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8/9/2019 Matematika Za 1 Razred Gimnazije Banja Luka

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Маака Iа а ај

Рак Вкћ, .

2010-11. .. .


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Садржај

I .............................................................................................................................. 5 1. .......................................................................................................... 5

1.1 .............................................................................................................. 6 1.2 ............................................................................................................. 6

1.3 .............................................................................................................. 8 1.4 ...................................................................................... 10 1.5 .......................................................................................................... 11

2. .......................................................................................................................... 13 2.1 .......................................................................................... 13 2.2 ............................................................................................................ 16 2.3 ................................................................................................................. 17

3. .................................................................................................................... 18 3.1 ............................................................................................................ 18

4. .................................................................................................................. 21

4.1 ............................................................................................................ 21 4.2 ........................................................................................... 24 4.3 ..................................................................................... 25

4.4 ................................................................................................................. 26 II .......................................................................................................................... 28

5. ......................................................................................................... 28

5.1 ................................................................................... 28 5.2 .......................................................................................................... 30 5.3 ..................................................................................................... 34

6. .................................................................................................................. 36

6.1. ....................................................................................................... 36

6.2. ....................................................................................................... 38 6.3. ....................................................................................... 40 6.4. ................................................................................................. 44

7. ....................................................................................................................... 46 7.1. ...................................................................................... 46 7.2. ................................................................................... 48 7.3. ......................................................................................... 49

8. ....................................................................................................................... 51 III .................................................................................................................. 55

λ. ................................................................................................................... 56

λ.1. , ............................................................................. 57

λ.2. .......................................................................................... 59 λ.3. .................................................................................... 60

λ.4. ........................................................................................ 61 λ.5. .................................................................................. 62

10. .................................................................................................................... 62

10.1. .................................................................... 63 10.2. ....................................................................... 64


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10.3. ...................................................................................... 66 10.4. .............................................................................................................. 67

11. .......................................................................................................... 67 11.1. ................................................................................. 68

11.2. .......................................................................... 70

11.3. ....................................................................................... 73

11.4. .................................................................................. 75 11.5. ...................................................................................... 77

IV ................................................................................................................... 80 12. ......................................................................................................... 80

12.1. .................................................................................... 81 12.2. ........................................................................ 83

13. ............................................................................................................. 86 13.1. ............................................................................................... 87 13.2. ..................................................................................... 89 13.3. ........................................................................................................... 92

13.4. .................................................................................................... 93

14. ................................................................................................................ 93

14.1. ........................................................................................... 94 14.2. ...................................................................................................... 97 14.3. ................................................................................................ 98

15. ............................................................................................ 102 15.1. ....................................................................................................... 102 15.2. ............................................................................................................ 105

V ....................................................................................................................... 107 16. Ј ............................................................................................................ 108

16.1. ..................................................................................... 109 16.2. ............................................................................................... 113 16.3. ................................................................................. 118

17. ....................................................................................................... 121 17.1 ................................................................................................... 121 17.2. .................................................................................. 123 17.3. ............................................................................. 126

VI ..................................................................................................................... 130 18. ................................................................................................. 130

18.1. ........................................................................................ 130 18.2. ................................................................................. 133 18.3. Ј ............................................................................................ 137

1λ. ............................................................................................................... 139

1λ.1. ....................................................................................... 139

1λ.2. ........................................................................................ 145 1λ.3. Ј .......................................................................... 148

VII ................................................................................................................ 153 20. .......................................................................... 154

20.1. Ј ..................................................................................... 154 20.2. ..................................................................... 156


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20.3. .................................................................................................... 159 21. .................................................................................... 162

21.1. ................................................................................. 162 21.2. ....................................................................................... 163

21.3. ........................................................................................ 164

VIII ..................................................................................................... 170

22. .............................................................................................. 170 22.1. ................................................ 170 22.2. ................................................................... 173 22.3. ...................................................................................................... 175

23. .......................................................................................... 178 23.1. .......................................................................................... 179 23.2. ....................................................................................... 181

24. ........................................................................................... 184 24.1. ............................................................................... 184 24.2. ................................................................................................. 188


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I

1. А , . , , , .

μ , , . , .

1. (. )ν

2. (. ).

, . , . , „“ , . ,

, , . , , .ђ, .

, , , . , . ђ

, . ђ , , ђ .

, . , , . „ “ , , . .


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() = (. „“), () = („-“). 1 0. .

1.1

1.1.1 „2+3ο5“ . „2+3ο6“ .

1.1.2 „ 180њ“ .

1.1.3 „ “ .

1.1.4 „ “ , , . , 4 6 , .

1.1.5 „Ј “ . , , . , , . , , !

1.2

, („“) („“). , , , , ,

.

( „-“) . , .

A, B A B („ “, „ “) . , . , „ “ „ “ „ “.

A, B A B („ “, „ “) . , . , „ “ „ “ „ “.

, :


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, μ ,

.

A, B A B („ B„) . . , „ “ .

A, B A B (. „ B“) , B . , .

, „ “ „ “ „ “.

A, B A B (. „ B“) , . (),

().

, μ

, , . . .

, , .

1.2.1. :

) (a b) a b; ) (a b) a b.

: )

a b a b (a b) a b a b T T T TT T T T


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T T T T T T T T T

1.3

1.3.1 μ

() (2,3456 – 1,2345 = 10/9); (b)

5

3

7

5

13

11

19

17;

(c)

12

1

4

1

3

1

6

1

3

1

2

1; (d)

11

9

15

13

15

13

19

17.

1.3.2 μ

(a)

2

22

n

m

n

mnm ; (b)

1

1

2

2

1

1

n

m

n

m

n

m

n

m.

1.3.3 (a) ; (b) ; (c) ; , .

, μ

(a) A & B = (A B) (A B);(b) A B = B A;(c) A B = (A B) (A B).

1.3.4 :

(i) A (B A);

(ii) ((A B) A) A;(iii) (A B) ((B C) (A C)).

(i), .

, (. B A). , (. = T) ( ). ,

, B (B = T) ( = ). , , , . . .

1.3.5 :(iv) A B A;(v) A B B;(vi) (A B) ((A C) (A B C)).

(vi) μ

B C A B A C A B C (A C) (A B C)


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, , (vi) .

1.3.6 :

(vii) A A B;(viii) B A B;(ix) (A C) ((B C) (A B C)).

1.3.7 :

(x) (A B) (A B);(xi) (A B) (B A);(xii) (A B) ((B A) (A B)).

1.3.8 :

(xiii) A (A B);(xiv) (A B) ((A B) A);(xv) (A B) ((A B) B).

1.3.9 , . .

, μ

abba a, b „“;

cbacba )()( a, b, c ;

)()()( cabacba a, b, c

.


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10

џ 1854. „ “

, , ,, , / .

.

1.4

.

, , , . , .

:1.

ν

2. A B (Modus ponens) Bν

3. ()

. , , |- A,

„ “.

ђ . . , . .

:

1. (Modus ponens). B ,

B;2.

(Demonstratio per enumerationem – ). C B C , (A B) C;

3.

(Deductio ad absurdum – ђ ). B B , A.

. , (1) B

. , B = ,


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. , B = . , .

:

1.

A |- B A ( B );2. A B |- (B C) (A C) ( AB ... );3.

A B, B C |- B (A C) ( AB BC ... );4.

|- (A B) (B A) (- )ν 5. |- (A (A B)) (A B) (- );6. |- ((A B) A) ((A B) B);7. |- (((B C) C) (A C)) (B (A C));8. |- (A (B C)) (B (A C));9.

|- (A B) (A) (B) ( );10.

|- (A B) (A) (B).

. (1) (i), . (2) (iii), . (3) (ii), . ђ, , , ђ ђ , .

, () (). , „ “ μ

(xN)(yN)(x < y), , x = 1.

1.4.1. ς

) (xR)(yR)(x < y); ) (xR)(yR)(x < y);) (aZ)(bZ)(a + b = 0); ) (aZ)(bZ)(a + b = 0); R , Z .

