ΠΡΟΤΕΙΝΟΜΕΝΕΣ ΑΣΚΗΣΕΙΣ ΓΙΑ ΜΑΘΗΤΙΚΟΥΣ ΔΙΑΓΩΝΙΣΜΟΥΣ 601...

http://www.mathematica.gr/forum/viewtopic.php?f=109&t=15584

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: xr.tsif 2

7

601 - 700

: xr.tsif


: xr.tsif 3

601 ( )

: 2n 2n 1 2n 2A 10 10 10 ,

(n N* ).

602 ( )

() : x,y 0 x y , : x y

xy2

.

() : 2010 2013

1 2 3 ... 20104

.

603 ( )

, 6 .

,

, .

;

604 ( )

,

:

1001 1001 1000 1000 1001x 1001 ,y 1000 1000 ,z 1001 1000 .

605 ( )

x,y N , 2 yx 2069 2 x , x,y .


: xr.tsif 4

606 ( )

: 2 2m m m m 8 2

x 5 9 ,y n 2(15p 1) , *m,n,p N .

xy .

607 ( )

n , :

n n 1 n 2 n 3A 2 3 5 7

, .

608 ( )

1,2,3,... :

1 2 3A (1,2),A (3,4,5,6),A (7,8,9,10,11,12),...

11

A .

609 ( )

: 2011 2010 4020 4023 4024x 5 5 ,y 2 2 2 .

: (

) Bernoulli,

a 1 *n N , : n(1 a) 1 na .

610 ( )

x,y,z :

2x y 2

2x

2 11 2y 1 z 1

4 9 3y 1 3z 1

, .


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611 ( )

ab , :

7 ab ba abba :11 1 .

612 ( )

7,8,9 1,4,7 .

613 (Socrates)

: a,b,c 0 2 2 2

(a bc)(b ca)(c ab) abc(a b)(b c)(c a) .

:

,

, (

).

,

:

1 2 n 1 2 n

x (x ,x ,...,x ),y (y ,y ,...,y ) n-

1 2 nw (w ,w ,...,w ) n-

1 2 nw ,w ,...,w

1 2 n

y ,y ,...,y , , :

n- x y , 1 2 nx x ... x

1 2 ny y ... y ( ), :

1 1 2 2 n n 1 w1 2 2 n n(x y )(x y )...(x y ) (x )(x w )...(x w ) .

, n- x y , ,

, .


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614 (Socrates)

x,y,z zx y 2

(2 1)(2 1) 2 1 .

615 (Socrates)

m n 3m 4n 100 .

|m n | .

http://forum.gil.ro/viewtopic.php?f=38&t=5617&start=0.

616 (Socrates)

a,b,c 5 5a |b ,b | c 5c | a . 31

abc | (a b c) .

617 (Socrates)

(x 1)yz 12

(y 1)zx 4

(z 1)xy 4

.

618 (Socrates)

a,b,c . ab bc ca 1

(a 1)(b 1)(c 1) 4 .

619 (Socrates)

a,b,c 2 2 2 3 3 3

a b c a b c a b c .

( )

1 1 1

a b c 2a b c

.


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620 (Socrates)

.

.

621 ( )

x,y 0 , x y 5 5x y x y , : 4 4x y 1 .

622 ( )

20137 .

623 (Socrates)

a

| x | x | a | | x | x | a | 2 .

624 (Socrates)

x,y 3 3x y 2xy . 2

3x,y 2 .

625 (Socrates)

a,b a 2b b 2a

. 3 / a,b .

626 (Socrates)

a 2x (a 2)x 1 3 | x | .

627 (Socrates)

n 2n 1 3n 1

. 5n 3 .


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628 (Socrates)

.

.

629 (Socrates)

1

2;

630 (Socrates)

n a b

(a 1)(b 2) n 2 .

2n .

631 (Socrates)

1,2,...,9 . ,

a b , 2 2a 7ab 3b .

8 .

.

2010 ;

632 (Socrates)

2

.

633 (Socrates)

2 2 2 21(a b c) a b c 2(a b 1)

3 .


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634 (Socrates)

C {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

S {4,5,9,14,23,37} .

