Α ΄ Α ΄ Μαθηματικό περιοδικό για το ΙΟΥΛΙΟΣ - ΑΥΓΟΥΣΤΟΣ - ΣΕΠΤΕΜΒΡΙΟΣ 2013 ευρώ 3,00 Ιστορία της Γεωμετρίας Ιστορία της Γεωμετρίας Βαλκανιάδα Νέων 2013 Βαλκανιάδα Νέων 2013 Διακρίσεις για την Ε.Μ.Ε. Διακρίσεις για την Ε.Μ.Ε. 89 Ελληνική Μαθηματική Εταιρεία Μαύρες τρύπες Μαύρες τρύπες ΠΛΗΡΩΜΕΝΟ ΤΕΛΟΣ ΠΛΗΡΩΜΕΝΟ ΤΕΛΟΣ Ταχ. Γραφείο ΚΕΜΠ.ΑΘ. Ταχ. Γραφείο ΚΕΜΠ.ΑΘ. Αριθμός Άδειας 4156 Αριθμός Άδειας 4156 ΕΛΤΑ Hellenic Post Hellenic Post ΕΚΔΟΤΩΝ ΕΝΤΥΠΟ ΚΛΕΙΣΤΟ ΑΡ. ΑΔΕΙΑΣ 1099/96 ΚΕΜΠ.ΑΘ. ΕΝΤΥΠΟ ΚΛΕΙΣΤΟ ΑΡ. ΑΔΕΙΑΣ 1099/96 ΚΕΜΠ.ΑΘ. 30 ο 30 ο Καρδίτσα 8-9-10 Νοεμβρίου 2013 υκλείδης Γυμνάσιο Ε 95 Ε.Μ.Ε. Χρόνια 0 5 25 75 95 100 0 5 25 75 95 100 0 5 25 75 95 100 0 5 25 75 95 100
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; 1 : ( ! ). , . ( ) : 2 2 2 26 4 2 (3 2) . 1: , . ( 1) . 2: , ( ), .. ( )= ( ) . .. 2 2( ) =( ) . :
22(x 2y) (2y x) (2y x) (x 2y) (x 2y) (x 2y) = (x 2y)( (x 2 y) . 2 : 2 : 2 2 ( )( ) . 1 : 2 2 2 24x (x 3) (2x) (x 3) 2x (x 3) 2x (x 3) =(2x x 3)(2x x 3) (3x 3)(x 3) (x 1)3(x 3) 3(x 1)(x+3). 2 : 2 2 2 220x 45xy 5x(4 9y ) 5x 2 (3y) 5x(2 3y)(2 3y). 3 : , . - , . 1: 2 2(x 2) x 4 =(x 2)+(x 2)(x 2)=(x 2)( x 2). , - . 2: (x 2) 3 x 2 3 x (x ). - .
. 2x 1 , 2 2x y ( )
i) 2x xy x(x y) . ii) 2 23 3 3( 1)=3( 1)( 1) iii) 3 3 2 2 2 22x y 8xy 2xy(x 4y ) 2xy x (2y) 2xy(x 2y)(x 2y). iv) 2 2 25( 2 ) 20 5 ( 2 ) 2 5( 2 2)( 2 2). v) 2 2(2x 3y) (x y) (2x 3y x y)(2x 3y x y) (3x 2y)(x 4y). 3: 4 . : 3 23 32 +3 (+)3 332 +32 3 ()3. 53 302+6040 =5(362+128)= 5(3322+32223)=5(2)3 4: 2 2 -, . - 5 . 33+622 = 3(2+2) (2+2)= (2+2) (3) .
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89 .1/26
5: - . 1: 2+2+2 2 = (+)2 2 =(+)(++)
2: x22+21 , - , 5
x2(22+1)= x2( 1)2=[x(1)][x+(1)]=(x+1)(x+1) (6 8) .