1.5

Ј . , , . , ,

. , .

.

A B ()μ


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.

A B, μ

A B μ

A B μ

1.5.1 : A (B C), A (B C).

1.5.2 , :

(1) A (B C) = (A B) ( C); (2) A (B C) = (A B) (A C).

А B

А

B

B

A

B


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(Claude Elwood Shannon, 1916. – 2001), .

21- - . ђ 1939.

1λ48. ,

.

2. , . . , . .

џ (1845 – 1λ18), ђ .

1874. 10 . .

ђ, , . . .

, , .

2.1

S , . - , .


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, B -- , Њ. :

S = {, , ј}, = {, }, B = {, , }.

, |S| = 3, |E| = 2, |B|= 3. , ђ . „“ B,

B. ђ, „“ S B, „“ S E, „“ E, B. μ

S, B; S, E;

E, B.

. ,

P = S B

S B x P x S x B.

P = {x| x S x B},

. P S B. P = {}. P E , . P E =.

. ,

U = S B

S B x U x S x B.

U = {x| x S x B},

. U S B. U = {, , ј, , }.

() μ


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.

S U

S U S U. x S x U.

2.1.1 А B . A = B A B B A.

2.1.2 А B . :1.

A A B B A B;2. A B A A B B.

. ,

R = S \ B

S B x R x S x B.

R = {x| x S x B},

. R S B. R = {, ј}.

S U U

S. U \ S. U S' = {x| x S}, . S S.

. S,

(S) {A| A S}.

(S) = {, {}, {}, {ј}, {, }, {, ј}, {,ј}, S}.


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(S) 23 .

A, B ђ .,

A B = {(x, y)| x A y B}.

2.2

2.2.1 A (B C) = (A B) (A C) A, B, C .

2.2.2 30 ђ μ , . 1λ ђ, 14, 13 ђ . 6 , 5, 7. 2 . ς

P, L, I.

2 . 4, 3, 5 , , .

, 8, 5, 3. , 30.

2.2.3 μ

A, B, C (A B)’ ο A’ B’ (A B)’ ο A’ B’.

. , . , .

A B A B (A B)’ A’ B’ A’ B’


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2.2.4 ,

μ A (B C) = (A B) (A C), .

. ђ ( x, y). ( x, y) A (B C) x A y (B C) x A ( y B y C). ,

(x A y B) (x A y C) (( x, y) A B) (( x, y) A C) ( x, y) (A B) (A C). , , .

2.3

2.3.1 A, B, C μ A C = {a, b, c, d, e, f, g},B C = {d, e, f, g}, A C = {e}, B C = {e, f}, A \ B = {a, b, c}, B \ C = {d}.

2.3.2 M, N, P μ M N = {1, 2, 3, 4, 5, 6},N P = {2, 4, 5, 6, 7, 8}, P M = {1, 3, 5, 6, 7, 8}, M N = {5, 6}, N \ P = {7, 8}.

2.3.3 B μ

(a) A B A B = A; (b) A B A B = B;(c) A (A B) = A; (d) A (A B) = A.

2.3.4 A B S.

() A B = |A B| = |A| + |B|;

(b) |A B| = |A| + |B| - |A B|;(c) |A \ B| = |A| - |A B|.

2.3.5 S (a) 10; (b) 12; ,

(S)?

2.3.6 () ν (b) .

2.3.7 A, B, C

(a) A \ (B C) = (A \ B) (A \ C); (b) A \ (B C) = (A \ B) (A \ C).

2.3.8 , (a) A (B \ C) = (A B) \ (A C);(b) A \ (B C) = (A \ B) C;(c) ( \ B)’ ο (B \ A)’ν

(d) A (B C) = (A B) (A C);(e) A (B \ C) = (A B) \ (A C).

2.3.9 B


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A B = (A \ B) (B \ A).(a) A = {a, b, c} B = {b, c, d, e}, A B.(b) .

(c) A B = (A B) \ (A B).

2.3 (1) A = {a, b, c, d, e}, B = {d, e, f}, C = {e, f, g}; (2) M = {1, 3, 5, 6}, N= {2, 4, 5, 6}, P = {1, 3, 5, 6, 7, 8}. (5a) 2

10 = 1024.

3. ј

, . ђ ђ .

, B A B. , .

A B:

= {(b,1), (b,2), (c,2)},

. (b,1), (b,2), (c,2) , b1, b2, c3. , -, , B, - (

). . (b,1) (1,b).

.

3.1

3.1.1. . ђ .

: = {(a,a), (a,e), (b,b), (b,d), (b,e), (c,d), (d,c), (e,c)}.

3.1.2 = {a, b, c} B = {1, 2}. ς

: A B = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}. |A B| = |A||B| = 32 = 6 , ђ . 26 = 64, , 63

( ). B A 63 . , 126 .


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3.1.3 Z ο д…, -1, 0, 1, 2, 3, …ж. ђ (m,n) , m < n, .

= {(m,n) Z Z | m < n}.

Z.

3.1.4 , , xy „ “.

3.1.5 R. = {(x,y) | x, y R, x2 + y2 = 1, y > 0}. R. , , (0,0) 1 .

. .

( ) μ

μ (x) xx.

μ (x,y) xy yx.

μ (x,y) xy yx x = y. μ (x,y,z) xy yz xz.

, , , , .

, ђ. ђ, m n ђ , .

, , . „ , “.

, .

, , . () . . „ “ , „“ , „“ .

, , . , . ђ , . Ј ,


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, () , . .

[x] = {y | xy}, , . μ

1.

x A, [x] ;2. в [x], [x] = [y], x, y A;3.

x, y A, [x] = [y], [x] [y] =;4. .

.

.

3.1.6 N ο д1, 2, 3, …ж. , 3, 0, 1, 2.

(x,y) , .

xy x y (mod 3),

. , . , , , , . μ

Д0Ж ο д3, 6, λ, …ж – . Д1Ж ο д1, 4, 7, …ж – .

Д2Ж ο д2, 5, 8, …ж – .

[3] , [4] , . . {[0], [1], [2]}.

[0],. . , . ,

xy x – y 0 (mod 3),

. , .

3.1.7 . xy , . , . , , , . . [2] [1] д2, 4, 6, …ж, д1, 3, 5, …ж.


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4. ј . , , .

f: A B,

f(a) = 1, f(b) = 2, f(c) = 2.

f = {(a,1), (b,2), (c,3)ж , .

μ

(xA)(!yB) y = f(x),

. .

f X Y ђ X Y, X , .

D(f) = X, (x,y1), (x,y2) f y1 = y2.

, . , .

(Leibniz, 1646-1716)

„“ .

(Clairant) 1734. f(x), (Euler, 1707-1783) .

(Dirichlet, 1805-185λ), , .

4.1 , „“ Y, . , . , μ

(yY)(xX) y = f(x),

. .


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, 1-1 B,. , . , μ

f(x1) = f(x2) x1 = x2 .

4.1.1 , 3 B . , f . ђ

, 2 B , b c .

4.1.2

1

4

4

3

4

2

3

1f , . f(1) = 3, f(2) = 4, f(3) = 4, f(4) = 1

{1, 2, 3, 4}2 = {1, 2, 3, 4}{1, 2, 3, 4}. , .

4.1.3 Z Z f = {(n, 2n-3) | n Z}. f(n) = 2n - 3, n . , 2n – 3 n.

, , m ο 2 n. , 2n – 3 = m n = (m – 3)/2, .

ђ, ,

f(n1) = f(n2) 2n1 – 3 = 2n2 – 3 n1 = n2 ,

. .

4.1.4 f = x2

. , . f: R [0,+].

. , .


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f = x2 , . f: x x2 , .

, , , -. , B, , -. .

ο 2, = 4 (2, 4) -.

ђ, , (+2)2

= (-2)2

= 4, . 2 4.

, .

4.1.5

1

4

2

3

4

2

3

1g , .

