Public Page11 19

7/21/2019 Public Page11 19

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Æ

x

102009 + 25

2 − 102009 − 25

2= 10x

9

200

220

12

C D AB

C

A

CB

DA

P

AC

BD Q QP

AB

27 + C 555+ C 1371+C

C

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ABC C BC = a CA = b

a ≥ b ABDE B C FG CAHI

ABC F I EH P F I

DG Q DG EH R

P QR

b

a

1

x − 1 + 1

x − 2 + 1

x − 6 + 1

x − 7 = x2 − 4x − 4

x

102009 + 25

2 − 102009 − 25

2= 10x

9

200

220

12

A C D

AB P CB DA Q

AC BD P Q AB

27 + C 555 + C

1371 + C

C

ABC C BC = a CA = b

a ≥ b

ABC

ABDE

B C FG

CAHI

P Q

R

F I EH F I DG DG EH

b

a P QR

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1

x − 1+

1

x − 2+

1

x − 6+

1

x − 7= x2 − 4x − 4

x2 + bx + c = 0

b c

5 4 c b

2

4

5

4

(x−5)(x−4) = x2−9x + 20 = 0

b

b = −9

2

4

(x − 2)(x − 4) = x2 − 6x + 8 = 0

c

c = 8

x2− 9x + 8 = 0

(x−1)(x−8) = 0

1

8

100

199

1

999

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1xy

1x1

1

9

11y

1

9

1xx

x

1

9

27

100

199

1 999

1

99

9

11 22 . . . 99

100 199 27

27 200

299

900 999

9 + 9 ·27 = 252 1 999

1 999

252

999 =

28

111

ABC

AB = AC

M

BC

P

BM

BC P BA K CA T P K + P T

P

P K + P T

P

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△ABC AB

AC M A ⊥ BC

P T ⊥ BC M A||P T

M A||P K △M BA

△P BK

B

P K

P B =

M A

M B P K =

P B · M A

M B

M A||P T △CP T

△CM A

P T

P C =

M A

M C

P T = P C

·M A

M C

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A

C B

K

P M

T

M B = M C = 1

2BC

P K + P T = P B · M A

M B +

P C · M A

M C =

(P B + P C ) · M A

M B

= BC · M A

M B = 2M A

P K + P T P

M A

ABC B BA BC

3 : 2

BD CA

10 CA

2x

3x

CB AB y

y + 10 CD DA

z

BD

2y + 10

CA

△ABC

(2x)2 + (3x)2 = (2y + 10)2

13x2 = 4y2 + 40y + 100

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C D A

B

z

y y + 10

2x3x

.

..

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..

.

..............................................

7/21/2019 Public Page11 19


△BDC

△BDA

y2 + z2 = 4x2

z2 + (y + 10)2 = 9x2

z 4x2−y2 = 9x2−(y+10)2

5x2 = (y + 10)2 − y2 = 20y + 100

x2 = 4y + 20

13x2 = 52y + 260

13x2 = 4y2 + 40y + 100

52y + 260 = 4y2 + 40y + 100

4y2 − 12y − 160 = 0

y2 − 3y − 40 = 0

(y + 5)(y − 8) = 0

y > 0

y = 8

CA = 2y + 10 = 26

10

10

1

3

6

5

x y

10

10

10 − x 10 − y

x+6y

3(10−x)+5(10− y) = 80− 3x−5y

x = 9 y = 4

33

33

33

32

64

x + 6y + (80 − 3x − 5y) = 80− 2x + y ≤ 64 2x − y ≥ 16

x y 10

(x, y)

(8, 0) (9, 0) (9, 1)

(9, 2) (10, 0) (10, 1) (10, 2) (10, 3) (10, 4)

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32

x + 6y ≤ 32

80 − 3x − 5y ≤ 32

(10, 4)

3x + 5y ≥ 48

33

f 1 = f 2 = 1

n

≥ 2

f n+1 = f n + f n−1 1 1 2 3 5 8

13 21 34 55 89 144 . . . 100

f 1 = f 2 = 1 f 3 = 2 f n = f n−1 + f n−2 f m

m 3

S n =

nk=1

f 3k

S n = 1

2

n

k=1

(f 3k+f 3k) = 1

2

n

k=1

(f 3k−2+f 3k−1)+f 3k = 1

2

3n

k=1

f k

f 1 = f 3 − f 2 f 2 = f 4 − f 3 f 3 = f 5 − f 4

f r−1 = f r+1 − f r

f r = f r+2 − f r+1

n

r

k=1

f k = f r+2 − f 2 = f r+2 − 1

S n = 1

2(f 3n+2 − 1)

S 100 = 1

2(f 302 − 1)

S 100

f m = 1√ 5

(αm − βm) α = 12

(1 + √ 5) β = 12

(1 + √ 5)

S 100 = 1

2√ 5

α302 − β302

− 1

2

7/21/2019 Public Page11 19


p

q

p

q

q < 100

3

7

3

7

0.428571 . . .

3

7

q = 99

3

7

42

99 = 0.424242 . . .

43

99 = 0.434343 . . .

3

7

42

99

3

7

42

99

0.004

3

7

14

42

98

1

−1

41

98

43

98

3

7

1

98

0.01

42

99

42

99

3

7

43

99

3

7

p

q

3

7

p

q− 3

7

=

7 p − 3q

7q

= |7 p − 3q|

7q

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0

p

q

3

7

1

p q

p q 1

1

7q q

1

7 p − 3q = ±1

q 7 p = 3q±1

q

p

q = 99

7 p = 3(99)±

1 = 297±

1

7

q = 98

7 p = 3(98) ± 1 = 294± 1

7

q = 97

7 p = 3(97) ± 1 = 291± 1

7

q = 96

7 p = 3(96) ± 1 = 288 ± 1

287

7

q = 96

p = 41

1

4196 − 3

7

= 1

7 · 96 = 1

672

1

2

27 · 99 = 2

693

2

1

672

3

7

p

q =

41

96

sin x

x − 1

6x3 +

1

120x5

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Public Page11 19