A B

() A B

() A B C

() A S .

() B S .

635 (Socrates)

S(n) ( ) n .

, S(2012) 2 0 1 2 5 .

n 0 n S(n) 9990 .

636 ( )

A(n) 1 3 5 ... (2n 1) B(n) 2 4 6 ... (2n) ,

*n N . : A(2)A(3)...A(n) 1

B(1)B(2)...B(n) n 1

.

637 (Socrates)

1

| a 2 b |2(a b)

a,b .

638 (Socrates)

p p 45 4p

.


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639 (Socrates)

6 . ,

A B ,

.

A 10 '

B , 9 .

; .

640 (Socrates)

2

2 2 2

1 1 1 1 1 1 1

2a bc 2b ca 2c ab 9 a b c

,

a,b,c .

640 (G.Bas)

:

a,b,c 0 2

2 2 2

1 1 1 a b c

2a bc 2b ca 2c ab ab bc ca

.

Cauchy Schwarz.

641 (Socrates)

2

3 3 3

1 1 1 (ab bc ca)

a bc b ca c ab 6

,

a,b,c abc 1 .


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642 (Socrates)

a b c a b c

a b b c c a 2

,

a,b,c 2 2 2a b c a b c .

643 (Socrates)

4 5 6x y (x y) 2

x,y [0,1] . ;

644 (Socrates)

2 2 2

a b c1

a 2 b 2 c 2

,

a,b,c abc 1 .

645 (Socrates)

2 2 2 2 2 2

x y z

2(x y ) 2(y z ) 2(z x )

2 2 2 2 2 2

2 2 2 2 2 2

4x y 4y z 4z x9

x 4y y 4z z 4x

,

x,y,z .

9 ;

646 (Socrates)

2 2

2(x y 2) y(x y 2)

x (y 1) y (x 1) xy 1

.


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647 (Socrates)

2 2 2

2x 2y 2z3

1 x 1 y 1 z

x,y,z 1 1 1 1

x y z xyz .

648 (Socrates)

11, ,

96 . 13 .

649 (Socrates)

abcd

abcd 1 (ac 1)(bd 1) (1) .

650 (Socrates)

a,b,c

2 2 2a b c 26, a b 5, b c 7 .

651 (Socrates)

) 1,2,3,...,10 , ,

5

;

) 1,2,3,...,20 ,

10

;


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652 (Socrates)

8 9 10

{x} x [x] (1) .

653 (Socrates)

10 , , ,

, x,y (1 x)(1 y)

0 x y9

.

654 (Socrates)

, s , 7 .

.

s

24.

655 (Socrates)

0 1 2 na ,a ,a , ,a , :

[*] n

a 5

n 1

a

( n 1

a

.)

[*] n 1 na a9

.

0

a ;

656 (Socrates)

1 2 2009x ,x , ,x { 1,1} ,

1 2 2 3 3 4 2008 2009 2009 1x x x x x x x x x x 999 ;


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657 (Socrates)

n 3

2

n 3Z

n 7

.

658 ( )

: 2[x] 4 3x , x R .

659 (Socrates)

n n7 147

.

660 (Socrates)

m,n m n 1

2 22(m n ) 1 . m n .

661 (Socrates)

1000

.

662 (Socrates)

1 2 3 4 5x ,x ,x ,x ,x

2 1 3 2 4 3 5 4 1 5| x x | 2 | x x | 3 | x x | 4 | x x | 5 | x x | .

1 2 3 4 5x x x x x .

663 (Socrates)

24 .

.

.


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100 ;

664 (Socrates)

100 1 2 100

M ,M ,...,M .

, A B ,

1 2 100AM B,AM B,...,AM B 241 .

665 (Socrates)

268 .

20 75 .

17,83 144 3,4 9 ,

210 .

666 (Socrates)

: 3 3 3

| x | | y | 1340

x y 2010xy 670

.

667 ( )

: 20141

A (1 11 111 1111 ... 11...1) 1081

,

11...1 2013 .

668 (Socrates)

a,b 1 . 3a 3b ,

a,b .

