p = q =1 w − m = mMw ( w − m ): m = p : qw > mp > 0 q > 0 AMBmw | MB | = mmM − m | AB | = wwMB pqMAB AMBqp + qAMpMBq AMMBpqABp | AM || MB | | AM | : | MB | = p : qpq MAB

AB M

|AM| : |MB| = p : q p q

|AM| |MB|

AM MB

p q

AB p + q

AM p MB q

A M B

q p

p q

M AB

A M B

m

w −m |AB| = w w

MB |MB| = m M m

(w −m) : m = p : q w > m p > 0 q > 0

p = q = 1 w −m = m

M w

∗

w1 w2 m w1 > m > w2 p q w1 w2 m

(w1 −m) : (m−w2) = p : q w1 > m > w2 p > 0 q > 0

w2 = 0 ^∗∗

p q m

p : q p : q ∈ {w1 : w1,w1 : m,w1 : w2,m :w1,m : w2,w2 : w1,w2 : m}

p : q = w1 :w1,p :q = w1 : m p : q = w1 : w2.

(w1−m) : (m−w2) = w1 : w1 w1−m = m−w2 =⇒m = ¹₂(w1+w2) ma w1 w2 1

2(w1 +w2)

(w1 − m) : (m − w2) = w1 : m m(w1 − m) = w1(m − w2) =⇒ m² = w1w2 =⇒ m = √w1w2

mg w1 w2 √w1w2

(w1 − m) : (m − w2) = w1 : w2 w2(w1 − m) = w1(m − w2) =⇒ m = _w^2w_1+w^1w2₂ = 1 ²

w1+_w¹

2

mh w1 w2 2w1w2 : (w1 + w2)

= 2 :

w11 + _w¹₂

ma mg mh

∗∗

AB BC

|AB| |BC| AC

H M

D H

BD ⊥ AB E

DM BE ⊥ DM _{A M B C}

E H D

|DM| = |AM| = ¹₂

|AB|+|BC|

|DM|

|AB| |BC|

ACD ^∗∗∗

|BD|² = |AB| · |BC| |BD| =

|AB| · |BC|

|BD| |AB| |BC|

DMB DEB

|BE|² = |DE|

|DM| − |DE|

= |DE||DM| − |DE|²

|BE|² = |BD|² − |DE|²

|DE|

12

|AB|+|BC|

− |DE|² = |DE||DM| − |DE|² = |BD|²− |DE|²

= |AB||BC| − |DE|²

|DE| = _|AB|+|BC^2|AB||BC|_| |DE|

|AB| |BC|

w1 w2

n w1 w2 wn n ≥ 2

wi

i 1 2 3 4 5

wi 3 4 6 5 4 ⇒ma = 4,4

ma−wi wi

ma 1,4 0,4 −1,6 −0,6 0,4

m_a ma = ¹₅(3 + 4 + 6 + 5 + 4) = 4,4 ma

wi

ma 0

w1 w2 wn n ≥ 2 ¹_n(w1 + w2 + ... + wn) ma

e i ai wi

i

i 0 1 2 3 4 5

ai 2000 2240 2419,20 2878,85 3022,79 3173,93

wi 1,12 1,08 1,19 1,05 1,05

ai i = 1, 2, ..., 5 a0 = 2000 a1 = w1a0 a2 = w2a1 = w1w2a0 a5 = w5a4 = w1w2w3w4w5a0

mg a^∗_i = ai i = 1, 2, 3, 4

a^∗₁ = mga0 a^∗₂ = mga^∗₁ = m²_ga0 a^∗₅ = m⁵_ga0

mg a^∗₅ = a5 mg

m⁵_ga0 = w1w2·...·w5a0 mg = √⁵w1w2 ·...·w5

mg 1,1

w1 w2 wn n ≥ 2 √ⁿ w1w2 ·...·wn

mg

200 ^km_h

i 1 2 3 4 5

vi i ^km_h 185,4 198,7 201,1 199,2 218,4

v v

v = .

L L

185,4 L

198,7 L

218,4

v

v = 5L

185,4L + _198,7^L + ... + _218,4^L = 5

185,41 + _198,7¹ + ... + _218,4¹ ≈ 200,016.

mh

w1 w2 wn n ≥ 2 1 ⁿ w1+_w¹

2+...+_wn¹

mh

11 + ¹₂ + ¹₃ + ...

n (n+2)

2

11

n

+

=

2

n + n + 2

1 n + 1

1+¹₂+¹₃+¹₄+...