: 202+60+45 2. 23+22 3. x27x+10 4. (22)+(22)+(22)
1. 202+60+45 = 5(42+12+9)=5(2+3)2 2. 23+22=( )= 22+22 =(2)(2)=(2)() 3. x27x+10=( )= x22x5x +10= x(x2)5(x 2)= (x2)(x5)
- : x23y2+2xy=(1 ) = x2y22y2+2xy=(xy)(x+y)+2y(xy)=(xy)(x+3y) x23y2+2xy=(2 ) = x2+2xy+y24y2=(x+y)2(2y)2=(x+y2y)(x+y+2y)=
(xy)(x+3y) x23y2+2xy=(3 , x) = x2+2yx3y2 (=16y2, x1,2=y 3y) -
x23y2+2xy=(xy)(x+3y) 4. (22)+ (22)+ (22)=22+ 22+ 22 =( ) =
22+ 22+ 22+ =22+2 2+2+2 = =()()2 ()+()=()(2+)=()[()()]= ()()().
1. .. 11, 25 : 112 = 121 252 = 625, - 422 = ;. (10.4+2)2
102.42+2.10.4.2+22=1600+160+4=1764. 5 X (10X +5)2=102. X2+2.10. X.5+52 (10X +5)2=100X2+100X+25=100 X(X+1)+25 100 X(X+1) 25 25.
852 = 4 (8.9)=72 25 7225. 2. -
. 2, 3, 4 5 : 2.3.4.5=120 1 121
121=112. x, x + 1, x + 2, x + 3 : x(x+ 1)(x + 2)(x+3) + 1 =x(x2 + 3x + 2)(x + 3 )+ 1 = [x(x2 + 3x) + 2x](x + 3 ) + 1 = [x2(x + 3) + 2x](x + 3) + 1 =[x (x + 3)]2+ 2x(x + 3) + 1 =[x (x + 3) +1]2
3. 2 + 1, 2( + 1), 2 + ( + 1)2 -; . : 2+(+1)2=22+2+1>22+2=2(+1) 2 + ( +1)2 =22 + 2 + 1 > 2 +1
2+(+1)2. - :
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y x .
x, y ; (1 ) , 0x y
2 2
2 2 2 2
2 2
2 2 8 2 2 4 8 0
4 4 2 4 2 0 4 4 2 2 1 0
2 2 1 0, , 0.
xy y x xy xy y x y x xy
xy xy x x y xy y x y y y x x
x y y x x y
: 2 0 2 1 0 1, 2, , 0.x y y x x y x y (2 )
2 2 8xy y x xy (1)
2 , , 0,a b ab a b ( a b ) , , 2a xy b , 2a y b x ,
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89 .1/36
2 2 2xy xy , (2) 2xy , 2 2 2y x xy , (3) 2y x . (2) (3) , 2 2 8xy y x xy . 2xy 2y x , 1, 2.x y 2 n , . () n, n 2n. (). n, 5n. () n . n n . n , 1 2
3 6n n n n .
1 2 2 1 2 12 1 4 56
n n nn n n n n
. () n . , .
1 2 3!
4 4! 4 ! 24n n n n nn
n .
1 2 3 5 1 2 3 12024
n n n n n n n n 31 2 3 2 3 5 4 5 6 1 6 7. n n n n n
3 p x, y , , p , x, y : p x 2 x y 1 . (1) x y 21 , x, y,p (1). 2 0 2,x x 1y , 2,1x y . 2 0x , , p , 2 1p x x y : 1y 1.p x p y
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89 .1/37
,p x , x px x , :
2 1 2 1 1 2
1, 1 2 2, 1 1 1, 1 2,, 1 2 , , , 1 , 2 ,
p px px y px x y x p yx p y x p y x y p x y px p y p x p y p x y p p x y p p
1,p y 1 ,y py y , (1) : 2 2 1 2 1, 1 2 2, 1 1 p x xpy x xy x y x y x y 1, 1 () 1, 0 ().x y x y 21x y , :
, , 1x y p p , 2 1 21 11p p , , , 11,10,11x y p .
, 2 ,x y p p , 3 21 7p p , , , 14,7,7x y p . 4 c(O,R) ( O R), AB . 1c (K,KO) ( K OA KO R ) , 1c (K,KO) . K c(O,R) Z ( ). Z AB N M . EMZN , c(O,R) .