. ђ μ

2

4

1

3

3

2

4

11g .

, .

4.1.6. .

4.1.7 f(x+1) = x - 2 .

!

ђ, , .

: ο 1 f(2) = -1, ο 2 f(3) = 0, ο 3 f(4) = 1, ο 4 f(5) = 2, ο 5 f(6) = 3,

... .

x+1 = t, x = t – 1, f(t) = t – 3,


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f(x) = x – 3. t. , .

, , . f(x) = x – 3, . , (4.1.7) ,

.

r f(x) = x – 3, x – 3 = r. x = r + 3, r

f -1(x) = x + 3.

f -1 , . + 3, . , .

4.2

f -1

f -

„“. , . .

, („“)

( B) ( ).

„“ ( B) , !

3.

, („1-1“ ),

(2 B) (b, c ). , B A, .

( 4.1.6), , . .

4.2.1 X Y f : X Y. S X, f(S) {f(x) | x S} S f.

A = {a, b, c} f(A) = {1, 2}.

4.2.2 f f(A B) = f(A) f(B), A B D.

: f . B D.

y f(AB). y = f(x) x AB. y f(A)f(B).


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f(AB) f(A) f(B). , y f(A) f(B). y f(A) y f(B). ,y = f(a) a A, y = f(b) b B. f(a) =f(b), a = b. , y f(AB). f(A) f(B) f(AB). f(AB) = f(A) f(B).

, f(AB) = f(A) f(B) D. . , x, y D f(x) = f(y) x y. ο {x} B = {y}. AB =, f(AB) =. ђ, f(A) f(B) = {f(x)} . f(AB) f(A) f(B), . , .

4.3

f : A B g : B C, .

h : A C,

h(x) = (g o f)(x) = g(f(x)).

, g.

μ

h(a) = (g o f)(a) = g(f(a)) = g(1) = ,h(b) = (g o f)(b) = g(f(b)) = g(2) = ,h(c) = (g o f)(c) = g(f(c)) = g(2) = .

4.3.1

14

43

32

21f ,

24

43

12

31g .

) f o g ν ) g o f ν ) f 2 = f o f ν ) g2.

) ) μ

f o g g o f

(f o g)(1) = f(g(1)) = f(3) = 4 (g o f)(1) = g(f(1)) = g(2) = 1

(f o g)(2) = f(g(2)) = f(1) = 2 (g o f)(2) = g(f(2)) = g(3) = 4

(f o g)(3) = f(g(3)) = f(4) = 1 (g o f)(3) = g(f(3)) = g(4) = 2

(f o g)(4) = f(g(4)) = f(2) = 3 (g o f)(4) = g(f(4)) = g(1) = 3

)

34

13

22

41gf ν )

34

23

42

11f g ; )

24

13

42

312f .

. , .

4.3.2 f(x) = 2x + 1, g(x) = 2x – 3, h(x) = 3x – 1,

(x R). ) ,f o (g o h) = (f o g) o h. ) f o g o h.


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: ) [f o (g o h)](x) = f((g o h)(x)) = f(g(h(x))) [(f o g) o h](x) = (f o g)(h(x)) = f(g(h(x))). ) f(g(h(x))) = f(g(3x – 1)) = f(2(3x – 1) – 3) =2(2(3x – 1) – 3) + 1 = 12x – 9.

4.4

4.4.1 f : R R , :

) f(x) = 2x2 – x + 1 ; b) f(x) = x3 + 2x ; c) f(x) = |x| – 3x + 2 ; d)3

2)(

x

xxf .

4.4.2 A = {1, 2, 3}.a) . b) .

4.4.3 , f : R R :

a) 23

12

)2

1

(

x

x

x

x

f ; b) 32

23

)2

1

(

x

x

x

x

f ; c) 23

32

)1

2

(

x

x

x

x

f ;

d) xx

f xf 3)1

()(2 ; e)xx

f xf 1

)1

(2)(3 ; f) 2)1

(3)( xx

f xf .

: ) tx

x

2

1

t

tx

1

21,

21

213

11

212

)(

t

tt

t

tf t

t

45

51

,

x

xxf

45

51)(

. d) x = t, t

tf tf 3)

1()(2 ;

x = 1/t, ttf tf 3

)()1

(2 .

tvu 32 t

uv 3

2 , )(tf u , )1(t

f v t

. t

tu 12 ,

xxxf

12)( .

4.4.4

1

4

2

3

3

2

4

1f ,

1

4

3

3

4

2

2

1g

1

4

4

3

3

2

3

1h .

a) f 2 , g 3 h 3 ; ς

b) e je u v, f o u = h v o g = h .c) (f o g)-1 = g -1 o f -1.

4.4.5 f : X Y A, B X. :a) f(AB) = f(A) f(B);b) f(AB) f(A) f(B);c) f(A\B) f(A) \ f(B), .


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4.4.6 f : X Y, S Y f -1(S) = {x X | f(x) S}. A, B Y.μ

a) f-1

(AB) = f -1(A) f -1(B);b) f -1(AB) = f -1(A) f -1(B);c) f -1(A\B) = f -1(A) \ f -1(B).

4.4.7 A, B ђ f : A B. A B, |A| = |B|. :a) ν

b) , . Z N;c) , . Q N;d) -1 1, . R (-1, 1).

: d) , x

xxf

1)( R (-1, 1).


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II , .

μ (R), (I), , . (Q).

(Z), (N).

.

5. Цј ј N ο д1, 2, 3, 4, … ж Z = {0, 1, -1, 2, -2, 3, -3, … ж,

f : N Z,

. , () .

N Z, kard N = kard Z, |N| = |Z| =0.

0 – , „“ , „“ .

5.1

ђ 5.1.1:

(Z#) , . S Z#, x S x y y S. , .

5.1.2: S Z#, no

S. S μ 1. n no , n Z

#, n S n + 1 S;2. no m < n , n Z

#, m S n S.

Tada {n Z# | n no } S.


29/191

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5.1.3

n μ2

)1(...321

nn

n .

.

, n = 1. 1, 1,

2

)11(1 , . ђ 1. ! , , n = 4.

2

)14(44321

,

10, . ! ђ, , .

.

(n no) S(n) S(n+1), μ

S(n) :2

)1(...321

nn

n

S(n+1) :2

)2)(1()1(...321

nnnn .

S(n+1) S(n)

(n+1). )1(2

)1(

n

nn, ђ

2

)2)(1( nn, . ,

, .

, .

5.1.4 (n 3) 2n + 1 2n .

μ

n = 3, 23 + 1 23 , , 7 8. n = 4, 24 + 1 24 , , λ 16. n = 5, 25 + 1 25 , , 11 32.

(n no = 3) S(n) S(n+1), μ

S(n) : 2n + 1 2n

, S(n + 1) : 2(n+1)+ 1 2n+1

. S(n+1) S(n). 2(n+1)+ 1 = 2n + 2 + 1 = (2n + 1) +2

2n + 2 , S(n) 2n +2n , n 3 2 2n

= 2n+1 , S(n+1) .

, 2n + 1 2n n 3.


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30

5.1.5

...},3,2,1{n μ

) )5(mod02)1(3 12 nn , . 5ν

) 6

)12)(1(...321 2222

nnnn ;

) 17 n 6ν

) nn 2 .

5.2

() 5.2.1.

2 2, . 4 4. 8

8. , 2n n N 2n.

3 3. λ λ. , 6λλλ3 λ 6+λ+λ+λ+3 λ. , 3n 3n.

5 5. 25 25. , 123...λ75 25, 75 25. , 5n n 5n.

7 7. , 203 7 20 – 2·3 ο 14, 14 7.

11 11. , 1 358 024 67λ 11 (3+8+2+6+λ) – (1+5+0+4+7) = 28 – 17 ο 11 11.

13 13. , 351 13 35 – λ·1 ο 26, 26 13.

. ђ, . , . 4 4.

, 57575746 4 14426 4,

57575736 4 12326 .

5.2.2. n .


