10 ZANIMLJIVIH ZADATAKA O BROJU 2014.pdf

7/24/2019 10 ZANIMLJIVIH ZADATAKA O BROJU 2014.pdf

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10 2014

a , b , c , . . . , m , n , k

a+b+c+. . .+m+n+k = abc . . . mnk= 2014 . 2014 = 2 19 53,

2 + 19 + 53 + 1 + 1 + . . .+ 1 + 1 = 2 19 53 1 . . . 1 1 . 1 1940

20142014 + 2014 916370 0 . 916370 = 2 5 7 13 19 53

20142014

+ 2014 = 2014 (20142013

+ 1)

2014 = 2 19 53 20142014 + 2014 2 19 53

20142013 + 1 5 (1)2013 + 1 50, 20142014 + 2014 5

20142013 + 1 13 (1)2013 + 1 130, 20142014 +2014 13 2014 7

2 (

2)3

7

1

20142013 + 1 7(2)2013 + 1 7(2)3671 + 1 7(1)671 + 1 70, 20142014 + 2014 7 20142014 +2014 2 5 7 13 19 53 = 916370 .


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17

4

28 48 2014

a b a b 0

a, ab, b, 1

a,

1

ab,

1

b, a , a b , b , . . .

6 6

a ab b 1a 1

ab1

b =

a2b2

a2b2 = 1.

6 1

17 4 17 = 2 6 + 5

a ab b 1a 1

ab=

a2b2

a2b =b = 4 .

28 48 28 = 4 6+4

a ab b 1a

=a2b2

a =ab2 = 48 , b= 4,

16a= 48, a= 3 .

2014 = 6 335 + 4, 2014

a ab b 1a

=ab2 = 3 42 = 48.

z 1, z2013 z2014

|z| = 1. |z2013| =|z|2013 = 1 |z2014| =|z|2014 = 1, z2013 z2014


3/11

z z2013 =1 z2014 =1,

1, z2013

z2014

,

z2014

z= 12

+i3

2 .

|z| =1

2

2+3

2

2= 1 ; z3 = 1, z2013 = (z3)671 = (1)671 = 1

z2014 =z2013 z =z =12 i

3

2 =1, z = 1

2+i

3

2

.

2014

n p1, p2, . . . , pk n p1 1 p2 2 pk k

n= p11p2

2 . . . pkk .

n

N(n) = (1+ 1)(2+ 1) . . . (k+ 1) .

2014 = 2

19

53, 2014 :

1. 2014 = 1 2014,2. 2014 = 2 1007,3. 2014 = 19 106,4. 2014 = 38 53,5. 2014 = 2 19 53.

2014

1. n1= 22013,

2. n2= 21006 3,3. n3= 2

105 318,4. n4= 2

52 337,5. n5= 2

52 318 5 .


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n5

2014

252 318 5 .

11 . . . 1 2014

22 . . . 2 2014

11 . . . 1 2014

22 . . . 2 2014

= 11 . . . 1 2014

102014 + 2 11 . . . 1 2014

=

= 102014 1

9 102014 + 2 10

2014 19

=(102014 1)(102014 + 2)

9 =

= 102014 1

3 10

2014 + 2

3 = 33 . . . 3

2014

33 . . . 3 2014

4 .

k

2013k | (201220132014 + 201420132012)

a= 2013 k

ak

|((a

1)a

a+1

+ (a+ 1)aa1

) .

a

(a 1)aa+1 1 a.(a+ 1)a

a1

1 a. a (a 1)aa+1 + 1

(a+ 1)aa1 1

k n N Sn tn

(1 +a)an

= 1 +Sn an+1 aSn . . . ()

(a 1)an = 1 +tn an+1 atn . . . ()

n= 1 : (1 +a)a = 1 +

a

1

a+

a

2

a2 + +

a

a

aa =


5/11

= 1 +a21 + a2+ a3a+. . .= 1 +S1 a2 . a a|a

2

aS1 S1= 1 +

a

2

+a

3

a+. . . .

n

(a+ 1)an+1

= (1 +Sn an+1)a = 1 +

a

1

Sna

n+1 +

a

2

Sn

2a2n+2 +. . . =

= 1 +an+2

Sn+

a

2

Sn

2an +. . .

