Amc 2010 Senior Ans

8/3/2019 Amc 2010 Senior Ans

1/14

(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


2/14

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


3/14

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


4/14

15. 21 2357111317xyzx


5/14

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


6/14

(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


7/14

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


8/14

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


9/14

(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


10/14

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


11/14

(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


12/14

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


13/14

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


14/14

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

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Amc 2010 Senior Ans