e th e p a th c h th e p a t h le n g t h t o e a c o rn e r p o in t ) 1 ) 1 + 1 ) 9 + 3
T h e p a t h le n g t h s a r e c o o rd in a te s o f th e c o t o ( 5 , 4 ) is 8 1 u n it s . F o f t t h e p a t h i s s 8 2 u n
a n g e s d i r e c t i o n . S i n c e ( 5 , 3 ) i s i n t h e c h o f t h e u p p e r r i g h t “ c o r n e r s ” . P a th le n g th + 2 + 2 + 3 = 9 + 4 + 4 + 5 = 2 5
c o n s e c u t iv e o d d s q u a r e s . I n f a c t , t h e y a r e t h e s q u a r e s o f t h e s u m o f t h e r n e r p o i n t s . T h e c o r n e r p o i n t c l o s e s t t o ( 5 , 3 ) i s ( 5 , 4 ) a n d t h e p a t h l e n g t h o l l o w i n g t h e s p i r a l , ( 5 , 3 ) i s t h e n e x t l a t t i c e p o i n t r e a c h e d , s o t h e l e n g t h i t s .
F O L L O W - U P S : I n I n t h e o t h e r 3 q u a d r a n t s w h a t p a t t e r n s a r e f o r m e d b y t h e p a t h ’ s l e n g t h s t o t h e c o r n e r s ?
1 5
M E T T H O D 1 : S t r a t e g y : F i n d t h e s p e e d o f e a c h h a n d i n d e g r e e s p e r m i n u t e . T o T h e m i n u t e h a n d m o v e s 3 6 0 o i n 1 h o u r a n d t h u s m o v e s 6 o p e r m i n u t e . T h e h o u r h a n d m o v e s 1 1 2 a s f a r a s t h e m i n u t e h a n d e v e r y h o u r a n d t h e r e f o r e
m o v e s 1 1 2 a s f a r e v e r y m i n u t e , i . e .
1 2
º p e r m in u t e . A t 8 : 0 0 t h e h o u r h a n d is 2 4 0 º
a h e a d o f t h e m in u te h a n d . I n t h e n e x t 2 4 m in u te s t h e h o u r h a n d m o v e s a n a d d i t i o n a l 1 2 º a n d t h e m i n u t e h a n d m o v e s 1 4 4 º . A t 8 : 2 2 4 t h e e a n n g l e b e t w e e n t h e h a n d s i s 2 4 0 + 1 2 – 1 4 4 = 1 0 8 º . M E T H O D 2 : S t r a t e g y : S t a r t a t 1 2 : 0 0 a n d s e e h o w f a r e a c h h a n d h a s r o t a t e d . F r o m 8 : 0 0 t o 8 : 2 4 , t h e m i n u t e h a n d h a s r o t a t e d 26 40 = 25 o f t h e w a y a r o u n d t h e c l o c k . T h a t i s ,
i t h a s r o t a t e d 5 2 o f 3 6 0 º = 1 4 4 º . T h i n k o f 1 2 : 0 0 a s 0 º , 3 : 0 0 a t 9 0 º , a n d 6 : 0 0 a s 1 8 0 º . T h e n t h e m in u t e h a n d is p o in t i n g t o 1 4 4 º . M e a n w h i le , t h e h o u r h a n d , w h i c h n e e d s 1 2 h o u r s t o r o t a t e 3 6 0 º , r o t a t e s 3 0 º e v e r y h o u r . T h u s , a t 8 : 0 0 , i t w a s p o i n t i n g t o 2 4 0 º . A t 8 : 2 4 , i t h a s r o t a t e d a n o t h e r 25 o f 3 0 º = 1 2 º a n d i s p o i n t i n g t o 2 5 2 º . A t 8 8 : 2 4 t h e a n g l e b e t w e e n t h e h a n d s s i s 2 5 2 – 1 4 4 = 1 0 8 º .
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M E T H O D D 1 : S t r a t e g y : S t a r t w i t h K a y l a , w h o K a y l a i s t h e t h i r d p e r s o n . K a y l a i s 2 p l a c e s i n f r o n t o f E l i . E l i i s 4 p l a c e s b e h i n d S a r a . S a r a i s 3 p l a c e s i n f r o n t o f A b b y . A b b y i s i n t h e c e n t e r o f t h e l i n e . A b b y i s t h e 4 t h p e r s o n i n l i n e . T h e r e a r e 3 p e o p le in f r o n t o f h e r a n d 3 s t u d e n t s i n n t h e l i n ee .
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M E T H O D D 2 : S t r a t e g y : C r e a t e t h e l i n e , k e e p i n g A b b y i n S a r a is 3 p la c e s in f r o n t o f A b b y . … , E li is 4 p l a c e s b e h in d S a r a . … , K a y l a i s 2 p l a c e s i n f r o n t o f E l i . … , K a y l a i s t h e 3 r d p e r s o n i n l i n e , s o n o a d d it io n a l s p a c e s a r e n e e d e d in f r o n t o f S a r a . t h e l i n e .