31/191

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31

) 23 ν ) 47 3ν ) 56, n = 7; ) 98, n ο 12. , 10, 11, 12, ... A, B, C, … .

: ) 23 2 . , 2

. 21100123 . 10 . 23 : 2 = 12 1

12 : 2 = 6 06 : 2 = 3 0

3 : 2 = 1 1

1 : 2 = 0 1

211001

01234 2120202121 23100816

5.2.3. ( ). x, y Z y 0. q r x = qy + r, 0 r < |y| .

, 7, 3 Z 7 ο 23 + 1, 7 ,2 , 3 , 1 . , , .

. y > 0. je y 1. S = {x – uy | u Z, x – uy 0}. ђ (5.1.1) r S r =x – qy q Z. x = qy + r . .

S , 0 r , r < |y|. , r |y| = y . x – (q+1)y = (x – qy) – y = r – y 0. r – y S, r S . r |y| = -y , . , p r x = p|y| + r, 0 r < |y| , . x =py + r , q = - p x = qy + r, 0 r < |y| , ђ. .

q r .

q’ r’ x = qy + r = q’y + r’, 0 r , r ’ ξ еy|. r – r’ ο (q – q’)y . |r – r’е ο еq – q’ееy| . -|y| < r – r’ ξ еy|,

|r’ – r| < |y|. 0 |q – q’е ξ 1. q – q’ ,

, |r’ – r | ο 0. , q – q’ ο 0 r’ – r ο 0, . q = q’ r’ = r . q r .

5.2.4 x , ρ 0, q r x = qy + r, 0 r < y .

, q r x =

qy + r, 0 r < |y|. |y| = y , x = qy + r, 0 r < y .


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32

(nzd) d μ

1.

d | x d | y (d x d y, . x, y d);2. (cZ) c|x c|y c|d.

5.2.5 , . , d. ,

m, n d = mx + ny.

5.2.6 , nzd, d, m, n d = mx + ny:) , = 45, 126; ) 70, 154; ) 132, 156; ) 143, 165.

) nzd(45, 126) = 9 μ

126 = 2 45 + 3645 = 1 36 + 936 = 4 9 + 0,

9 = 45 - 136= 45 – 1 (126 – 2 45)= 3 45 + (-1) 126.

) d = 14, 70/154 2, 7ν

14 = m70 + n154 14, m = -2, n = 1; ) d = 12, m = 6, n =-5; ) 11, 7, -6.

, p 1 p.

, a b 1, . nzd(a, b) = 1.

5.2.7 a b .

a b m, n Z ma + nb = 1.

. a b . nzd(a, b) ο 1. 5.2.3 m n ma + nb = 1., ma + nb = 1 m n. d = nzd(a, b). d a b, . d|a d|b d|(ma + nb),

d|1, d = 1.

5.2.8 a, b, c Z a 0. , a|bc a, b , a|c.

. a, b , m, n Z ma + nb = 1. mac + nbc = c. a|bc a|a a|( mac + nbc) a|c.

5.2.9 a, b, p Z p . p|ab p|a p|b.


33/191

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33

. p|a . p a. p , p a .

p|b.

5.2.10 a1 , a2 , a3 , …, an , p Z p .

p| a 1a2a3…an p|ai i, 1 i n.

.

54 2 33 . , 496125 34 53 72 . .

5.2.11 ( ) k > 1

nc

n

ccpppk 21 21 ,

p1 , p2 ,…, pn , c1 , c2 ,…, cn .

5.2.12 () .

. p1 , p2 ,…, pn .

1...21 npppx . pi 1 x, i ο 1, 2, …, n.

, p p|x . , n + 1 . , ,

.

(n) n n

, . }1),(|{)( nmnzdnmmkardn .

, 1)2( , 2)3( , 2)4( , 4)12( .

5.2.13 1) n, 6 μ

) nn 3 ;

) nn 73 ;

) nn 113 .

2) n μ

) nn 5 5ν ) )67)(52)(1( nnn 6ν

) nn 7 7.

3) n A B:


34/191

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34

) 532 nnA , 2 nB ;

) 862 nnA , 3 nB ;

) 1582 nnA , 4 nB .

1) ) ) , , , 2, 3, 4, 5, . ν ) 6.

2) ) )(nf 5

k n 5 , 15 k n , 25 k n , 35 k n 45 k n , k ,

)1)(1)(1( 2 nnnn ν ) , 6

ν ) )1)(1)(1)(1( 22 nnnnnnn .

3) ) 1)2(54 22 nnn , ) 1)4(158 22 nnn .

5.3

5.3.1 a b n n

a – b )(mod nba „a b n“.

(1777 - 1855) 1801. 24. „DisquisitionesArithmeticae“, ,

.

, , , , ,, . Њ .

Ј μ , .

5.3.2

)(| ban , . n a – b nmba , mZ. ђμ )(mod nnmaa , mZ. a b n a b

n, )(mod nba )(| ban . μ )5(mod121364651 .


35/191

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35

5.3.3 1) μ

) )3(mod2834 , 34 – 28 = 23;) 7 ≡ - 1 (mod 4), 7 + 1 = 8 = 2 · 4;

) -19 ≡ - 4 (mod 5), jer je -19 – (-4) = (-3) · 5;) 23 ≡ 0 (ЦШН 23) , jer je 23 – 0 = 1 · 23.

2) a, b, c n μ

μ )(mod naa ;

μ )(modnba )(modnab ; μ )(modnba )(modncb )(modnca .

. a – a = 0 = 0 · n;. a – b= m · n b – a = (-m) · n;

. a – b= m1 · n b – c= m2 · n a – c= (m1 + m2) · n.

3) )(modnba )(modndc μ

1. )(mod ndbca ;

2. )(mod ndbca ;

3. )(mod ndbca ;

4. )(modnba mm , m ;

5. )(mod nkbka , k .

1. 2. a – b = m1 · n c – d = m2 · n (a ± c) – (b ± d) = (m1 ± m2)n;

3. (m1 m2) n2

= (a - b)( c - d) = ac – bc – ad + bd == ac – bd – bc + bd – ad + bd == ac – bd - b(c - d) - d(a - b) == ac – bd - (b m1)n - (d m2)n ac - bd = ((m1 m2) n + b m1 + d m2)n.

4. 5. 3.

, b = c = k.

4) 908019901980 37 10ς

4. 3. 7

2

= 49 = 50 – 1 -1 (mod 10) 74

=(72)2 (-1)2 1 (mod 10) , 32 = 9 = 10 – 1 (mod 10), . 34 1 (mod 10).

4|1980 4|80, )10(mod01137 908019901980 .

5) μ ) 3 13ν ) 11 13ν ) 6 7ν ) 3 11.


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36

) 3, 33 = 27 = 213 + 1, . )13(mod133 ,

3. 33 , )13(mod1399 , 3ν ) 1;

) 1ν ) 333 4025 (mod 11).

6. , ,, , . . , Q μ

}|{ ΝΖQ nmn

m.

, .

6.1.

.

6.1.1. μ ) 0,333... ) 0,272727...

) 3,121212... ) 12,0131313...

: ) ...333,0x 10

...333,310 x , xx 310 , 39 x , 9

3x ,

ν ) 100 3/11ν )

33

43 ; )

990

1312 .

Ј , , ... 5

1,5

2, ... .

, () (), () .

, . , , , , . , . .

,

2

11

2

12

2

3

,

4

21

4

6 . ,

( ) , .


37/191

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37

6.1.2. . ) . . ) , .

) 25 2. ) 25.

μ ) 2/5ν ) 7/15ν ) 13/11ν ) -3/4, -4/3.

6.1.3.

) 33

22

517

517

,

224

44

42025

45

,

4224

66

3355

35

.

) n:

i. 75

32

n

n

; ii. 314

421

n

n

; iii) 54

43

n

n

.

: ) 12/229; 1/9; 16; ) i. , a, b k ρ 1 2n + 3 = ak, 5n + 7 = bk. 5 2 1 = 5ak – 2bk, . k 5a – 2b , . ii.

