Rmathtronc

February 26, 2018 | Author: Youssef NEJJARI | Category: N/A
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‫‪ ‬ﺑﺴﻢ‪ ‬ﺍﷲ‪ ‬ﺍﻟﺮﺣﻤﻦ‪ ‬ﺍﻟﺮﺣﻴﻢ‬ ‫‪ ‬ﻭﺍﻟﺼﻼﺓ‪ ‬ﻭﺍﻟﺴﻼﻡ‪ ‬ﻋﻠﻰ‪ ‬ﺃﺷﺮﻑ‪ ‬ﺍﻟﻤﺨﻠﻮﻗﻴﻦ‪ ‬ﻣﺤﻤﺪ‪ ‬ﺳﻴﺪ‪ ‬ﺍﻟﻤﺮﺳﻠﻴﻦ‪ ‬ﻭﻋﻠﻰ‪ ‬ﺁﻟﻪ‪ ‬ﻭﺻﺤﺒﻪ‪ ‬ﺃﺟﻤﻌﻴﻦ‬ ‫‪ ‬ﺃﻣﺎ‪ ‬ﺑﻌﺪ‪ ٬ ‬ﻳﺴﺮﻧﻲ‪ ‬ﺃﻥ‪ ‬ﺃﻗﺪﻡ‪ ‬ﻟﻜﻢ‪ ‬ﻫﺬﺍ‪ ‬ﺍﻟﻌﻤﻞ‪ ‬ﺍﻟﻤﺘﻮﺍﺿﻊ‪ ‬ﻭﻫﻮ‪ ‬ﻋﺒﺎﺭﺓ‪ ‬ﻋﻠﻰ‪ ‬ﻣﻠﺨﺼﺎﺕ‪  ‬ﻣﻊ‪ ‬ﺗﻘﻨﻴﺎﺕ‬ ‫‪ ‬ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ ‬ﻟﻤﺴﺘﻮﻯ‪ ‬ﺍ‪ ‬ﻟﺠﺬﻉ‪ ‬ﺍﻟﻤﺸﺘﺮﻙ‪ ‬ﻋﻠ‪ ‬ﻤ‪ ‬ﻲ‪ ‬ﻣﺠﻤﻌﺔ‪ ‬ﻓﻲ‪ ‬ﻛﺘﺎﺏ‪ ‬ﻭﺍﺣﺪ‬ ‫‪ ‬ﻭﻫﻲ‪ ‬ﻟﻸﺳﺘﺎﺫ‪ ‬ﺣﻤﻴﺪ‪ ‬ﺑﻮﻋﻴﻮﻥ‬ ‫‪sefroumaths.site.voila.fr ‬‬

‫‪ ‬ﺗﺠﻤﻴﻊ‪ ‬ﻭﺗﺮﺗﻴﺐ‬ ‫‪ALMOHANNAD‬‬

‫ــــ دئ اــــ ﺏـــــــــ ت‬

 (2 . 0 ". 6 -! !3!  . 0 , !  ". 0 . a ". c $) b ". c  a ". b $ !%& . 6 -! !3! 8  ". 1  -! . 9 ".   . 1 , ! "# 1  -

(* (* (* (* (* (*

25,11,9,5,4,3,2 () &* + (3  " (a $ # α r 7..... 7 α 3 7 α 2 7 α1 7 α 0 $ -

{0,1, 2,3, 4,5, 6, 7,8,9}

%-!  -! -& α r α r −1 ...α 0 6 -  ............ 7 α1 ! "# 7 α 0 ! "#

- (b : - a = α r α r −1 ...α 0  -!    α 0 ∈ {0, 2, 4, 6,8} $ !%& 2  6.-!  . a (* 3 / α 0 + α1 + α 2 +  + α r $ !%& 3  6.-!  . a (* 4 / α 0α1 $ !%& 4  6.-!  . a (*

α 0 ∈ {0,5} $ !%& 5  6.-!  . a (* 9 / α 0 + α1 + α 2 +  + α r $ !%& 9  6.-!  . a (* $

a !%& 3  6.-!  . (* 11 / (α 0 + α 2 + α 4 + ......) − (α1 + α 3 + α 5 + .....)