= 1 +Sn+1an+2

aSn aSn+1. () () aa a

(a+ 1)aa1 1 (n= a 1 ()) aa+2 a

(a 1)aa+1 + 1 (n= a+ 1 ()) k(=a) = 2013 .

x n

x2n 4 = 3 (4 + 42 + + 42013) .

3 (4 + 42 + + 42013) =

= (1 + 4)(4 + 42 + + 42013) == 4 + 42 42 + 43 43 + 44 42013 + 42014 = 42014 4 ,

x2n4 = 420144 , x2n = 42014 . x2n (22)2014 = 0, (xn)2 (22014)2 = 0,

(xn 22014)(xn + 22014) = 0, xn = 22014 = 221953 .

x= 2, n= 2014x= 22, n= 19 53 = 1007x= 219, n= 2 53 = 106x= 253, n= 2

19 = 38

x= 238, n= 53x= 2106, n= 19x= 21007, n= 2x= 22014, n= 1 .


6/11

(x, y)

(x+y+ 2014)2 =x2 +y2 + 20142 .

x2 +y2 + 20142 + 2xy+ 2x 2014 + 2y 2014 =x2 +y2 + 20142

xy+ 2014x+ 2014y= 0 (x+ 2014)(y+ 2014) = 20142 .

20142 = 22 192 532 20142 3 3 3 = 27 2 27 = 54

54

x+ 2014 = 1, y+ 2014 = 20142 (x, y) = (2013, 4054182) (x, y) = (2015,4058210)

x+ 2014 = 2, y+ 2014 = 2 192 532 (x, y) = (2012, 2026084) (x, y) = (2016,2030112)

x+ 2014 = 22, y+ 2014 = 192 532 (x, y) = (2010, 1012035) (x, y) = (2018,1016063)

x+ 2014 =

19, y+ 2014 =

22

19

532

(x, y) = (1995, 213484) (x, y) = (2033,215498)

x+ 2014 = 2 19, y+ 2014 = 2 19 532 (x, y) = (1976, 104728) (x, y) = (2052,108756)

x+ 2014 = 53, y+ 2014 = 22 192 53 (x, y) = (1961, 74518) (x, y) = (2067,78546)

x+ 2014 = 22 19, y+ 2014 = 19 532 (x, y) = (1938, 51357) (x, y) = (2090,55385)

x+ 2014 = 2 53, y+ 2014 = 2 192 53 (x, y) = (1908, 36252) (x, y) = (2120,40280)

x+ 2014 = 22 53, y+ 2014 = 192 53 (x, y) = (1802, 17119) (x, y) = (2226,21147)


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x+ 2014 = 192

, y+ 2014 = 22

532

(x, y) = (1653, 9222) (x, y) = (2375,13250)

x+ 2014 = 2 192, y+ 2014 = 2 532 (x, y) = (1292, 3604) (x, y) = (2736,7632)

x+ 2014 = 19 53, y+ 2014 = 22 19 53 (x, y) = (1007, 2014) (x, y) = (3021,6042)

x+ 2014 = 22 192, y+ 2014 = 532 (x, y) = (570, 795) (x, y) = (3458,4823)

x+ 2014 = 2 19 53, y+ 2014 = 2 19 53 (x, y) = (0, 0) (x, y) = (4028,4028)

x+ 2014 = 532, y+ 2014 = 22 192 (x, y) = (795,570) (x, y) = (4823,3458)

x+ 2014 = 22 19 53, y+ 2014 = 2 19 53 (x, y) = (2014,1007) (x, y) = (6042,3021)

x+ 2014 = 2 532, y+ 2014 = 2 192 (x, y) = (3604,

1292) (x, y) = (

7632,

2736)

x+ 2014 = 22 532, y+ 2014 = 192 (x, y) = (9222,1653) (x, y) = (13250,2375)

x+ 2014 = 192 53, y+ 2014 = 22 53 (x, y) = (17119,1802) (x, y) = (21147,2226)

x+ 2014 = 2 192 53, y+ 2014 = 2 53 (x, y) = (36252,1908) (x, y) = (40280,2120)

x+ 2014 = 19

532, y+ 2014 =

22

19

(x, y) = (51357,1938) (x, y) = (55385,2090)

x+ 2014 = 22 192 53, y+ 2014 = 53 (x, y) = (74518,1961) (x, y) = (78546,2067)