∗∗∗∗

{1, 2, 3, 4, 5, 6} 3

2 6 4 3 6

11 + ¹₂ + ¹₃ + ¹₄ + ...

∗

≈ 14,392 1,51·10⁴³

> 100

12 + ¹₃ > 2· ¹₄ = ¹₂ ¹₄ + ¹₅ + ¹₆ + ¹₇ > 4· ¹₈ = ¹₂

18 + ¹₉ + ... + ₁₅¹ > 8· ₁₆¹ = ¹₂

1 1+

„1 2 + 1

3

« +

„1 4 + 1

5 + 1 6 + 1

7

« +

„1 8 + 1

9 + ... + 1 15

«

+ ...>1 +1 2+ 1

2+1 2+ ...

11² + ₂¹2 +

31² + ₄¹2 + ...

∗

11² + ₂¹2 + ₃¹2 + ₄¹2 + ...

1 1² +

„ 1 2² + 1

3²

« +

„ 1 4² + 1

5² + 1 6² + 1

7²

« +

„ 1 8² + 1

9² + ... + 1 15²

« + ...

<1 + 2· 1

2² + 4· 1

4² + 8· 1

8² + ... = 1 + 1 2 + 1

4 + 1 8 + ...

2 < 2

∗∗

P(x) = x· 1− x

π 1 + x

π 1− x

2π 1 + x

2π 1− x

3π 1 + x 3π

·...

sin 0 ±π ±2π

±3π

x sinx = x ·

1− _π^x22 1− _4π^x22 1− _9π^x22

·...

sinx

x³ x⁵ x⁷

sinx

x x³

x sinx = x¹−x³

π1² + _4π¹2 + _9π¹2 + ...

+x⁵(...)−...

x¹ − x³

π1² + _4π¹2 + _9π¹2 + ...

+ x⁵(...) − ...

= x¹− ^x₆³ + ₁₂₀^x5 −...

x³

1 1² + 1

2² + 1

3² + ... = π² 6 .

ζ(s)

ζ(s) = ₁¹s + ₂¹s + ₃¹s + ... s ≥ 2

ζ(4) = ₁¹4 + ₂¹4 + ₃¹4 + ... = ₉₀^π

ζ(s) s ≥ 6

ζ(s) p1 = 2

p2 = 3 p3 = 5 ζ(s) =

1− _p¹s 1

₋₁

1− _p¹s 2

₋₁

1− _p¹s 3

₋₁

·... s > 1 s > 1

n 2n n > 1

s

ζ(s)

π(x) < x

• ≥ 4

•

a ¹₁ + ¹₂ + ¹₃ + ¹₄ + ... = a

b c

11 + ¹₃ + ¹₅ + ¹₇ + ... = b ¹₂ + ¹₄ + ¹₆ + ¹₈ + ... =c ^∗∗∗

a = b+c

a b c b c

a = b+c > 2c = 2·₁

2 + ¹₄ + ¹₆ + ¹₈ + ...

= ¹₁ + ¹₂ + ¹₃ + ¹₄ + ... = a

a a > a

11 + ¹₂ + ¹₃ + ¹₄ + ... = ∞

1 + ¹ + ¹ + ¹ + ... = ∞ ¹ + ¹ + ¹ + ¹ + ... = ∞

e

•

•

•

k^{2 ∗} k

1 n

n n = 2008

s ≤ n

1 2 3 s

s s

s

s s

s g u

g g g 0

g u u g u 0

u u g −2

1 2 3 s

1 2 3 s s = 2007 s = 2008

1004 s s = 2005

s = 2006 1003 s

s = 4m− 1 s = 4m

s = 4m+ 1 s = 4m+ 2 m s ≤ n

∗

n n! + 1 n! 1·2·3·...·n

n

n < 10^t t

n! +m m > 1 n m n! +m m ≥ 1

n = 1 2 4 nⁿ + 1

n nⁿ + 1

∗ †

n ≤ 100 nⁿ + m m 1 20

n ≤ 100 m ≤ 20 nⁿ +m

m 1 2 3 4 6 7 9 10

n 1, 2, 4 1, 3 2 1, 3, 7, 43 1 2, 4, 6, 32 2 1, 3, 7, 9, 39

m 12 13 14 15 16 18 19 20

n 1, 5 2, 4, 16, 42 3, 23 2, 4 1, 3, 13 1, 19, 49 2, 10, 30 3, 53

m 100

1≤ n ≤ 200 −100 ≤ m ≤ 100 n ≤ 15 m ≤ 20

n ≤ 12 m ≤ 20

6·8 = 48

5·5+4·6 = 25+24 = 49

5·6 + 4·5 = 30 + 20 = 50

10 cm 5·√

3 cm 5+8·5·√

3+5 = 10+40·√ 3≈ 79,28 < 80

10 : 4 = 2,5 3· 4 = 12

90^◦

1

2

24−21 = 3

x y z x ≤ y ≤ z

x +y + z = x · y ·z xyz

yz1 + _xz¹ + _xy¹ = 1 x ≤ z _xy¹ ≥ _yz¹ y ≤ z

xy1 ≥ _xz¹ 1 = 1

yz + 1

xz + 1

xy ≤ 3 xy.