669 (Socrates)

) 2 148 149a 1 3 3 ... 3 3 99

b 1 5 ... 5 .


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) 2013 2013200 243 2013 2013216 225 .

670 (Socrates)

abcd 2 2 2 2 6 6 6 6

a(a b c d)(a b c d )(a 2b 3c 4d ) abcd .

671 (Socrates)

1,2,3,4,...,2011,2012

11.

672 (Socrates)

2 2 2

1 1 11

x yz 3 y zx 3 z xy 3

x,y,z 2 2 2x y z 9 .

673 (Socrates)

a,b,c 2 2a 4b c . 2a 2b

.

674 (Socrates)

20132 2 ,

20142 .

675 (Socrates)

d A,B .

2013 , A,B

AB .


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A

B ;

676 (Socrates)

a,b,c 2 2 2a b c 20122012 ;

677 (Socrates)

2 2a 2b b 2a a | 2a 1 2b | .

678 ( )

: 0 a x a 1 (1) , a x

. : 2 2

x a x a 1a x a

2a 1 2a 1 8a 4

.

(*) To a (1) , x

.

679 ( )

n N , 5n n 30 .

680 ( )

N (6x y z)(x 6y z)(x y 6z) , x,y,z N .

() N .

() 5 | N , 250 | N .

681 (Socrates)

1 2 n n 1x x ... x x

n n 1 1 2 nx x 2(x x ... x )

.


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682 (Socrates)

:

x(x 1) yz

y(y 2) zx

z(z 3) xy

.

683 (Socrates)

2 2 2

2 2 2

a b c 3

a b b c c a 2

,

a,b,c a b c 3 .

684 (Socrates)

2 2 2x xy xyz 4xyz 4 ,

x,y,z .

685 (Socrates)

a,b,c 2 2 2

x (a 1) 4b, y (b 1) 4c, z (c 1) 4a .

x y z 2013 , a,b,c .

686 (Socrates)

n

(a ),n 1 1

1a

6

n 1 n

n 1 1a a

n 3 2

.

2010

a .

687 (Socrates)

X ( ) 9

.


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A,B,C,D,E 2 2 2 2 23cm ,9cm ,1cm ,2cm ,8cm

, X .

http://forum.gil.ro/viewtopic.php?f=38& ... bad5725d55

688 (Socrates)

a n 0 b a6 1 bb b , b n .

http://forum.gil.ro/viewtopic.php?f=36&t=5809&start=0

689 (Socrates)

p q r 25pq r 2004

pqr 1 .

p q r ;

http://erdos.fciencias.unam.mx/omm/primer18omm.pdf.

690 (Socrates)

m,n N p 5 2 n

m(4m m 12) 3(p 1) .

691 (Socrates)

9,10 11 .

692 (Socrates)

x,y,z

2 2 2 3x y z xy yz zx 2(x y z)

2 .


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693 (Socrates)

n

(a ) na 12n p 12 , p

.

.

694 (Socrates)

(x,y)

2 23 1(x y ) xy 83

8 4 .

695 (Socrates)

A {1,2,3,...,2012}

,

A .

696 (Socrates)

a,b,c

2 2 2 2 2 2 3 3 3a b a c b c 4 a b c , ab ac bc 2 abc .

;

697 (Socrates)

80%.

80%.

80%.


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698 (Socrates)

[x] [2x] ... [10x] 3 .

699 (Socrates)

a,b,c,x,y,z 0

ax by cz 2 ab 2 bc 2 ca a b c

y z z x x y 2

.

700 (Socrates)

(a,b) a b3 7 .

ΠΡΟΤΕΙΝΟΜΕΝΕΣ ΑΣΚΗΣΕΙΣ ΓΙΑ ΜΑΘΗΤΙΚΟΥΣ ΔΙΑΓΩΝΙΣΜΟΥΣ 601...

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Transcript of ΠΡΟΤΕΙΝΟΜΕΝΕΣ ΑΣΚΗΣΕΙΣ ΓΙΑ ΜΑΘΗΤΙΚΟΥΣ ΔΙΑΓΩΝΙΣΜΟΥΣ 601...