314

171

n

n

17

12

11

n

. ,

, .

6.1.4.

) 10099

1...

43

1

32

1

21

1

; )

168421

16

1

8

1

4

1

2

1

1

1

1

aaaaaa

.

: ) 10099

99100...

32

23

21

12

10099

99

10099

100...

32

2

32

3

21

1

21

2

100

1

99

1...

3

1

2

1

2

1

1

1

100

99

100

1

1

1 ; )

32

1

32

a

,

.

, , () μ 1, 2, 3, ... . .


38/191

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38

, μ 0, 1, -1, 2, -2, 3, -3, ... . ђ , . , , (). – .

6.1.5. () .

: 2-. , . . .

6.2.

. , . μ

dc

ba , dcba :: .

, μ

cbda .

,21

14

3

2 143212 , . 42 ο 42.

6.2.1. μ

) 153

5 x ; )

3

2

5

7

x

x;

) 9

1

4

3

x

x

x

x; ) 3

6

7

5

x

x

x

x.

: ) 25; ) 31; ) 31; ) -27.

6.2.2. dcba :: μ

) ddcbba :: ; ) ddcbba :: ;) dcdcbaba :: ; ) dcdcbaba 54:3254:32 .

6.2.3. 5w

c

v

b

u

a μ

) wvu

cba

432

432

; ) 222

222

32

32

wvu

cba

.

: ) 5; ) 25.

wvucba ::::

, k w

c

v

b

u

a .


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6.2.4. :) 2:7:6:: yyx ;

) 5:11:15:)(:)( xxyxy ;

) 2:)(:)(10:: 2222 yxyxyxyx ;) 2:3:::33 yxyxyx .

: ) 63, 21; ) 33, 66; ) (0, 5), (5, 0)ν ) 2/7, 1/7.

6.2.5. μ

) 13:7: yx , 5:7: zy ; ) 4:9: nm , 5:6: pm ;

) 5:3:2)(:)(:)( baacbc ; ) 7:10:11)(:)(:)( wvwuvu .

: ) 35:92:49:: zyx ; ) 15:8:18:: pnm ; ) 5:3:2:: cba ; )

3:4:7:: wvu .

nGGm :: mnG , ,

. ,

2

nmA

nm

H11

2

m

n .

6.2.6. m n.) 4 6.

) . ) G, A H

m n AHG .

) AGH ,

nm .

: ) 4 λ 946 . ) ,

)(:)(: kbkaba b

a

kb

ka

b

a , . . ) Ј

mnnm

mnnmAH

22

. ) 02

nm .

, , .3 abc .


40/191

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40

. , ACABABAC :: ,

yxxyx ::)( .

1y , . 1 xAC

1::)1( xxx , 12

xx .

12 xx , ¼ 4

5

4

12 xx ,

4

5

2

1 2

x .

2

51x .

. 2

51

1λ0λ. 450 ....

6.2.7. …

287498948481.61803398 .

)

...1

11

11

11

.

) ...111 .

: )

...111

11

11

x .

xx

11 . „“ 12 xx . )

...111 x . ,

xx 12 , . „“ .

6.3.

ђ . , . .

. , , , .

6.3.1. . ) 126 2 20 . λ0 ς


41/191

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41

) 6 . 7 ς

: ) , 126 140 , , λ0 .

ђ , . ђ .

μ 140 ο λ0 μ 126. ο 100 , . 1 40 .) , .

, . : 6 = 4 : 7,

7

33x , . 3 26 .

6.3.2. ) 14 17 . 5 6 . ς ) 6 5 . 2 ђ 3 ς

: ) 14 12 , 8 . , . μ 12 ο 14 μ 8,

12

8

14x , . 21 .

) , 1 6 5 , 3/5 λ .

, , . , .

: 5 = 6 : 9ο 3/5 μ 1 ,

μ 5 , ,

, , . 19

)5 / 3(6

5:

x . ο 2 .

6.3.3. ) . ς


42/191

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) 6 , 8 λ 18 . 7 , 6 8ς ) 5 , 6 , 8 140 . 6 , 5 , 7

ς

: . ) 1,5 1,5 1,5 , , . 1,5 3 3 . , , . 3 3 6 . , 6 , . 1 3 2 ., , . 1 1 2/3 .

) 9861

V (

), 8672 V .

xVV :18: 21 , 18986

867 x , .

14x . ) , , , . .

x : 140 = 6 : 5= 5 : 6

= 7 : 8

865

756140:

x , ,

. , ο 160 .

6.3.4. . ) 500 2, 3 5. ) Ј ђ, 3 (ђ) 2% , 1,5% 1,5 % , . ς ) 800 20% .

: ) x, y z. 500 zyx

5:3:2:: zyx .

k x 2, k y 3

k z 5

, 500532 k k k , 50k . 100x , 150y 250z

. ) ђ λ5% 3000, . 2850 , 60 , 45 , 45 . ) 220, 264 316 .

6.3.5.


43/191

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) Ј 700 . 6 7 , 5 8 , 3 12. ς ) 800 , 400, 500 600 . ς

) λ00 40%, 30% 30% , 7 , 11 , 2 .

: ) , 42, 40 36, 118 . , 42/118, 40/118 36/118 700 , . 24λ, 237 214 , . ) 4/15, 5/15 6/15 800 , . 213, 267 320 . ) . , . 2/11 λ00 , . 164 .

40% μ 30% μ 30%, 66, 4λ 4λ . 5 . 5/λ λ00 40% μ 30%, .

2347

4900

9

5 , 175

7

3900

9

5 .

4 , 4/11 λ00, . 327 . , , 66 + 234, 4λ + 175 + 327 4λ , . 300, 551 4λ , .

6.3.6. . ) ђ 200 1,1 300 1,2 . ς

) 60 30 λ0 50 ς ) 65 %, 60 % 45 % λ00 50 %ς

: ) 500 , x 5002,13001,1200 . , ο 1,16 . ) Ј 90 yx 90503060 yx , = 60, ο 30 .

50 60 30.) , 50 % 50, 100 150


44/191

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. x, y z .

900 zyx

150:100:50:: zyx 150, 300

450 , .

.

6.3.7. . ) λλ% . 98% . ς ) 1λ8 2 . λ8%.

: ) , v s .

100

99

sv

v. r

rv , λ8%. 100

98

srv

rv . r ο 0,4λ5

4λ,5% , 50,5% . ) λλ% . 1 % , 2%. 2 2% , 100 . λ8 . , 100 .

100

pTGK , K , G

, T , p .

365 ( 366) , 360 30 . , 7λ

0,216365

79T , 0,219

360

79T .

6.4.

(, ) .

. , . , .

6.4.1. . ) .


45/191

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) . .

) n N .

: ) , , , , . ) 24 . ) nn ...21! , .

„ “.

6.4.2. . ) () . ) () . ) . ) k- n- ς

: ) , , , , , . ) k n . k n . n , n .

n-1 , n(n-1) . , , - n-k +1 ,

)1(...)1( k nnnVnk .

6.4.3. (). ) 0-5?) 0-4?

: ) , 0, 1, ..., 5,

. , , 5 . , , 5 .

4 . 554 = 100. ) 4432 = 96.

6.4.4. . ) 0-5?) 0-4?

: ) , .

1-5. , 0-5. 566 ο 120 . ) 500.

6.4.5. . ) 2- 5 ς ) 3- 5 ς ) k- n ς ) 10- μ 2-, 8-ς


46/191

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46

: ) (, ) 5- . 5

, 4, 54 . , , .

, 2! ο 12. , 2 5 10

21

4552

C . )

10321

34553

C . )

k

k nnn

k

VC

n

k n

k

...21

)1...()1(

!. ) , 45,

10810

2)8...3(21

)3...8(910

21

910CC

.

7.

7.1. 3

34 . „()

“, „ “. ( ) ,

. (1,732)2 3 3 1,732 3 4 1,587 (1,587)3 4.