8/11

x+ 2014 = 2 19 532, y+ 2014 = 2 19 (x, y) = (104728,1976) (x, y) = (108756,2052)

x+ 2014 = 22 19 532, y+ 2014 = 19 (x, y) = (213484,1995) (x, y) = (215498,2033)

x+ 2014 = 192 532, y+ 2014 = 22 (x, y) = (1012035,2010) (x, y) = (1016063,2018)

x+ 2014 = 2 192 532, y+ 2014 = 2 (x, y) = (2026084,2012) (x, y) = (2030112,2016)

x+ 2014 = 22 192 532, y+ 2014 = 1 (x, y) = (4054182,2013) (x, y) = (4058210,2015)

x2 y |z| = 20142 .

2014 2014 = 2

19

53

x 1, 1,2, 2,19, 19,53 53

x =1 y |z| = 20142 = 22 192 532 20142 = 22 192 532 3 3 3 = 27 x =1 2 27 2 = 108 x z

x= 1, y= 1, z = 22 192 532,x= 1, y= 2, z = 2 192 532,x= 1, y= 19, z= 22 19 532,x=

1, y= 53, z=

22

192

53,

x= 1, y= 22, z = 192 532,x= 1, y= 2 19, z= 2 19 532,x= 1, y= 2 53, z= 2 192 53,x= 1, y= 192, z= 22 532,x= 1, y= 19 53, z= 22 19 53,


9/11

x= 1, y= 532, z= 22 192,x= 1, y= 2

2

19 z= 19 532

,x= 1, y= 22 53, z= 192 53,x= 1, y= 2 192, z= 2 532,x= 1, y= 2 532, z= 2 192,x= 1, y= 2 19 53, z = 2 19 53,x= 1, y= 192 53, z= 22 53,x= 1, y= 19 532, z= 22 19,x= 1, y= 22 192, z= 532,x= 1, y= 22 532, z= 192,x= 1, y= 22 19 53, z = 19 53,x= 1, y= 2 192 53, z = 2 53,

x= 1, y= 2 19 532

, z = 2 19,x= 1, y= 192 532, z= 22,x= 1, y= 22 192 53, z= 53,x= 1, y= 22 19 532, z= 19,x= 1, y= 2 192 532, z= 2,x= 1, y= 22 192 532, z= 1,

x =2 y |z|= 192 532 192 532 3 3 = 9 x =2 2 9 2 = 36 x z

x= 2, y= 1, z = 192 532,x= 2, y= 19, z= 19 532,x= 2, y= 53, z= 192 53,x= 2, y= 192, z= 532,x= 2, y= 19 53, z= 19 53,x= 2, y= 532, z= 192,x= 2, y= 192 53, z= 53,x= 2, y= 19 532, z= 19,x= 2, y= 192 532, z= 1,

x =

19 y

|z

|= 22

532 22

532 3

3 = 9

x =19 2 9 2 = 36 x z

x= 19, y= 1, z= 22 532,


10/11

x= 19, y= 2, z= 2 532,x= 19, y= 53, z= 2

2

53,x= 19, y= 22, z= 532,x= 19, y= 2 53, z= 2 53,x= 19, y= 532, z= 22,x= 19, y= 22 53, z= 53,x= 19, y= 2 532, z= 2,x= 19, y= 192 532, z= 1,

x =53 y |z| = 22 192 22 192 3 3 = 9 x =53 2 9 2 = 36 x z

x= 53, y= 1, z= 22 192,x= 53, y= 2, z= 2 192,x= 53, y= 19, z= 22 19,x= 53, y= 22, z= 192,x= 53, y= 2 19, z= 2 19,x= 53, y= 192, z= 22,x= 53, y= 22 19, z= 19,x= 53, y= 2 192, z= 2,x= 53, y= 22 192, z= 1.

108+36+36+36 =216


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10 ZANIMLJIVIH ZADATAKA O BROJU 2014.pdf