Θαλής 2010-2011
date post
29-Mar-2016Category
Documents
view
216download
0
Embed Size (px)
description
β ΜΚΔ16,32,248 8 = = . Από την υπόθεση έχουμε: α 86 υ 48 υ,όπου υ = ⋅ + = + ακέραιος με δυνατές τιμές από 0 μέχρι και 7 . Δοκιμάζοντας τις δυνατές τιμές του υ στην παραπάνω σχέση διαπιστώνουμε ότι μόνο για υ 1 = , ο αριθμός α 49 = που προκύπτει, είναι πολλαπλάσιο του 7. Άρα έχουμε α 49 = και β 8 = . 2 3 2
Transcript of Θαλής 2010-2011
( ) 34
106 79 . 3616532 - 3617784 - Fax: 3641025
e-mail : info@hms.gr www.hms.gr
GREEK MATHEMATICAL SOCIETY 34, Panepistimiou (leftheriou Venizelou) Street
GR. 106 79 - Athens - HELLAS Tel. 3616532 - 3617784 - Fax: 3641025
e-mail : info@hms.gr www.hms.gr
71
, 30 2010
1. 2 3 5x 3 4 2 : 4 2= + 2 3 2y 4 5 4 7 3= + . () x y . () , x y . ()
2 3 5x 3 4 2 : 4 2 9 4 8 : 4 32 9 32 : 4 32 9 8 32 33= + = + = + = + = . 2 3 2y 4 5 4 7 3 4 25 64 7 9 100 64 63 99= + = + = + = .
() x, y . ( ) 33,99 33= , 33= . 2. , . 6. , 7, 16, 32 248. , 16, 32 248.
16 32 24816 0 80 0 8
,
( ) 16, 32, 248 8= = . : 8 6 48 , = + = + 0 7 . 1= , 49= , 7. 49= 8= .
2
3. . . . 0 70= 0 130= , : ) . ) o . . / / : o1 70= = ,( , ). / / , : o o o1 180 130 50= = = . ( 1 , , ). ,, , :
o o o o o o 180 180 180 70 50 60+ + = = = = . . , :
oo
1 70
352 2
= = = . , / / , : o1 1 I 35= = , 1 1 I , , I .
1
, : o
o1
50 25
2 2= = = .
E / / , : o2 1 I 25= = , 2 1 I , E, . 4. . 80 , 120 . 2600 . 10% , : . .
3
. 150% . . 120+80=200 , 2600:200=13 . 120 120 13 1560 = . 101560 156
100 = .
. x ,
150 3xx100 2 = .
3x 260080 x 120 2600 80x 180x 2600 260x 2600 x 102 260
+ = + = = = = .
3 10 152 =
, 120 15 1800 = 101800 180
100 = .
4
1. ( )2x y 3 2+ = 6 44 63 3y w
5 5
= ,
: 7x 10y 3w 87= + . ( )2x y 3 2 3 4 12+ = = =
6 44 6 24 24 24 24
2424
3 3 3 3 3 5y w5 5 5 5 5 3
3 5 1 1.5 3
= = = = = =
.
:
( ) ( ) 7x 10y 3w 87 7x 7y 3y 3w 87
7 x y 3 y w 87 7 12 3 1 87 84 3 87 0.= + = + +
= + + = + = + = 2. , : () 4, () , () 5 , () 1. xyzw 1000 x 100 y 10 z w= + + + . , () w 0 4 8= , () z 0 2 4,= . , () y 1 5= . :
x100, x124, x148, x500, x524, x548 . () 4500, 4524, 4548 , 0.
3. 0 120= . x y , , , . 0 120= , 0 60= 2 3 , : . . . . . .
5
. 1
o2
120= = , o1 60= . o1 60= . .
2
. (x) E (y) , : 0 o1 3 90 120 90 30= = = =D D . : = ( ),
o2 2
120= = o1 3 30= = . , = . 60 = =D , 0 30= = , . . . H :
2 2 2 = + ( )2 22 2 34= +23 12 4
4 = = .
H :
2 2 2 = + , ( )2 22 3 2 32 = + 2 48 4 3 = = . 12 8 3+ . 12 , 3 2 3
3+ .
4. 2 . 1 ( )C , , , . 2 ( )C , . . ( ) ,, ( )C , ( )C , . . ( ) ( )1 2 1 2 ,
6
( )( )12
135
.
. x ( )C , x ( ) ,, .
3
() 2 2 2 2 2 2 2= + = = ,
( )( ) ( )2 2 2 2 2 ,, 2 2 = = = . () 1 2 , .
( ) ( ) ( ) ( )2 2 2 2 21 2 2 2 4 2 4 = = = . 2 , ,
( ) ( ) ( )2 2 2 2 22 2 4 4 = = = . ( )( )12
2 4 4
= : ( )( ) ( ) ( )12
2 4 13 725 2 4 13 4 23 72 3,1304 4 5 23
= > > > > , , 3,14 . () x 2< < ,
( ) ( )2 2 2 2 2 2 2 22
2 2 2
2 x x 2 x x
3 32x 3 x x .2 2
= = = = =
.