μ 2 = 1,414... ; 3 = 1,732... ; 4 = 2 ; 5 ο 2,236... ν .

. ( 6.1.1.) . .

7.1.1. 2 .

: , n

m2 , m n

. Ј . ђ,

22 2nm .

, m , m = 2k, k = 1, 2, 3, ... , k ( m).

, m 22 2)2( nk , . 22 2k n .

n . , m n , .


47/191

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47

, ђ, 2

.

μ

yxxy y

x

y

x ,

. ђ,

xx

x

x

x

x

1.

7.1.2. μ

) 4520 ; ) 156 ; ) )234)(234( ; ) 18

23.

: ) 5954 5352 55 ; ) 5332 592

109 103 ; ) 22 )2()34( 44448 ; ) 29

23 1

23

23 .

7.1.3.

μ ) 13

1

; ) 6223

34

.

: )

13

13

13

1

2

13 ; )

2362

2362

2362

34

1824

612188

6224 .

7.1.4. ABC CD. AC = 15 cm CD = 10 cm. AD = x, AB = y BC = z.

: 222 1015 x ,

. 55x cm. ACD ABC

15 μ x = y : 15 = z : 10. 59y cm

56z cm.


48/191

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7.2.

, ( ) () . a, b c

(a + b) + c = a + (b + c), (a · b) · c = a · (b · c).

, (2 + 3) + 5 ο 5 + 5 ο 10 2 + (3 + 5) ο 2 + 8 ο 10. , (2 · 3) · 5 = 6 · 5 ο 30 ђ 2 · (3 · 5) = 2 · 15 = 30.

, . acabcba )(

. ( ) ( ). 2 · (3 + 5) = 2 · 3 + 2 · 5, 16.

, .

7.2.1. μ

(a + b)2 = a

2 + 2ab + b

2 , ν

(a + b)3 = a

3 + 3a

2b + 3ab

2 + b

3 , ν

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 , .

7.2.2. b – b μ

(a – b)2 = a2 – 2ab + b2 , ν

(a – b)3

= a3

– 3a2

b + 3ab2

– b3

, ν (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4 , .

7.2.3. μ

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

( ) μ

(a ± b)5 = a

5 ± 5a

4b + 10a

3b

2 ± 10a

2b

3 + 5ab

4 ± b

5.

7.2.4. μ


49/191

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a2 – b2 = (a – b)(a + b), ν

a3 – b3 = (a – b)(a2 + ab + b2), ν

a3 + b3 = (a + b)(a2 – ab + b2), .

7.2.5. , μ

) x

5

+ x + 1; ) x

4

+ x

2

y

2

+ y

4

; ) (x + y + z)

3

– x

3

– y

3

– z

3

.

: ) 1225 xxxx )1()1( 232 xxxx

)1)(1( 232 xxxx ; ) 224224 2 yxyyxx 2222 )()( xyyx

))(( 2222 xyyxxyyx ; ) )())(( 3333 zyxzyx

))(())())(()(( 2222 zyzyzyxzyxxzyxxzyx

))()2333)((( 22222 zyzyyzxzxyzyxzy

)3333)(( 2 yzxzxyxyx ))()()((3 yxzyxxzy

))()((3 zyzxyx .

7.2.6. 1. μ

) 12 2 xx ν ) 123 2 aa ν ) 145 2 bb ν ) 167 2 cc .2. μ

) 22 5136 baba ν ) 22 31914 vuvu ν ) 22 7910 yxyx .

3. μ

) 32 a ν ) 253 x ν ) vu 23 ν ) 2332 .

4. μ

) ...124 2 xx ν ) ...2309 2 aa ν ) ...129 2 mnm ν ) ...61212 .

5. μ bcacabcbacba 222)(

2222 .

6. μ

) 12 a ν ) 3x ν ) 23 ν ) b52 .

7. μ

) ...128 23 xx ν ) ...9 23 aa ν ) ...2933 ν ) ...36324 c .8. :

) 22)( cba ν )4

1)

2

15( 2 ν )

9

4

5

12

2

x ν ) 213 2 .

9. , μ

) 125)1( 3 a ν ) 343)3( x ν ) 16 x ν ) 13333 baba .

10. μ

) 133 1255 nn aa ν ) 124 813 nn bb ν ) 3284 235 nnn ccc ν ) 51256 36 nn xx .

7.3.

, , .


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. .

7.3.1. ) 5

3

75

73

35

21

; )

12

2

x

xx

)1)(1(

)1(

xx

xx

1x

x, 01x .

. () , () .

7.3.2. ) 7

4

5

3

3

2

105

60

105

63

105

70

105

67; )

1

3

1

21

xxx

)1)(1(

)1(3

)1)(1(

)1(2

)1)(1(

)1)(1(

xxx

xx

xxx

xx

xxx

xx

)1(

)(3)(212

222

xx

xxxxx

xx

xx

3

2 152 .

() () . μ

bd

ac

d

c

b

a ,

db

ca

d

c

b

a

:

:: .

, .

7.3.3. ) 35

6

7

2

5

3 ,

5

2

7

5:

35

10 ; )

111 2

2

x

x

x

x

x

x,

1

1

1

1:

1

123

2

xx

x

x

x

x

x, 1x .

. μ

c

d

b

a

d

c

b

a

: .

7.3.4. ) 13

9:

10

7

90

91

9

13

10

7 ; )

1:

1 x

x

x

x

x

x

x

x 1

1

)1(

)1(

xx

xx

( )1

1

x

x, x 0.


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, ( ) , ( ) μ

bc

ad

dc

b

a

d

c

b

a: .

7.3.5. μ

)

aa

a

a

1

1

1

; )

21

11

1

11

b

bb

; )

13

2

9

4

127

8

2

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3

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b

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; )

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n

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b

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.

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a

aa

a aa

aa

2

1

3

2

aa

aa

2

13

3

24

, 1,1,0 a ; ) 1 – b, b

±1; ) 13

2

b

a, b 0; )

)1...(1

)...1)(1(

321

12

nnn

n

n

bbbb

bbbb)1( bb

n , b 0.

7.3.6. μ

1. ) 23

2

2

2

1

12

xx

x

x

x

x

x; )

65

7

3

2

2

32

aa

a

a

a

a

a .

2. )

nm

nm

nm

m

m

n

n

m1

22 ; )

x

y

y

x

yxyxx

yx 11112

2

.

3. )

u

v

v

u

vu

v

vuuvu

u2:

22

; )

nm

nm

nm

nm

nm

nm

nm

nm: .

4. )

bc

cba

bca

c

acbca

abcba

21:

2

2

2 222

222

222

;

)

vw

uwv

wvuwvu 21:

11:

11 222.

8.

1. n:) nnn 45 33 120ν

) nnn 23 23 6.

2. n :

) )67)(52)(1( nnn 6ν

) 12 n 3.


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3. 7 , λ. ς

4. 7

λ 2μ3. ς

5. n ,

) 3322 5454 nnnn 10ν

) nnnn 21122 3232 6.

6. , m )(modmyx ,

n )(mod myx nn .

7. μ

11 f , 1

2 f , n

nnn

f f f

12

.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,... . n:

) n

f 3

ν

) n

f 4

3ν

) k nk f f nk || .8. . . ς

9. . 42. ς

10. , 78% , 20% , 1% . 1000 ς

11. 2007. 2410, 7187, 4211 37. ς

12. 450 5 λ0 . 7 75 ς .

13. 5 7 115 . 3 3 ς


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14. 5,5 6,25 . λ00 6 ς

15. 10,10 , λ,65 .

120 λ,80 ς

16. , 27% 370, 40% 250ς

17. . , . , ς

18. 5000 8% . , ς , .

ς

19. 7000 6% 5 . , ς ς

20. 3 .

21. 32 .

22. μ

) )234)(234()234( 2

ν ) )81(2)122(2

1

.

23. 20 cm 12 cm.