ZEE1 , : 1 1 90o E . (1) 1OZ ZEE1 , 11 E2O ZO :
1 12 1
180 180 2 902 2
o ooE . (2)
(1) (2) , : 21
. , :
1) ( o90 ). 2) ( 21 ). 3) ( , )c( 1 , ). , , K . )A( , :
, , . .
, : 1 2 .
1
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89 .1/38
EM , : 21 . , : NZM , , . , , . T ( EM ) EZ . T Z OT EZ . T .
88 12. . . N ab , N , :
100 10M Nab N a b . : 100 10 10N M N N a b N a b ab N ab . ,a b , N 98. 98N , 98ab
9898 98 101M 98. N 98. 13. n
n 1 n 3 n 5 n 2011 n n 2 n 4 n 2010 n , 2 4 2010n n n n , 1 3 5 2011n n n n , 1 3 5 2011 2 4 2010n n n n n n n n . , n , . 17. n m2 2 , n,m . 3n m , 32 2 2 2 1 2 9n m m m , m ,
2m k n , : 222 2 2 9 2 3n m k k . , , 2n m n k 22 2n k . 12. ABCD M,L,N K AB.BC,CD DA , . BMN MNC . : () DKL CLK . () ABCD . 2010-11 () , , M L N K . , AB BC CD DA , , KM DB LN KN AC ML , KMLN .
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89 .1/39
ABCD ABD ACD , : AMK ABD ACD KND (1) (1) BMN MNC , 180 180KMN AMK BMN KND MNC KNM . (2) KMN KM KN , KMLN , NKL NLK . (3)
, KM DB LN KN AC ML , DKN NLC . (4) (3) (4)
DKL DKN NKL NLC NLK CLK . () KM KN ,
2 2DB KM KN AC . ACS BDS ,AC DB DBS DBA ACD ACS , AD
180 180BDS BDC CAB CAS . ACS BDS ,
AS DS . SAD SDA 180 SAD 180 SDA DAB CDA 180ABC CDA 180ABC DAB BC DA .
18. ,x y ,x y 2 214 65 10 34 7x x y y . 19. , , ,a b c d
1a b c db c d c d a d a b a b c
,
2 2 2 2a b c d
b c d c d a d a b a b c .
20. , ,a b c , : 3 3 3 2 2 2a b c a b b c c a
b c c a a b . .
14. n
2 7 162 3 4 5 24
n nnn n n n
. 15. ,p q ,p q p q ,p q p q . 16. ,x y ,
2 2
3 3
4x xy yx y .
13. . . , 2 .
1
-
89 .1/40
, 2
, .
( , ) Fermat, Bernoulli, Pascal . . , 20 Borel Neumann (1928) .
, .
, : ) ) )
, . , , , ) )
. , . .
, ,
. . (), . 2 . .
. , . (. ) . :
1 2 1 f11 f12 f1v 2 f21 f22 f2v f1 f12 fv
1, 2, , 1, 2, , . f11, , f . , . . . -- :
1 () 2 () 3 () 1 () 0 1 -1 2 () -1 0 1 3 () 1 -1 0
,
, . . , , .
, :
1 2 3 min 1 4 7 2 2 2 3 8 2 2 3 1 1 1 1 4 5 1 -4 -4 max 5 8 2 2
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89 .1/41
.. 2 (. 2 ), , 2 (. 3 ). , , . .. 2 (. 2 ) , 8.
, (min) , , , . ( min(max) = max(min) ), (von Neumanns Minimax Theorem). min(max) = 2 = max(min) (2, 3) (1, 3), , (. 2 ). :
! minimax
( ), . , , . .
(Neumann): V V . , V.
: V = . , , . . ( ) . , ( ) . ( ).
2x2 . ..
: x
1, 1-x A2. , y 3, 1-y 4. , : V(A, B3) = 2x 5(1x) = 7x 5 V(A, B4) = 2x +0(1x) = 2x V(A, B3)=V(A, B4) 7x5=2x x=5/9, 1x=4/9.
, : V(B, A1) = 2y 2(1y) = 4y 2 V(B, A2) = 5y +0(1y) = 5y V(B, A1) = V(B, A2) 4y2=5y y=2/9, 1y=7/9. , (x y) V(A,
B1) V (A, B2) V(B, A1) V(B, A2), V=10/9. , ,
() 10/9 ( 1,11%) 1,11 . 1: 2x2. 2: : (5/9, 4/9) (2/9, 7/9) . 5 1 4 2 2 3 7 4 9 . 3: . 4: , . = . . -- . 5: 2 ) ) .