{

}

α1α 0 ∈ 00, 25,50, 75 $ !%& 25  6.-!  . a (*

 . $"% &* (4 . $    1 $ $  b  a $ - ! " "   1 "#   , b  a $  -  3! 4-! ".-! . a ∧ b  PGCD (a , b )  -  . 5 4

. / )' 0   - (5 . a ≥ b   IN * $ b  a $ : 6 - 6  # ;#  PGCD (a, b )    $   ".-! , PGCD (a, b ) :     ?-! @%, A * $  a r1

b q1

r1

r2

q2

q3

r1

r2

...

...

...

...

...

rn

0

       (I IN = {0,1, 2,3, 4,5.......}

IN * = {1, 2,3, 4,5.......}

 –         (II   a         (1 . k ∈ IN   a = 2 k    a         (2 . k ∈ IN   a = 2 k − 1  a = 2 k + 1 

 (3 .   ! "# $ !%&    $ (a .   ! "# $ !%&    $ (b .  a + b $) $  b  a $ !%& (* (c .  a + b $) $  b  a $ !%& (* .  a + b $)  b  $  a $ !%& (* .  ab $) $  b  a $ !%& (* (d .  ab $) $  b  a $ !%& (* .  ab $)  b  $  a $ !%& (* *+!  , $) $  $  b  a $ !%& (e . 

    (III . $ $  b  a $ - ! " (1   a $ !%& b  - /0 a  -! $& . . k ∈ IN   a = b k 

 (2 .    /0 0 (* . 0 , ! /0 - 0 (* $) c /0 b  b /0 a $ !%& (* . b  - /0 a

 # $"% !   (3 . $    1 $ $  b  a $ - ! " /0 2 , b  a $  - 23! 4-! /0-!  PPCM (a, b )  -  . 5 4 "   1 . a ∨b

 (4

PPCM (a, b ) = a $) b  - /0 a  -! $ !%& (* PPCM (a, a ) = a (*

 & ' (IV . $ $  b  a $ - ! " (1 ". b  -! $&  7 b  6.-  # a  -! $& .    a  b /0 a $ !%& a . b / a  . k ∈ IN   a = b k

    (V -    a    - !  ! " (1 . a  1 . $#

 (2 .   8  - a  -! , B. - (a p ≤ a B. -! p 6 -3! !3! 8   . -  1 a $) a ". !3! @%,  $ !%& . - a $) a ". C !3! @%, 8  ; !%& , 100 $ 23! 6 -3! !3! (b 47 ,43 ,41 ,37 ,31 ,29 ,23 ,19 ,17 ,13 ,11 ,7 ,5 ,3 ,2 .97 ,89 ,83 ,79 ,73 ,71 ,67 ,61 ,59 ,53 7  , p ≠ 2 -   (c . - E - 1  -! (d 2

 0 2 3 (4  $ ." (3  3=‫@ئ ا‬+‫ ا‬A ( ABC )  . +   ;  R

A

B

  

C



x

 %& & ' ( !  % (3

π

cos (π − x ) = − cos x

2

π −x π

π 2

sin (π − x ) = sin x

(b

tan (π − x ) = − tan x

(c

x 0

AB AC BC = = = 2R   SinC SinB Sin  A

−x

(a

 x  %& & ' ( !  % (4

π  cos  − x  = sin x (a 2  π  sin  − x  = cos x (b 2  1 π  tan  − x  = (c 2  tan x

π

π

2

2

π

−x

x 0

0

 +, ' (   (5 x

0

cos x

1

π

π

π

π

6

4

3

2

3 2

2 2

1 2

0 1

sin x

0

1 2

2 2

3 2

tan x

0

1 3

1

3

×

2π 3

3π 4

5π 6

−1 2

− 2 2

− 3

3 2

2 2

1 2

− 3

−1

π

2



−1 0

1 3

0

π

3π 4

2

2π 3

π

1 3

3

2

2 2

4

1 2

5π 6

2 2

2

-1  3 2

1 2

π 6

 2

π

π

1 2

0

3 2

0 1

 - .& #/   % (6 B <   -% >> ( ABC )  (a C

AB AC BC Sin  A= AC

Cos  A= A



B

Tan  A=

BC AB

‫اوال اد‬ . I L. 