1 ≤ _xy³ y {1, 2, 3} x = 1

x ≤ y x ≤ ³_y

(x,y)

y = 2 x = 1 z = 3 x = 1 y = 2 z = 3

↑

↓

x ^↑ y

↓ 99 = 1 + 13x −9y 9(y + 11) = 1 + 13x

1 + 13x 9

y ≥ 0 13x ≥ 98 x ≥ 8

x = 11 y = 5 99 = 1 + 13·11−9·5

0 3 7

1 4 8

2 5 1

2 5 9

3 0 4

3 6 2

4 1 5

4 7 3

5 2 6

5 8 4

6 3 7

6 9 5

7 4 8

8 5 1

8 5 9

9 6 2

10 − 3 = 7 16·7 = 112

(2007 + 2007) + (2007−2007) + (2007·2007) + (2007 : 2007)

2008 = 2008

x = −1,x = 0

T(x) := (x+x) + (x−x) + (x·x) + (x : x) = 2x+ 0 +x²+ 1 = (x+ 1)².

T(x)

x+1 = x + 1

x = 2007

x −1 = x = 0

0 9

1·10³+ 9·10² + 5·10¹+ 8·10⁰ 1958 10

0 7

8

S₁ = 1−2 + 3−4 + 5−6 + 7−... + 2007−2008 S₂ = 1 + 2−3 + 4−5 + 6−7 + ...−2007 + 2008

e e

e

499 500 501 502 502 503 504

505

S S

p = 5 p⁴ − 1 10

0

1 + 2007²⁰⁰⁹ 2008

Z = 2009²⁰⁰⁸ −2007²⁰⁰⁸ 16

Z

2 251

7571

141%

200%

2, 4, 6, 8, ... , 50 1

n = 0 n = 1 n = 2

2008 √

2011·2009 + 1 √

2009·2007 + 1 √

2007·2005 + 1

(a+ 1)·(a−1) + 1 = √

a²−1 + 1 = √

a² = a

√2011·2009 + 1 = 2010 √

2009·2007 + 1 = 2008

√2007·2005 + 1 = 2006

2010+2008+2006

3 = ⁶⁰²⁴₃ = 2008

12 1

3 1

4

n sn = ⁿ

i=1 1i

(sn)

k n 2^k−1 < n ≤ 2^{k 1}_n ≥ ₂¹k

2^k−1 2^k

s₂^k =

2^k i=1

1

i = 1 +

k j=1

⎛

⎝ ²

j

i=2^j−1+1

1 i

⎞

⎠ ≥ 1 +

k j=1

⎛

⎝ ²

j

i=2^j−1+1

1 2^j

⎞

⎠

= 1 +

k j=1

2^j−1 1

2^j = 1 +

k j=1

1

2 = 1 + k 2. (sn)

123671

x²

yz − y²

xz − z²

xy = 3 x²z

y − 2z y = 2 x,y,z ≥ 1

x > y

2y ≥ x² −2

x²

yz = y²

xz + z² xy + 3 z²

xy > 0 x²

yz > y²

xz x³ > y³ x > y y 2y = x²z −2z = (x²−2)z ≥ x²−2 z ≥ 1 x²−2 > 0

2x > 2y ≥ x²− 2 2x > x² −2 0> x² −2x + 1−3

3> (x −1)²

x ≥ 1 x = 1 x = 2

x = 1 −z

y = 2

x = 2 2z

y = 2 y = z

4 y² − y

2 − y

2 = 3 y³ + 3y² = 4 y = 1

x = 2,y = 1,z = 1

n

n = 2007

n > 3

n = 4 n = 6 n = 8 n > 8

•

•

•

v = ^s_t s = tv s = 2,5·(vB +vF) s = 3,75·(vB − vF)

vf

2,5·(vB +vF) = 3,75·(vB −vF)