μ

24. ) 9)64)(4( 22 xxxx ν ) 16)85)(5( 22 yyyy ;

) 24)105)(5( 22 aaaa ν ) 120)227)(7( 22 bbbb .

25. ) 14610 xxx ν ) 12 23456 uuuuuu ;

) 11664 36 yy ν ) 12 23456

vvvvvv .

26. ) )()()( 222 bacacbcba ;

) )()()( 222 bacacbcba ;

) 333 )()()( accbba .


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27. μ

) ))(()()( 222222 yxbabxaybyax ;

) ))(()()( 222222 yxbabxaybyax ;

) ))(()()( 222222 yxbabxaybyax ;

)

222222

)()())(( aybxbyaxbayx .


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III . , , .

.

p P . P .

ν , . , .

p ђ A B,

. A B . T p p AB .

AB , , () , . , , , .

. , . .

.

. , . .

p . p .

.

, , , , , .

, ,

p. p p , p . , . ()


56/191

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. , .

. ,

. .

ђ 6. ... . (.), 6. 3. ... .

13 џ , , .

, 17. , , . , . 18. . , 1λ. ,

, , , , .

1λ. μ , (David Hilbert, Grundlagen der Geometrie,18λλ), . ђ ђ, . 1λ57. , , . 8. . .

9. А ђ.

.


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57

. , , .

Ј ђ ,

.

. 1λ. μ

1. (1-8);

2. (1-4);3. (1-5);4.

ν 5.

(1-2).

9.1. ,

ђ , . .

I-1 . I-2 .I-3 . . I-4 , . . I-5 .

I-6 , .I-7 , .I-8 .

A, B, C,...; , , a, b, c, ...ν , ,

, , , ... ν .

I-7 , I-8 . I-1-3 I I-4-8 I.


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(I-1,2), A B a = b .

, , , ... ,, , , .

9.1.1. a b .

: , . , a b , A B. I-1,2 . , , . a = b. ,

, ђ .

. I-3 ђ .

9.1.2. ђ .

I-4.

ђ .

9.1.3. ђ .

: p C , . I-3, p A B, I-4, A, B C

. A B p

, I-6 . , p C A, B C,

I-4,5 . , ђ p

C.

9.1.4. ђ .

: }{Aqp , . I-3 p

A B, q A C.

I-4 A, B C . I-5

.


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9.1.5. ν .

: А, I-7

B. А B I-1,2 p I-6 , А B ., p.

, p , p , I-5 .

, .

9.1.6.

.

: , I-6 , . . ,

.

. I-8 ђ .

9.1.7.

1. . ς

2. . ς

3. ђ ς

4. ђ 5 4 ς

5. ђ ς

9.2.

"ђ" , .

II-1 B ђ A C, A, B, C B ђ ђ C А.


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II-2 А C B AB, C ђ A B.II-3 ђ . A, B C ,

B ђ A C. II-1 B ђ C A.

ђ ђ .

II-4 A, B C a ABC ν AB, AC, BC.

. 1

. II-4 , μ .

ABC , , BC. AC BC.

9.3. , .

III-1 A, B , , А' ', B' ' А', AB А'B', μ AB А'B'.III-2 А'B' А"B" АB, А'B' А"B", μ , ђ .

III-3 АB BC ,, А'B' B'C' ' ν AB А'B' BC B'C', AC А'C'.

1 Moritz Pasch: Vorlesungen uber neuere Geometrie, Leipzig 1882.


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, AB BA. , . , . .

(a, b) (b, a)

b . Oa Ob , . , .

ђ

(), . O

A B AOB, B. .

III-4 (p, q) α a' α' ђ

α' a '. p' a ' 'ν α' q' (p, q) (p', q') (p',q') К', (p, q) (p', q'). .

9.4.

9.4.1. , .

9.4.2. .

IV ( ). А μ , ђ А, А .

9.4.3. ђ .

: IV, a, b .

, ,

, I-5 .

p p . p .

b А a a, А a.


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b . , ђ .

, , . (12.4.1.) ,

(12.4.2) . 12.1.4. , . .

9.4.4. .

9.5.

V-1 ( ). AB CD , n, CD n A B B.V-2 ( ).

ђ I-III V-1.

10. , , . ђ . , .

, a b t . A

B a b. 1, 2, ..., 4.

AB, 1, 2, 3 4.

, 3, 4, 1 2.

,

3 3.

, 3 1.

, 1 2.


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10.1.

,

180 (). 10.1.1. .

180 . ( ) . , . , . . 10.1.2.

180. 10.1.3. .

10.1.4. 360.

10.1.5. μ

) 360.) 540.) n- 180)2(n .

) n- 360.

: ) n- n д3, 4, 5, …ж. А1 n-3 , A1A3, A1A4, A1A5, …, A1An-1.

n-2 А1 А2 А3, А1 А3 А4, А1 А4 А5, ..., А1 Аn-1 Аn. 180)2(n .

n-2 А1 . ђ, , Аn.

)

. 1, 2, 3, ..., n, 1, 2, 3,

..., n, . 1 + 1 = 180, 2 + 2 = 180, 3 + 3 = 180, ... , n + n = 180. 1+ 2+ 3+...+ n= (180- 1) +(180- 2) + (180- 3) + ... + (180- n) = n180 + 1+ 2+ 3+...+ n = n180 + (n-2)180 = 360.

Ј

, 60.


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64

Ј . ().

.

10.1.6. .

: ABC, BC > AC, . a > b. BC D, C-D-B CD = b. ADC , A D

, . BAC =

ABD, = ABD, . > >

, > , .

10.1.7. .

, > , a > b.

b > a.

( ) . . 2:1 .

10.2.

10.2.1. CD ABC:

1. 2

ABCD C > 90.

2. 2

ABCD C = 90.

3. 2

ABCD C < 90.

: 1. ABC . AD > CD ACD > , DB > CD DCB > . > + , 180, > 180 - , > 90. 2. AD CD ( DB CD) , .

ACD = ( DCB > ). = + , = 90.


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λ0 ( ) , .

λ0, . ,

λ0. . .

10.2.2. .

: C B D. ABCD ,

360, + 2 = 180. ђ, 1 + 2 = 180.

10.2.3. ABC 3645'30'' 6225'30'' .) , C ς ) . ђ . ) B.

: ) , , 60. ,) , ђ .

. .

10.2.4.

135.: ABC

A B . , . + = 90. ABS,

, /2 /2, 45. , S 135. , S 135, B () 45, λ0. C.


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66

10.2.5. Ј 5 - 12 -. .

: , Ј

ABC.ABC = = 90

222

125 x , . ο 13 .

Ј ,

45 + = 112,38.

10.3.

( ) , . Ј , .

. .

10.3.1. . . 1. 2. 3.

1. AOB BOC,

. OBC = , = 2 . 2. , AD DB, 1 = 2 1 2= 2 2 ,

= 1 + 2 = 2 . 3. = 1 – 2,

AD BD.

10.3.2. ( ) .


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) . ) (), () . ) ђ .

: ) 2. CAD , ACD . BCD.

180. ) , tAB ACB .

10.4.

1. . .

2. .

3.

λ. .

4. ς

5. ђ .

6. ABC CD AB 105. CD = BC.

7. AA’ BB’ ABC AB

C.

8. H ( ha hb) ABC,

AHB +C = 180.

11.

, ( )

, , . , .

ђ , , , , ,


68/191

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68

, . ђ, , ( )μ

) ν )

ν ) ν ) , .

() .

AB1C AB2C , (),

b.

1 2 1 2.

11.0.1. Opq M. P М Op Q M Oq MP = MQ, OM Opq.

: , OMP OMQ

(P = Q = 90), , OM.

MP = MQ, OMP OMQ, POM = QOM.

11.1.

11.1.1. ђ .

: , ABC A1B1C1 μ

11CBBC , 11CAAC

11DCCD

, . a a1, b b1 h h1. , .D = 90. a b.