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T h e r e a r e 7 s tu d e n t s in
M E T H O D D 1 : S t r a t e g y : C a l c u l a t e t h e r e q u i r e d v a l u e s . T h e m e d i a n i s 8 + 2 6 = 7 . T h e m o d e i s 6 . T h e m e a n i s 3 + 6 + 6 + 6 8 + 1
, … , … , …
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a n d m e d ia n o th e m e d ia n a s e t ? [ 2 0 ] ( 2 e e o f th e n u m
f a n d ) T b e
s e t o f fi v e d i a n o th e r is h h e m e a n , m e r s a r e 9 , 1 3 ,
ffe r a lf d ia a n d
I t e m s i n pp a r e n t h e s e s a r e n o o t r e q u i r e d .
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M E T H O D 1 : S t r a t e g y : F i n d t h e u n s h a d e d a r e a i n s i d e e a c h c i r c l e . T h e a r e a s i n s i d e t h e t w o c i r c l e s a r e 2 5 π a n d 9 π r e s p e c t i v e l y . T h e u n s h a d e d a r e a i n s i d e c i r c l e A i s 2 5 π − 7 2 = 4 3 2 . T h e u n s h a d e d a r e a i n s i d e c i r c l e B i s 9 π − 7 2 = 1 1 2 . T h e π
t o t a l u n s h a a d d e d d a r e e a i s 4
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s q c m .
M E T H O D D 2 : S t r a t e g y : S t a r t w i t h t h e t o t a l a r e a o f t h e t w o c i r c l e s . T h e s u m o f t h e a r e a s o f t h e c i r c l e s i s 2 5 π + 9 π = 3 4 π . T h i s , h o w e v e r , c o u n t s t h e s h a d e d a r e a t w i c e , o n c e f o r e a c h c i r c l e . T h e t o t a l u n s h a d e d a r e a i s 3 4 π − 2 ( 7 2 ) π = 2 7 π . π
2 0
M E T T H O D 1 : S t r a t e g y : C o u n t i n a n o r g a n i z T o f o r m a t r ia n g le , t w o p o in t s m u s t b e c o t h e r . S u p p o s e t w o p o in t s a r e c h o s e n p o in ts in t h e t o p r o w A , B , C , D . T w o p o A C , A D , B C , B D , C D ) . F o r e a c h o f t h e s m a y b e a n y o f t h e 4 p o in ts in t h e b o t t o t r ia n g l e s u s in g 2 p o i n t s f r o m t h e t o p r o w
e d w a y .
h o s e n fro m in ts m e 6 p a m ro w a n d 1
fro m th e a y b ir s o . T h fro m
L ik e w is e , t h e r e a r e 2 4 tr ia n g le s u s in g 2 p o i n t s f r o m 4 8 t r i a n g l e s s c a a n b e f o r m e d . M L v v
E T a b e e rte e rte
H O D D 2 : S t r a t e l a ll 8 p o in ts x B b u t n o t A x D b u t n o t A
1 ro w to p ro w e c h o s e f p o in ts e re a re th e b o
a n d . L n in , th th e tto m
th e b o tto m
o n e fro m th a b e l th e fo u 6 w a y s (A B e th ir d v e r te n 6 × 4 = 2 r o w . ro w
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, a s s h o w n . L i s t a l l 1 8 t r i a n g l e s c o n t a i n i n g v e r t e x A , a l l 1 4 t r i a n g l e s c o n t a i n i n g , a l l 1 0 t r ia n g l e s c o n t a i n i n g v e r t e x C b u t n o t A o r B , a n d a l l 6 t r ia n g l e s c o n t a i n i n g o r B o r C . A t o t a l o f 4 8 t r ia n g le s c a n b e f o r m e d . B P ∆ A B Q
C P ∆ A C Q
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M E T T H O D D 1 : S t r a t e g y : G r o u p t h e n u m b e r s b y t e n s a n d l o o k f o r a p a t t e r n . In te rv a l
S a t is fa c to ry N u m b e r s
1 0 -1 9 2 0 -2 9 3 0 -3 9
1 2 , 1 3 , 1 4 , 1 5 , … , 1 9 2 3 , 2 4 , 2 5 , … , 2 9 3 4 , 3 5 , … , 3 9
M E T H O T h e re a d ig its (1 n u m b e r te n s d ig
8 7 6
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D 2 : S t r a t e g y : E l i m i n a t e a l l r e 9 0 tw o - d ig it c o u n tin g n u m 1 , 2 2 , e t c . ) . T h e n e l im in a t e s h a v e 2 u n e q u a l n o n z e ro d it. T h e r e a r e th e n 3 6 s u c h n
1 3 6
T h h e r e a r e 3 6 t w o - d i g i t n u m b e r s i n w h i c h t h e o n e s d i g i t i s g r e a a t e r t h a a n n t h e t e n s d i g i t .
( n u m b e r s )
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u n w a n t e d n u m b e r s .
b e rs . E th e 9 m ig its . In u m b e rs
I t e m s i n pp a r e n t h e s e s a r e n o o t r e q u i r e d .