⇐⇒ 2,5vB + 2,5vF = 3,75vB −3,75vF

⇐⇒ 6,25vF = 1,25vB

⇐⇒ vF = 1

5vB

s = 2,5(vB + ¹₅vB) = 3vB

t = _v^s_B = ^3v_vB^B = 3 [Stunden]

2

a0 b0 a0 : b0 = 1 : √ 2

an bn an+1 = ^b₂ⁿ

bn+1 = an

an : bn = 1 : √ n 2

a0 b0 c0 a0b0c0 = 1 m³ a0 < b0 < c0

an+1 = ^c₂ⁿ bn+1 = an cn+1 = bn an : bn : cn = a0 : b0 : c0 n

a0 : b0 : c0 = ^c₂⁰ : a0 : b0 b0

a0 = ^2a_c0⁰ _b^c0₀ = ^b_a⁰₀ c0 = ^b_a⁰₀²

b0

a0 = ^2a_c0⁰ = ^2a_b2²⁰ 0

b0

a0 = √³

2 _b^c0₀ = √³ 2

a0b0c0 = 1 1 = a0b0c0 = a0 · ^b_a⁰₀a0 · ^c_b⁰₀ ^b_a⁰₀a0 = 2a³₀ a0 = ³

12 = √3¹

2 b0 = a0√³

2 = 1 c0 = √³ 2

x1,x2,x3...

x1 = x2 = 1 x2n+1x2n−1−x_2n² = 1 x_2n+1² −x2nx2n+2 = 1

xn n = 3, 4, 5, 6, 7, 8

2, 3, 5, 8, 13, 21

∗

x_2n+1² −x_2nx_2n+2 ^(∗)= x_2n+1x_2n−1− x_2n^{2 (∗∗)}= x_2n−1² −x2n−2x2n = ... = x3x1−x₂² = 1

(∗) x_2n+1² − x2nx2n+2 − (x2n+1x2n−1 − x_2n² ) = x2n+1(x2n+1 − x2n−1) − x2n(x2n+2 −x2n) ^(•)= x2n+1x2n −x2nx2n+1 = 0

(∗∗) x2n+1x2n−1 − x_2n² − (x_2n−1² − x2n−2x2n) = x2n−1(x2n+1 − x2n−1) − x2n(x2n −x2n−2) ^(•)= x2n−1x2n −x2nx2n−1 = 0

(•)= xk+2 =

xk+1 +xk

12ⁿ − 1 11

13ⁿ−1 12

14ⁿ−1 13 n = 1, 2, 3, ...

n ∈ a

aⁿ −1 = (a− 1)(aⁿ⁻¹ +aⁿ⁻²+ ... +a¹ + 1)

a = 1 a = 2 a = 12 a = 13 a = 14

n ∈ aⁿ−1 a−1

(x + 1)ⁿ −1 (x+1)ⁿ−1 =

n−1 k=0

n k

x^n−k+1−1 =

n−1 k=0

n k

x^n−k = x·

n−1 k=0

n k

x^n−k−1

x ≡ m mod n

x :n m

a ≡ 1 mod (a− 1)

=⇒ aⁿ ≡ 1ⁿ ≡ 1 mod (a− 1)

=⇒ aⁿ− 1≡ 1ⁿ −1≡ 1−1≡ 0 mod (a−1)

A(1) 1

A(n+ 1) A(n)

n = 1 a¹−1 = a−1 = 1·(a−1) n ⇒ n+ 1 n

aⁿ⁺¹ −1 =aⁿ⁺¹ −a+ (a−1) = a·(aⁿ −1) + (a−1) aⁿ − 1 a − 1

=⇒ aⁿ⁺¹ −1 a −1

n

aⁿ −1 = (a²−1 + 1)·aⁿ⁻²−1 = ((a− 1)(a+ 1) + 1)·aⁿ⁻²− 1

= (a− 1)(a + 1)aⁿ⁻²+aⁿ⁻²−1

a² − 1

a¹−1 = a−1 a−1

a = 12 n = 2

12²−1 = 12·12−1 = 12 + ... + 12

12−

−1 = 11+12 + ... + 12

11−

= 11+11·12, 1

a

12ⁿ + 1 13 n ∈

n = 2k + 1 k = 0, 1, 2, ...

aⁿ + 1 =a^2k+1+ 1 = ((a+ 1)−1)^2k+1 + 1 = (a+ 1)(...)−1 + 1

= (a+ 1)(...), (a+ 1)|aⁿ + 1

12ⁿ+ 1

13 n ∈

1n n > 1 1 : n

L

1n

371 = 1 : 37 = 0,027 027 027... = 0,027 L

371

= 3

m ¹_n

p1 p L

1n

= m L

p1

= m m = 1 2 3

p = 2 L

1p

= p− 1 L

p1

(1, 10²) (1, 10³) (1, 10⁴)