. I L.  OP 

. I L. 3QP,

. I L.  OP 3QP,

. I L. 9I

f f f f f

A& A& A& A& A&

. T 0# U T ( x, y ) ≥ 0 @ "# (∗ T ( x, y ) > 0 @ "# (∗ T ( x, y ) ≤ 0 @ "# (∗ T ( x, y ) < 0 @ "# (∗ T ( x, y ) = 0 @ "# (∗

  ‫(  ا‬I          f    

(1

D f    p( x) Q( x) :   (2  ( )* + % . Q ( x) ≠ 0 ! "# $%& "# &  f ( x) 

f ( x) =

'()

D f = R − {  ( - ./ } , Q( x) = 0 f ( x ) = P ( x ) :  (3

\ 3 ] >3

0# 1 + % P ( x) ≥ 0 ! "# $%& "# &  f ( x) 

D f = P ( x) ≥ 0 ( 23&    456 7) , P( x)

. I L. 3QP, :  W! "# I -V L. &  f " #$% (3 3 * + % ? − x ∈ D f , D f  < H C f X  I L. 9I f (c . I -V Y )3& L. 3QP,  [ 5,9]  [1,3]  -@ Y _ `a. [ 3,5]

. 3@B f A& f (− x) = f ( x) @ "# (∗ . & f A& f (− x) = − f ( x) @ "# (∗ . & 5 3@B 5  5 :  < E (a C/D  x n ‫زوﺝ‬n −x = x (b (− x) =  n − x ‫ دي‬n  G H>, )I 9 C f F8,( ! "# $%& "#  f  (2 n

. J3 

f ( x) = ax + b  !  (4

) H>, )I 9 C f F8,( ! "# $%& "#   f  (3 . K. (

R L.  f A& a > 0 ! "# (a

R L.  f A&

a > 0 !

"# (b

R L. 9I f A& a = 0 ! "# (c .  3%9>   f  F8, (d

     ! (5 . 3@B  f < 9 (a . –I L. 3QP, f A& I L.  f W! "# (∗ . –I L.  f A& I L. 3QP, f W! "# (∗ . &  f < 9 (b . –I L.  f A& I L.  f W! "# (∗ . –I L. 3QP, f A& I L. 3QP, f W! "# (∗

. − I = [ −b, − a ] A& I = [ a, b ] ! "# (c

. ‫رف دا‬# (IV : \H b x0 Y  QP  3P )H% f  : \H : : "# (1   Q%  3% d_   x0 L. Z 9c I - Y f ( x) ≤ f ( x0 ) . f ( x0 )

. ‫( ﺕ! ات دا أو رﺕ دا‬III N3* I  I‬و ‪ A‬و ‪B‬‬ ‫= ‪. k‬‬

‫ﺏ‪ h ( A ) = A ' +‬و ' ‪ h ( B ) = B‬و ' ‪ h ( I ) = I‬و " ?* ا‪0a‬‬

‫‪  . (5b‬ی ‪ [ AB ] C> I‬إذن ' ‪. [ A ' B '] C> I‬‬

‫‪. O ' = t u (O ) P‬‬

‫ﺡ‪/‬‬ ‫ ‬ ‫‪ t u ( M ) = M ' (a‬ﺕ‪. MM ' = u 76‬‬ ‫‪ (b‬إذا آن ' ‪ t u ( M ) = M‬و ' ‪86 t u ( N ) = N‬ن‬

‫‬

‫‪ (III‬ا 'ــــ ﺙ‪ 6‬ا'ـــــري‬ ‫‪ (A‬ﺕ‪. !"#‬‬ ‫ )∆( " ? ا?‪ *V‬ا?‪$‬ري اي ‪$‬ر )∆(‬ ‫ه‪ $‬ا!‪ #‬اي   ﺏـ ) ∆ ( ‪ S‬واي یﺏ(‬ ‫آ*  ‪ ( P )  M‬ﺏ  ' ‪ M‬ﺏ‪+‬‬