1 2 3 1 -3 1 0 2 2 3 4 3 2 4 0 4 1 1 2
3 4 1 2 2 2 5 0
3 (y) 4 (1y) 1 (x) 2 2 2 (1x) 5 0
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89 .1/42
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1. . 2
7
. 30 - 20 . ; 2. . , 3 1 - . 20 . ;
3. 7 . - -. ;
4. , . , . ;
5.
. . -. 5 , - ;
6.
5 - . - - 28 -;
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89 .1/43
7. , 3 5. , 8 . - ;
8. () 1, 3, 9 27 . - 1 40 ;
9.
- . 25 10. 5 , 2 . 3 9 27 2 ,
27 2 29 ;
10. . ;
-
89 .1/44
1
2
1 1 1 . 1 , 2 . , 1, 2, 3, 4, 1 2 , , :
,
, , , , . . , . , , , , , , .
. , , , . , ,
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----------------------------------------------------- 2 ------------------------------------------------
89 .1/45
. : . / , /, , , , , , , , 1+2+.... 99 Gauss, Mobius, ( ), , , . (+) = +2+, Pascal, , , , , ESCHER, .
. NaCl . , . . : . ;
,
, . .
-
----------------------------------------------------- 2 ------------------------------------------------
89 .1/46
, -.
Gauss, Pascal, Mobius, Durer, Escher . . , , , . videos powerpoint / , . GeoGebra .
, , .
, . , , -, , , , , , , ,
-
----------------------------------------------------- 2 ------------------------------------------------
89 .1/47
. , .
,
. . 1 . 3 .
1 : , ,
.
. . .
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89 .1/48
============================================================================ 1.
. 1 , ( , , , ...). ; 2.
10:45 .. , 6 Km . 2 Km , 2 2 Km . - 2 Km/h , 3 Km/h 4 Km/h . 3 Km/h, 8:40 .. - 5 - . ; ;
3. # 1 # 1 = 2 3 # 5 = 34 6 # 9 = 117 10 # 14 = 296 15 # 19 . 4.
. - : , - , 64- . ; 5.
3m , 3m 3m 2 , 2 , - 2 - 6m , 6m 6m , ;
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89 .1/49
1.
37 ! -, -, , -:
1 10 (2, 3, 4, 5, 6, 7, 8, 9, 10, 11). 2 11 (1-3, 3-4, 4-5, 3-6, 6-7, 4-7, 5-8, 7-8, 7-10, 8-11, 10-11). 3 6 (2-4-5, 5-8-9, 3-4-5, 6-7-8, 4-7-10, 5-8-11). 4 3 (3-4-6-7, 4-5-7-8, 7-8-11-12). 5 3 (1-2-3-4-5, 2-4-5-7-8, 4-5-7-8-9). 6 2 (3-4-5-6-7-8, 4-5-7-8-10-11). 7 2 (2-4-5-7-8-10-11, 3-4-5-6-7-8-9). 2. - . (2) : 3 : 3 3. (4) : 3 - 3. , 3 - 9, 1, 2, 3. : i. 3 1 -1 1 3 ii. 6 2 -2 2 6 iii. 9 3 -3 3 9 (5) i iii , ii =9 - 6297926. 3. += -=. =0, :
2-1=1 4-2=3 6-3=3 8-4=4 10-5=5 =0. =5. =4 (50-45=5) - (50+45=95) =9. 4. 1 9 . 9.
1 3 6 2 =
9 8.
1 4 6 3 =
8 7 - , , 7 - , . 6.
1 2 9 3 =
6 5 , 7. 4.
1 3 8 6 =
4 3. 3 :
1 2 9 6 =
3 2.
1 3 6 9 =
2 1 - 3
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89 .1/50
. : 2, 3, 4, 6, 8 9 5. 6 - , .
- , , -.
- , , -.
- , , -.
-
, , .
- , , -.
.
........ !!!!!! H 88
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