7
1. -
( )22 x x 1x 1 x 1x 5x 14, .2 4 4
= + < 2x 5x 14 = ( )x x 5 14 = . , x 14. { }x 1, 2, 7, 14 . 7 -2. , 2x 5x 14 0 = , 1, 5, 14= = = , 2 4= =81 x 7= x 2= .
( )2 2 2x x 1x 1 x 1 2x 2 x 1 x x 3x 3 x 12 4 4
+ < + < < < . x 2= . 2. ,, , , :
4 3 2 2 2 2 2 3 2 4 2 2 2 2 4 2 2 = + + + .
( ) ( ) ( )( ) ( ) ( )
( )( ) ( )( )( ) ( ) ( )( )( )
4 3 2 2 2 2 2 3 2 4 2 2 2 2 4
4 3 2 2 2 2 2 3 2 4 2 2 2 2 4
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 22 2 2 2 2 2 2 2
2 2
2 2
2 2
2
2
= + + += + + + + = + + + + = + +
= + + = + = + ( )( ) .+ + +
3. :
x 4 x 1 2 5 x1 ,2 y 2 3y 3 3
= = + .
1 wy= , :
8
x 8w 2 x 2 8w x 2 8w x 2 8wx 4w 13 x 13 4w 2 8w 13 4w 8w 4w 13 2
11 x 2 2 11 x 24x 2 8x 2 8w 4 .11 114w 11 11 w ww 4 44
= = + = + = + = = + + = + = = + == + = + = = = =
( ) 4x, y 24,11
= 4. ( )= . () . () , = < . . ( )= ( )= . = , . ( = ). , 1 2 = = . , 3 4 = 3 4 , 1 2 , .
4
: 1. = ( ). 2. = ( ).
9
3. 3 4 = ( 1 2 , ). = = . : 1. = ( ). 2. = ( ). 3. = ( ). = . : 1. = ( ) 2. = ( ) 3. = + = + = 1 1 = . , 1 1 = , , . , .
10
1. x, y, z
2
2
2
x y z x 2
y z x y 2
z x y z 2,
= = =
x y z 6+ + = x, y, z . : x 2, y 2 z 2 , 2 2 2x y z, y z x z x y. + + +
2 2
2 2
2 2
x y z x 4x 4 4x y z 4y z x y 4y 4 x 4y z 4
x y 4z 4z x y z 4z 4
= + = = + + = + = = + , (1)
: x y z 6+ + = . x, y, z (1),
( )( )( )
5x x y z 4 5x 6 4 x 25y x y z 4 5y 6 4 y 25z x y z 4 5z 6 4 z 2
+ + = = = + + = = = + + = = = .
, x 2, y 2 z 2 , x 2> , x y z 6+ + > , . x y z 2= = = . 2. 1c (,) ( 1R = ) 2c (,) (
2R = ). 1c (,) . 2c (,) . . . . = o 30= , .
11
. ( ) 1c (,) , o 90= . ( ) 2c (,) ,
o 90= . , , / / = . o 90= = , / / . o 90= = .
5
. . o 30= , .
2= = = ,
, = , = , , . 3. x, y x y 4+ = , :
( ) ( )2 22x 1 2y 1 25x y+ ++ .
; :
12
2 24x 4x 1 4y 4y 1 25x y+ + + ++
( ) 1 1 : 4 x y 8 25x y
+ + + +
: 1 1 1x y+ .
.
( ) 1 1x y 4x y
+ + , x y 4+ = , ( ) 1 1 x y x yx y 2 2 2 4
x y y x y x + + = + + + = .
x y 2= = . ,
( )
( )2
2
1 1 x y1 1 xy 4 x 4 x 4 x 4x 4 0x y xy
x 2 0, .
++ +
x y 2= = , x y 2= = . 4. o( 90 )= . . , . E A , :
13
6 = .
1 1 2
= = .
, 1 2 2
= =
1 1 2
= = , 1 2 2= = . = , . :
1
2
=
, 1
2
=
. 2
= , = = ,
. , . , , .
14
1.
( ) ( ) ( )3 3 32 2 22x 3x 1 x 3x 2 7 x 1+ + + + = .
(1 ) 2 2a 2x 3x 1, b x 3x 2= + + = + + , 2a b x 1 = :
( ) ( )( ) ( )( )( )( )( )
3 33 3 2 2
2 2 2 2
2 2
2 2
2 2
a b 7 a b a b a ab b 7 a b
a b a ab b 7a 14ab 7b 0
![Coelho Paulo - †»µ† [2010] - ›™’‘—£ 2011](https://static.fdocument.org/doc/165x107/563dba40550346aa9aa3ffd9/coelho-paulo-2010-2011.jpg?t=1657044005)