‫‪M‬‬

‫‪ (B‬ا (  ا'' &ة‬ ‫‬

‫ﺏ?‪ *V‬آي ‪ S Ω‬إذا و‪ ( 6‬إذا آن ‪. M ' N ' = − MN‬‬

‫‪ (C‬ﺥ ( ــــ ت‬ ‫‪ P?3‬ا‪0a‬ت ا? ‪  2‬ﺏآ ﺕ! ‪ 0 1‬ﺏ"! ‪ *V?2‬ا?آي ‪P‬‬ ‫ﺕ ‪$‬ﺏ‪ k c‬ﺏـ ‪  ، -1‬ا )‪ (6‬و )‪ + (9‬ﺕ>!@ ‪.‬‬

‫) ‪. C '(O ', r‬‬

‫‪. O ' = S Ω (O ) P‬‬

‫ﺡ‪/‬‬ ‫‪ S Ω (M ) = M ' (a‬ﺕ‪. [ MM '] C> Ω 76‬‬

‫‪(b‬‬

‫إذا آن ' ‪ S Ω (M ) = M‬و ' ‪86 S Ω ( N ) = N‬ن‬

‫‬ ‫‬ ‫‪. M ' N ' = − MN‬‬

‫‬

‫‪. M ' N ' = MN‬‬

‫ﺕ‪$‬ن ا ن ' ‪ M‬و ' ‪$0 N‬رﺕ ا  ‪ M‬و ‪ 12 N‬ا‪$‬ا‬

‫‪(9‬‬

‫‬

‫‪$0 (9‬رة ا ا‪O‬ة ) ‪ C (O , r‬ﺏ‪f‬زا ‪ t u‬ه ا ا‪O‬ة ) ‪. C '(O ', r‬‬

‫‪ (II‬ا 'ـــــــ ﺙ‪ 6‬ا'"آـــ&ي‬

‫‪$0‬رة ا ا‪O‬ة ) ‪ C (O , r‬ﺏ?‪ *V‬ا?آي ‪ S Ω‬ه ا ا‪O‬ة‬

‫‬

‫إذا آن ) ‪ ( D‬ی‪$‬ازي * ‪ ) u‬ی  ‪ D3$ u‬ـ ) ‪86 ( ( D‬ن‬

‫) ‪. t u ( D ) = ( D‬‬

‫‬ ‫‬ ‫*( " ?* ا ی‪ΩM ' = k ΩM C‬‬ ‫*( إذا آن ‪. (5b) *? " [ AB ]  K C> M‬‬ ‫‬ ‫‬ ‫*( إذا آ ‪. (5c) *? " AM = α AB H‬‬ ‫*( إذا آ ‪ M H‬ﺕ ‪. (10) *? " O3 PW‬‬ ‫)  ی ‪ M ∈ E ∩ F‬إذن ) ‪( h ( M ) ∈ h ( E ) ∩ h ( F‬‬ ‫*( إذا آ ‪  H‬ی ا>‪ T‬ا‪. D2? " 22‬‬

‫‪ (6‬ا?‪ *V‬ا?آي ی‪ 12 A6‬ا?"‪ 6‬ی  ‪.‬‬ ‫إذا آن ' ‪ h (A ) = A‬و ' ‪86 h (B ) = B‬ن ‪A ' B ' = AB‬‬

‫‪ P?3‬ا‪0a‬ت ا? ‪  2‬ﺏآ ﺕ! ‪ 0 1‬ﺏ"! ‪g‬زا ‪  ،‬ا )‪(1‬‬ ‫و )‪ (2‬و )‪ (3‬و)‪ (4‬و )‪ (6‬و )‪ (8cde‬و )‪ (9‬و )‪ (12‬و )‪ (13abd‬و ی ‪:‬‬

