amc 2010 senior ans
TRANSCRIPT
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8/3/2019 Amc 2010 Senior Ans
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(11-12)1. 5 3
2010 20.10 1989.9 = D
2. 43 3 5
53 3
5
m
n
= = =
A3. P(x,y)
104 18
2
12 6 02
xx
y y
+= =
+ = =
P( 18 , 0)A
4. 87.5% 63 1% 63
87.5 100%100 63 72
87.5
=
87.5%= 78
63 78863 727 =
C5. 2 2
( )2
2 22 =24=12
1 1 1 1 1
2 2 2 2 16 = 6.25%
1
12
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8/3/2019 Amc 2010 Senior Ans
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64
4 1 6.2564 16
= = %
A
6.x8 10 24 28 23 9
7
x
x
+ + + + + +
= 102+x=7xx=17B
7. P Q 22 23 =49 QR 23 24 =916 4916
R 16 Q 9 P
49 4 5 = R 516
R 12R144 Q 981 P 636
81 36 45 = R 45 5144 16
=
E
8. 12
1 1 1 1 1 1
2 2 2 2 2 32 = 1 311
32 32 =
E
P Q R
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8/3/2019 Amc 2010 Senior Ans
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9. xy ( 2)( 2) 2 2 4x y xy x y = +
2 4 4 8
4 4 8 0( 4) 4( 2) 0
( 4) 4( 4) 8
( 4)( 4) 8
xy x y xy
xy x yx y y
x y y
x y
+ =
+ =
=
=
=
x y (i) 4 2x = x=6y=8 (ii) 4 1x = x=5y=12x+y=14 17 17
A10.0 1x< < 3 20 x x x< < < x x> x
3 2x x x x x < < < <
2x B
11. 3 2 2 22
7 7 7 (7 1) 77
7 7 7 (7 1) 7
x x x x xx
x x x x x
+ += = =
+ +
D12.RPS=xPQS=y PQ=PSPSQ=y PS=RSPRS=xPRQPRSPRQ= x+yQRP =QPSx+y= x+QPRQPR= yPRSPRQx=2yPRS 2x+y=180 5y=180y=36PSR=36
B
13. 3 4 5 3 4 10 10 7 14 22 2
a ba b a b a b a b
a b
+ = + = = =
2 2 2
2
2 63
2
a b b
ab b
+ = =
C14. 15 n=123456782
2 2
123456785 123456782 123456783 123456784 ( 3) ( 1)( 2)
3 3 2
2
n n n n
n n n n
= + + +
= +
=
A
y
x ( 2)( 2)x y
RSQ
P
x
x yy
yx+y
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8/3/2019 Amc 2010 Senior Ans
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15. 21 2357111317xyzx
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8/3/2019 Amc 2010 Senior Ans
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18.p 1p Snap(p1)=2(p1) Snap(p1)p=p2 2p Snap(p2) =3(p2)
E19. r OPR
2 2 2
22 2
2 ( 2 1)
OP r r OP r r
OQ r r r
= +
= =
= + = +
2r 2 2 2(( 2 1) ) ( (2 2 2 1)) (3 2 2)
4 4 4
r r r + + + += =
1 (3 2 2)
4
+=43 2 2+
E20. 2a b a b = +
2 2
2 2 2 3 4
2 2 3 4
4 3 2
2 2
( ) ( ) 2
( ) ( ) 2
2 2
2 0
( 2 1) 0
2 80, 1 2
2
a a a a a a a a
a a a a a a a a a a
a a a a a a
a a a
a a a
a
= + = +
= + + = + + +
+ = + + +
+ =
+ =
= =
a>0 2 1a =
C21. 2!=1!24!=3!46!=5!612!=11!12
2 2 2 2
2 6
3 2
1! 2! 3! 12! (1!) 2 (3!) 4 (5!) 6 (11!) 12
(1! 3! 5! 11!) 2 6!
(1! 3! 5! 11! 2 ) 6!