3 6
N u m b e r o f n u m b e r s
8 0 -8 9
2 1
li m i n a t e t h e 9 n u m b e r s w i t h e q u a l u l t i p l e s o f 1 0 . T h e r e m a i n i n g 7 2 h a l f o f t h e m t h e o n e s d i g i t e x c e e d s t h e .
2 3 ( c h e c k s )
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M E T H O D 1 : S t r a t e g y : M a k e a t a b l e . N u m b e r o f C h e c k s A : F l a t F e e e = $ 7 7 . 5 0
1
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1 0
$ 7 .5 0 $ 7 .5 0
$ 3 .2 0 $ 5 .0 0
2 0 2 1
$ 7 .5 0 $ 7 .5 0
$ 7 .0 0 $ 7 .2 0
2 2 2 3
$ 7 .5 0 $ 7 . 5 0
$ 7 .4 0 $ 7 . . 6 0
T h e r e f o r e , t h e l e a a s s t t n n u u m b e e r o f c c h h e e c c k s s u c h t h a t P l a n n A c o s t s l e s s s t h a n P l a n B i s s 2 3 . M E T H O D 2 : S t r a t e g y : U s e A l g e b r a . L e t N r e p r e s e n t t h e n u m b e r o f c h e c k s f o r w h ic h p la n A is c h e a p e r .
2 3
5 6 ( p e r c e n t )
2 4
3 6 3
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M E T H O D 2 : S t r a t e g y : C o m p a r e a r e a s u s i n g t h e a r e a f o r m u l a . L e t T h e o r i g i n a l a r e a o f t h e t r i a n g l e i s g i v e n b y A = 2 1 b h . T h e c h a n g e d a r e a o f t h e t r i a n g l e i s g i v e n
b y 2 1 ( 1 . 2 0 b ) ( 1 . 3 0 h ) =
1 2
( 1 . 5 6 ) b h ,
w h i c h i n t u r n e q u a l s 1 . 5 6 t i m e s 21 b h . T h e r e f o r e t h e a r e a o f
t h e n e w t r ia n g le is 1 . 5 6 t im e s a s g r e a t a s th e a r e a o f t h e o r ig in a l . T h is is 5 6 % a r e a o f t h e o r ig i n a l. T h e in c r e a s e is 5 6 % .
g r e a t e r t h a n t h e
F O L L O W - U P S : S u p p o s e t h e b a s e o f a t r i a n g l e i s d e c r e a s e d b y 2 0 % , a n d i t s h e i g h t i s d e c r e a s e d b y 3 0 % . B y w h a t p e r c e n t i s t h e a r e a o f t h e t r i a n g l e d e c r e a s e d ? [ 4 4 ]
2 4
S t r a t e g y : D e t e r m i n e h o w m a n y o f t h e fi r s t 3 0 0 n u m b e r s c a n ’ t b e u s e d .
In e a c h o f th e s e ts a o n e s d ig it o f 9 , a e n d in g in 9 9 h a s b t h e o v e r a ll lis t 1 t h n u m b e rs , 3 0 1 th r o 3 2 9 , 3 3 9 , 3 4 9 ). A d s o a d d o n o n e m o
2 5
S tr a te g y: D r a w
1 th ro u g h 1 0 0 , 1 0 1 th ro u g h 2 0 0 , a n d 2 0 1 th ro u g h 3 0 0 , 1 0 n u m b e rs n d 1 0 n u m b e r s h a v e a t e n s d ig it o f 9 . H o w e v e r , in e a c h s e t t h e n u m e e n c o u n te d tw ic e , s o e a c h s e t h a s 1 9 n u m b e r s th a t c o n ta in a d ig it r o u g h 3 0 0 , 5 7 n u m b e r s m u s t b e e lim in a t e d . R e f ill th e lis t w it h th e n e u g h 3 5 7 . T h e r e a r e 5 n u m b e r s in t h is s e t th a t c a n ’t b e u s e d ( 3 0 9 , 3 1 d o n 5 m o r e n u m b e r s , 3 5 8 t h r o u g h 3 6 2 . O n e o f th e s e , 3 5 9 , c a n ’t b e 3 6 3 . r e n u m b e r . T h e 3 0 0 t h n u m b e r o n S a a r a ’ s l i s t i s s
p o s s i b l e p a t h s .
B y d r a w in g s o m e p a th s , y o u m a y s e e t w o t h in g s : ( 1 ) t h e p a th c a n t u r n e i t h e r l e f t o r r i g h t , a n d ( 2 ) e a c h h o r i z o n t a l s e g m e n t h a s a n o d d l e n g t h w h i le e a c h v e r t ic a l s e g m e n t h a s a n e v e n le n g t h . C o n p o in o r ig a n d
s id e t. N i n , b 3 a
r f i r s t t h e it h e r { 1 u t {1 , 3 , n d 5 u n
e h o r iz o u n it a n 5 , a n d 7 its t o th
n ta l s e g m d 3 u n it s u n its } c a e l e f t , t h e
e n ts th a t } n o r {1 , 3 n . B y tra v e h o r iz o n t a
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a t c a n n its a th
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s t a r t i n g a t t h e e r i g h t , e n d a t
h a v e b e r o f 9 . I n x t 5 7 9 , u s e d ,