,

Q ADC A1D1C1 . D = D1, a = a1 h = h1, ,


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( 10.1.7.), . b h,

ADC A1D1C1. A = A1.

, a = a1 , b = b1 a b A = A1,

ABC A1B1C1.

, ( ) . . () 2:1 .

11.1.2. , . .

: ABC, ,

CD. = =C/2, AD = DB.

, CD

E C-D-E CD = DE. BDC =ADE ( ), BDC ADE. BC =AE = . AEC , , . AC = AE. AC = BC, .

ABC .

11.1.3. .

: AC = BC , EC = CD = , . C

AEC BDC, AEC BDC, AE = BD, . .

, , . AE = BD, AT= BT ο 2/3 , DT = ET ο 1/3 ,

ATD = BTE. ATD BTE. AD = BE, AC = BC, ABC .


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11.1.4. ( , 17. 1λ83. ) ABC A1B1C1 . D D1

AB A1B1. BCD =B1C1D1, AD = A1D1.

: BCD B1C1D1

( ), DB = D1B1 AB – DB = A1B1 – D1B1, AD =A1D1.

11.1.5.

1. ABC A’ B’ C’ μ ) b = b’ , ha = ha’ , = ’ ; ) hc = hc’ , = ’ , = ’ ;) a = a’ , c = c’ , tc = tc’ ; ) hc = hc’ , tc = tc’ , c = c’ .

2. μ ) ν ) .

3. .

4. M AB AM = BM. .

5. . μ ) ν ) .

6. , ( ) ( ) .

11.2.

, , , . , .

11.2.1. .

,


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. , ABCD AC.

ACB =CAD, CAB =ACD, ABC CDA.

. .

11.2.2. .

, AB CD BC DA ABCD, .

E ABE CDE ( )

. , , ( ). .

11.2.3. .

11.2.4. .

: ABC M N AC BC, . MN

. MN P P-M-N PM = MN. (3) ANCP . (2) CN PA . , (2)

ABNP . , MN AB AB = 2MN.

11.2.5. M N AB CD ABCD. DM BN AC .

: , DM BN E F.

AME CNF () AE= FC. EG || MB MBGE

EGF AME (), ђ AE = EF. DM BN

AC .

11.2.6. 2μ1 .


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: ABC () AS CM , . AS D. ABDC.

, CM BN (N

CD) AD T V AT = TV = VD. S

TV AT = 2TS, T AS 2μ1 A.

11.2.7. . . .

,

.

.

11.2.8. ( , 15. 1λ84.) ABCD AB > BC. B1 B AC, E AB1 CD. ADE CB1E.

: , CAB DCA . CAB CAB1

AB1 AB AC., DCA CAB1, AEC (AE = EC). , CB1 = CB = DA ђ DEA CEB1,

ADE CB1E .

11.2.9. DC ABCD M. BAM BC N. AM = DM + BN.

: DC P P-D-M PD = BN, .

ADP ABN, APD PAM, APD = ANB= 90 - , PAM =PAD + DAM = + (90 -2 ) = 90 - . , AMP . AM = MP = DM + DP

= MD + BN.


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. , , . , . Ј

, – .

. .

11.2.10.1. μ ) ν ) ν ) ν ) .

2. ђ .

3. .

4. AC ABCD A

C. ABC ACD.

5. , .

6. M N AD BC ABCD, AB

+ CD = 2MN.

7. , .

8. ABCD ( ) . AB CD BC AD, M N AC BD AC BD.

11.3.

10.3.

( 10.3.1.). () ( 10.3.2.) , ђ.

() . () .


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11.3.1. ABCD AB +CD = BC + AD. еAB – CD| = |AD – BC|.

.

ABCD M, N, P Q, AB, BC,CD DA. A, B, C D , . AM = AQ, BN = BM, CN = CP DP =

DQ. AB + CD = BC + AD.

11.3.2.

(180).

: ABCD. ( ) ( A C)

BD, . + = 180.

11.3.3. A.

B C. BAC (λ0).

: , AS A SA SB , SA SB . SA = SB = SC. , S A, B C, A

BC .

11.3.4.

1. 65, 105 115. ς

2. A, B C 1μ2μ3. ABC.

3. s – a, s – b

s – c, 2

cbas

.


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4. . .

5. . .

6. А BC D. ABC ABD ς

[30]7. O AB. A B C D. COD .

11.4.

(&10.). ђ,

180 ( 10.1.2.), ( 10.1.3.),

360 ( 10.1.4.). ђ, .

, , .

11.4.1. .

: ABC AO BO sA sB. OM,

ON OP BC,CA AB.

APO ANO

/2. OP = ON. ђ,

BPO BMO

/2, OP = OM. OP = ON = OM.

CMO CNO, , OM = ON.

OCN OCM, CO , . OP = ON = OM

.

11.4.2. .


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: ABC S sa sb BC CA. M N .

MBS MCS ,

. BS = CS. NCS NAS. CS = AS. AS = BS = CS.

, ABS S sc AB. ђ, S

ABC.

.

11.4.3. , .

: , A, B C ABC NP, PM MN BC, CA AB.

BMC, CNA APB ,

, . A, B C NP, PM MN.

(11.1.2.), MNP

H, ABC.

, . . (&11.1.). ( 11.2.6) 2:1 .

ђ, , , . ( 11.2.4.) .

11.4.4.

1. 2μ1 . 2. , . .


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. 1. , AN BM ABC T. MN , AB .

ABT PQ, ђ AB . MN || PQ MN = PQ. MNT PQT

MNT PQT, PT = TN QT = TM. P AT Q BT, 2μ1 A B.2. AN BM ABC T.

CP BM S , .

, BS = 2SM, BMBS

3

2 . , BT = 2TM

BMBT3

2 . BS = BT,

S T

. , CP , .

11.4.5.

1. ђ .

2. .

3. .

4. D AB ABC. E

BC DE BC. F DE. AE CF.

[ CD . G BE GE

BED GF || BD, GF CD. F CDG, DE GF. , CF CF DG. ђ, DG

ABE AE || DG AE CF.]

11.5.

(10.1.6. 7.).


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,.

11.5.1. ( ) , .

: BC ABC D-B-C DC = CA = b, . ACD

DAC =ADC = . BAC = , < BAD, BDA ’ c > c’.

: A =A’ C = C’, CB’

. s BCB’ AB D. DCB DCB’ ( ), DB = DB’., AD + DB’ ρ AB’,

AB > AB’.

11.5.3. , a = a’, b = b’ c > c’ > ’.

: C C’ μ = ’, < ’, > ’. 1) = ’, ABC A’ B’ C’ , c = c’, . 2) < ’ 11.2.2. 3) > ’ .

11.5.4.

1. ђ .

2. .

3. .

[ O ABC . , ABO: /2 /2 BO AO, BOC: /2 < /2 CO < BO. , CO.]


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4. .

5. ђ .

6. .

[ tc < b + c/2 tc < a + c/2]

7. , . .

8. ¾ .

9. μ) ν) .

10. ABC M. :

) ACB


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IV .

, , , (&13), (). ђ ђ (&14.).

, (μ "υ ῖ " – ) , . , , , , ђ.

12. Кј 2.

, .

. :

1μ ν 2: ν 3μ ν 4μ ν

5μ ν 6μ .

:

ν

(--) ν

(--) ν

(--) ђν (--) .

, , . μђ, ђ, , , , , , .

2 (42λ-384. ....)μ „

.“


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μ

, ν

, ν

,

ν , ђ

.

. . μ

. , .

. , .

. , .

12.1.

12.1.1. a, b ha, .

. () ABC. a, B C . A C b, BC ha.

. , k(B,a) B a Bx BC.

.

BC : Bx; C = Bx k(B,a). k 1(C,b).

n BC C; P, Q n k 2(C, ha). p q P Q BC.

А: p k 1(C,b), q k 1(C,b), . AiBC (i = 1, 2, 3, 4 ), .

.

8/9/2019 Matematika Za 1 Razred Gimnazij