‫‪(8e‬‬

‫‪   (e‬د ‪$0‬رة  ‪ M‬هك  ة ‪W‬ق  ﺏ‪: D‬‬

‫‬

‫آ*  ‪ (P )  M‬ﺏ  ' ‪M‬‬ ‫ ‬ ‫‪M‬‬ ‫ﺏ‪MM ' = u " +‬‬ ‫‪ (B‬ا (  ا'' &ة‬ ‫ﺕ‪$‬ن ا ن ' ‪ M‬و ' ‪$0 N‬رﺕ ا  ‪ M‬و ‪ 12 N‬ا‪$‬ا‬ ‫ ‬ ‫ﺏ‪f‬زا ‪ t u‬إذا و‪ ( 6‬إذا آن ‪. M ' N ' = MN‬‬ ‫‪ (C‬ﺥ ( ــــ ت‬ ‫‪ (6‬ا‪f‬زا ﺕ‪ 12 A6‬ا?"‪. 6‬‬

‫‪ !  (d‬أن ‪ Ω‬و ‪ I‬و ‪ ? " J‬ی‪ N‬أن ! أن‬ ‫‪. h( Ω , k ) ( I ) = J‬‬

‫' ‪M‬‬ ‫‪ (A‬ﺕ‪. !"#‬‬ ‫ ‪  Ω‬ا?‪ *V‬ا?آي اي آ ‪Ω‬‬ ‫‪Ω‬‬ ‫ه‪ $‬ا!‪ #‬اي   ﺏـ ‪ S Ω‬واي یﺏ(‬ ‫آ*  ‪ ( P )  M‬ﺏ  ' ‪ M‬ﺏ‪+‬‬ ‫‬ ‫‬ ‫" ‪ ΩM ' = −ΩM‬ی  ‪. [ MM '] C> Ω‬‬

‫‪ (A‬ﺕ‪. !"#‬‬ ‫‬

‫‬

‫ ‪ . DE u‬ا‪f‬زا ا ‪ u DDE‬ه‬ ‫' ‪M‬‬ ‫ا!‪ #‬اي   ﺏـ ‪ t u‬واي یﺏ(‬

‫‬ ‫‪u‬‬

‫ی‪$‬ن‬

‫)∆( واﺱ( ا   ]' ‪[ MM‬‬

‫)∆(‬

‫‪M‬‬

‫' ‪M‬‬

‫‪.‬‬

‫‪ (B‬ﺥ ( ــــ ت‬ ‫‪ P?3‬ا‪0a‬ت ا? ‪  2‬ﺏآ ﺕ! ‪ 0 1‬ﺏ"! ‪ *V?2‬ا?‪$‬ري ‪،‬‬ ‫ ا )‪ (1‬و )‪ (2‬و )‪ (3‬و)‪ (4‬و ‪ (6) (5‬و )‪ (8e‬و )‪ (9‬و )‪ (13abd‬و ی ‪:‬‬ ‫‪ (6‬ا?‪ *V‬ا?‪$‬ري ی‪ 12 A6‬ا?"‪. 6‬‬

‫‪(8e‬‬

‫*( إذا آن ) ∆ ( ⊥ ) ‪86 ( D‬ن ) ‪. t ( ∆ ) ( D ) = ( D‬‬ ‫*( إذا آن ) ∆(‪86 ( D ) //‬ن ) ‪. t ( ∆ ) ( D ) //(D‬‬

‫‪$0 (9‬رة ا ا‪O‬ة ) ‪ C (O , r‬ﺏ?‪ *V‬ا?‪$‬ري ) ∆ ( ‪ S‬ه ا ا‪O‬ة‬ ‫) ‪. C '(O ', r‬‬

‫‪. O ' = S ( ∆ ) (O ) P‬‬

‫ﺡ‪/‬‬ ‫‪ S ( ∆ ) ( M ) = M ' (a‬ﺕ‪ ( ∆) 76‬واﺱ( ا   ]' ‪. [ MM‬‬

‫‪ (b‬إذا آن ‪ S ( ∆ ) (M ) = M‬ﺕ‪M ∈ (∆) 76‬‬ ‫ا?" ‪  0 ( ∆ ) L‬ﺏ  ‪.‬‬

‫ااء ا‬ : 678 6+3+  6+M  +O  v  u 6+ 2"+3 41(e



( I



   u .v = − u . v    u .v = 0 S8+Q u ⊥ v (f   u .v = v .u (∗ (g       u .(v + w) = u .v + u .w (∗       u .(v − w) = u .v − u .w (∗      (α u ).v = u .( α v ) = α (u .v ) (∗      (u + v ) 2 = u 2 + v 2 + 2u .v (∗      (u − v ) 2 = u 2 + v 2 − 2u .v (∗       (u + v ).(u − v ) = u 2 − v 2 (∗