=
=
=
L L
L
L
m=6B
22. AC 26cm FADACDADEAFEEFCADC
P
Q
B
A
RO
r
r
A
DC
B
E
F
24
x
x
y
10
1610
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8/3/2019 Amc 2010 Senior Ans
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(AA) 26 52 202416 24 3 3
yy x y= = = = cm
x+y=24y=24xy
2=16
2+x
2=(24x)2
256+x2=57648x+x2 20
3x = cm
DAE=EAC=
tan10
x = 24 12tan2
10 5 = =
2
2 2
12 2tan
tan25 1 tan
205
1001
100
x
x
x x
= =
= =
2
2
1200 12 100
3 25 300 0
25 625 3600
6
25 65
6
x x
x x
x
=
+ =
+=
=
x>0 203
x = cm C
23. 25 1 1111
11110 1111 1110 0111 01110 01111 11101111 1111 1110 0111
E 1111011000010100111 1111001000011010111
A
DC
B
E
F
24
x
x
y
10
1610
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8/3/2019 Amc 2010 Senior Ans
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24. 161616 888 888PQ
2 2 2 2 2 2(8 2) 8 8 3PQ PR RQ= + = + =
8 3PQ = 8888 3 2 2(8 3) 192 196 14= < = 8 314 8 14 161616 888 8888 8 161616
16 888 14 9
B25.(1, 0)(1, 1)(0, 1)(2, 1)(1, 2)(3, 1)(0, 0)(3, 1)(1, 0)(2, 1)
PQ(0, 0)(3, 1)(1, 0)(2, 1)
1
10
PQ ABr PQ
OP OB= = = =
(3, 1) 110
(3, 2) RS1 1 2 1
2 2 2 13 13
RS AC RS
RT OC = = = =
110
C
P
R
Q
1
21
2
0 3
P
Q
B
A
1
21
2
0 3
C
R
S
T
A
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8/3/2019 Amc 2010 Senior Ans
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26. 11m n+ = 2 2 99m n+ = 2 2 2
3 3 2 2
( ) 2 121
2 22
11
( )( )
11(99 11)
968
m n m mn n
mn
mn
m n m n m mn n
+ = + + =
=
=
+ = + +=
=
968
27. 29 28
A B C D
E F G H I J
A=1 203H=2
1 B C D
E F G2 I J
E>BH=2E=B+8B+7 1 2B=0E=7 8 234I=3
1 0 C D 1 0 C D
7 F G 8 F G2 3 J 2 3 J
CF(C, F)(4, 7)(5, 9)(6, 9)DGJI=3
I=4
1 0 C D 1 0 C D
7 F G 8 F G2 4 J 2 4 J
(C, F)(3, 8)(3, 9)(C, F)=(3, 8)
1 0 3 5 1 0 3 5
7 8 6 7 8 9
2 4 9 2 4 6(C, F)=(3, 9)DGJ
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8/3/2019 Amc 2010 Senior Ans
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(C, F)(3, 7)(5, 9)DGJ 246
A B C D
E F G H I J
45EFG HIJ ABCD+ = (mod 9)E F G H I J A B C D+ + + + + + + + A B C D+ + + 9z y x = x
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8/3/2019 Amc 2010 Senior Ans
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2 240
6425
PRPT
PS= = = cm
P QR V
VPS
2 2
625a h+ =
PVQ 2 2 1600b h+ = 2 2 975b a = 2 225 ( )( ) 975ST a b b a b a = + = = ST=39 cm PT=64 cm
6429. 11 1 0( ) n nn nf x a x a x a x a= + + + +L 1 1 0(1) 6n nf a a a a= + + + + =L 6 1 1 1 0(7) 7 7 7 3438
n n
n nf a a a a
= + + + + =L 0 6 3438 7
0
1
2
3 4
3438 7 491 1 1
491 7 70 1 1
70 7 10 0
10 7 1 3 3, 1
a
a
a
a a
= + =
= + =
= =
= + = =
4 3( ) 3 1f x x x x= + + + (3) 81 81 3 1 166f = + + + = 166
30. 30 BYBYBY
a
Y
B
B
B
Y
Y
f
e
d
c
b
a
Y
B
B
B
Y
Y
R
Y
B
B
B
Y
Y
R
Y
B
B
B
Y
Y
R
f
e
d
c
b
a
f
e
d
c
b
a
f
e
d
c
ba
40
SRQ
P
T
b
h
b aaV
25 40
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8/3/2019 Amc 2010 Senior Ans
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(i) a
b BRB
c YRYRY
(ii) a
b BRB
Y
B
B
B
Y
Y
R
f
e
d
c
b
a
Y
B
B
B
Y
Y
R Rf
e
d
c
b
a
Y
B
B
B
Y
Y
R
f
e
d
c
b
a
Y
B
B
B
Y
Y
R Rf
e
d
c
b
a
R
Y
B
B
B
Y
Y
R
f
e
d
c
b
a
R
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8/3/2019 Amc 2010 Senior Ans
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a(
)
RBR
RYR
Y
B
B
B
Y
Y
R
R
R
R
R
R
Y
B
B
B
Y
Y
R R
R
R
R
RY
B
B
B
Y
Y
RR
R
RR
R
Y
B
B
B
Y
Y
R
R
R
R
R
R
B
B
B
Y
B
B
B
Y
Y
R
R
R
R
R
R
B
B
B
Y
Y
Y
Y
Y
Y
R
R
R
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8/3/2019 Amc 2010 Senior Ans
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BRB BYBYB
24 RYRYRYYBYBYB
RBRB
n 3 32
n
RBY RBY
3 32 2n n =
Y
B
B
B
Y
Y
R
R
R
R
R
R
B
B
B
Y
Y
Y
Y
Y
Y
R
R
R
B
B
B
B
B
B
Y
B
B
B
Y
Y
R
R
R
R
R
R
B
B
B
Y
Y
Y
Y
Y
Y
R
R
R
B
B
B
B
B
B
Y
Y
Y
R
R
R
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8/3/2019 Amc 2010 Senior Ans
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a RYRYRYb YBYBYBc RBRB RBY
3 2
2 2 3 23 3
3 32 23 2
3 2
2 2
a c n
a c na b n
a b n
b c nb c n
+=
+ =+
= + = + =+
=
4c n n n n= + = 4
nc =
73 4 12
n n nF a b c= + + = + = V n= 3
2
nE=
2F V E+ = + 7 3
212 2
n n
n+ = + 212n
= 24n = 24
24