∗

m n L

1n

= m

13 = 0,3 ₃₃¹ = 0,03 ₃₃₃¹ = 0,003 L

_m 1

i=13·10ⁱ⁻¹

= m m ∈

19 = 0,1 ₉₉¹ = 0,01 ₉₉₉¹ = 0,001 L

_m 1

i=19·10ⁱ⁻¹

= m m ∈

11 = 0,9 ₁₁¹ = 0,09 ₁₁₁¹ = 0,009 L

_m 1

i=11·10ⁱ⁻¹

= m m ∈ ,

m = 1

m p

L

p1

= m

L₁

n

L

p1

p n

19 7 17

100 7 17 19

23 29 47 59 61 97

1 000 10 000

1 000 ₁₆₈⁶⁰ ≈ 0,36 10 000

1229467 ≈ 0,38

0,40 0,35

(x,y) ¹_x + ¹_y = ¹₃

\{0, 3}

1x + _y¹ = ¹₃ 3xy x

x 1 x+1

y = 1

3 ⇐⇒ 3y+3x = xy ⇐⇒3y = x(y−3)⇐⇒ x = 3y

y −3 = 3+ 9 y − 3.

x (y − 3) 9 (y −3) ∈

{−9,−3,−1, 1, 3, 9} y ∈ {−6, 0, 2, 4, 6, 12} y = 0 y

y ∈ {−6, 2, 4, 6, 12} x ∈ {2,−6, 12, 6, 4}

(2,−6) (−6, 2) (12, 4) (6, 6) (4, 12)

n

n = 4 n = 7 n = 10

n = 4 n = 7 n = 10 n = 5

n = 6

2 : 3

n = 1

n = 2 n = 3

n = 6 n = 8

3 : 4 n = 5 n = 7 n = 8 n > 8

n > 8 3 n = 6 n = 7 n = 8

n > 8

n = 4 n > 5 n

A P B

Q

C D

α1

α2

ABCD |AB| > |CD|

AB D

AB P

AC Q AC

A α1 α2

α1 = 2α2 QC AD

|QC| = 2|AD| α1

α2

M

QDC = 90^◦ C D Q

M r =

|MC| _A ^α2 _B

α1

P Q

C D

M

β2 = α2

β1 = β2

CDM CD

δ = 180^◦−β1−β2

χ = 180^◦−δ

α2

A B

α1

P Q

C D

M β1 β2

χ δ

δ = 180^◦−2β2

χ = 180^◦−δ = 2β2

α1 = 2α2

=⇒ |AD| = |MD| = |MC|= ¹₂|QC|

=⇒ QC AD

|AD| = ¹₂|QC|

=⇒ |AD| = |MD|

=⇒ AMD AM

=⇒ α1 = χ

=⇒ α1 = 2α2

n

a b a

a = 0 a

b a 10−1 = 9

b = a 9· 9 = 81

aaabb

a ∗ ∗ ... ∗ ∗ n 1. 2. 3. (n−1). n.

a a b

2ⁿ⁻¹ a b

n a (2ⁿ⁻¹−1)n

a b a b

81·(2ⁿ⁻¹−1) n n = 5 81·(2⁵⁻¹−1) = 1215n

n = 6 81·(2⁶⁻¹−1) = 2511 n = 7 81·(2⁷⁻¹−1) = 5103

∗

0,514

3 : 5 e e

e 3

8

∗ †

38 ·8e+ ⁵₈ ·0e = 3e

38 ·0e+ ⁵₈ ·8e = 5e

16

N =

6

N1

43·9N ≈ 8·10⁻⁸ 43

N A

A

∗∗

K R

14 r K

πr²

πR² = _r

R

₂

A P(A)

0 1

1 0

A B

C A B P(C) =P(A) +P(B)

∗∗∗

Ω

Ω A A

P(A) P(∅) = 0 P(Ω) = 1

A1 A2 A3 Ai ∩Aj = ∅ i = j

P(A1 ∪A2∪A3 ∪...) =P(A1) +P(A2) +P(A3) + ...

σ

Ω Ω

P

A

∅ ∈A Ω ∈ A A1∪A2∪A3∪...∈ A A1,A2,A3, ... A σσ

(Ω,A,P)

Ω σ A

A P

∗∗∗

e