A

. $%

H

= AC. AK ∧

= AB. AC.cos( BAC )     AB. AC = 0 678 AC 5 AB  / 01 2"+3 41 (2

 (II

: + *  +WW ( ABC ) 

B

C

'()* +,- (2

[ AB ] T A XY  I

A

 +WW ( ABC ) 

BC 2 : + *  2 1 BC 2 AI 2 = ( AB 2 + AC 2 − ) 5 2 2

AB 2 + AC 2 = 2 AI 2 +

B

I

C

H  [ BC ] XY 

.  ./ 01& 23 4  )* &  (3 A '  A Z T*% [\+B +WW ( ABC ) (a : + *  . ( BC )  A     (L^ +8 TB_ ) AB + AC = BC 2

2

2

(∗

BA = BH .BC = BH .BC (∗ 2

A

CA2 = CH .CB = CH .CB (∗ B H

I

B A

B

AH 2 = − HB.HC = HB.HC (∗ 1 AA ' = BC (∗ 2 C : + *  . A Z T*% [\+B +WW ( ABC ) (b AC BA cos Bˆ = cos Bˆ = BC BC AC A tan Bˆ = AB : + *  . +WW ( ABC )  (c sin Aˆ sin Bˆ sin Cˆ C = = BC AC AB C

( AB)  C     H  ( AC )  B     K    AC  AB    ! "   B : * +, -&  AB. AC # $ %&" ' ()     AB. AC = AB. AH

& (1

BC 2 = AB 2 + AC 2 − 2 AB. AC.cos Aˆ AC 2 = BA2 + BC 2 − 2 BA.BC.cos Bˆ AB 2 = CA2 + CB 2 − 2CA.CB.cos Cˆ



.    AC  AB  (1

K

   !"# (III A



C



D



.    CD  AB  (1

( AB )  C     C ' 

C A

( AB )  D     D '    AB.CD = AB.C ' D ' + *  B

:     /&?@   / 01 " AB.CD 9+0 :;  . +AB+, +C+ AD5 E   F " G + + AD5 E  +# H+(0I JG K! L+

 2

.  J#& * AB

  %&+# AB. AB  %&"(a (2

 2 AB = AB 2 + *  (b   : 678 M  N(" +O  CD  AB 6+ 2"+3 41(a (3   AB.CD = AB.CD   : 678 6+3+  6+M  +O  CD  AB 6+ 2"+3 41 (b   AB.CD = − AB.CD   6+ 3 6+3 41 8 41 6+Q +  CD  AB  61 P" (a (4   AB ⊥ CD R" . * +  (CD)  ( AB )     AB.CD = 0 S8+Q AB ⊥ CD + *  (b 678 T D  C  B  A   2"+3 41

(5

  A B C D AB.CD = AB.CD   UV " 3 C  B  A   v  u  (a (6     AC = v  AB = u      v C u .v = AB. AC : + *     v :    v  u  (b ∧  A B u.v = u . v cos(u, v ) + *  u   2 u u2 = u (c      u .v = u . v : 678 M  N(" +O  v  u 2"+3 41 (d

   

.,'$ %' (b I (ABC )  AB  " & ( P )  ( D )  / 5 , ( P )  ( D )  "> " 5 ,0 . ( D )  ( P )  ( D′)     ( Q )  ( P )     6 A05 $10 . ( P ) ≠ ( Q )  "> " & ( Q )  ( P )  : / ( Q )  ( P )  B  A >  5 , (* . ( AB ) / * ( Q )  ( P )    ( ∆′′)  ( ∆′)  A "> " 5 , (* . ( ∆′) // ( ∆′′)  ( ∆′′) ⊂ Q  ( ∆′) ⊂ ( P ) , A   ( ∆ ) / * ( Q )  ( P )   . ( ∆′′)  ( ∆′) $ 3!     " K  J  I  ,-.    7  ( Q )  ( P )  0B   "> &  ." & = & / A' 

θ

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