MA21 - Resolução de Problemas.pdf

8/13/2019 MA21 - Resoluo de Problemas.pdf

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13/326

c

a

b

c2 =a2 +b2

a a

a= 1/2 (1/2)2 =

1/4 1/2


14/326

P

Q

P

Q

P c

a

b

Q c2 =a2 +b2

P

P

P

Q

P

Q

P

Q

P = Q

P

Q

a a


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19/326

A

B

A

B

d1, e1, v1

A

d2, e2, v2

B

A

B

v1> v2

d1+e1+v1=d2+e2+v2.

A

e1+ 3v1

B

e2+ 3v2

e1+ 3v1=e2+ 3v2.

3(v1 v2) =e2 e1 v2 v1=e2 e13

.

v1v2>0 e2e1>0

d1 d2=e2 e1+ (v2 v1) =e2 e1 e2 e13

=2

3(e2 e1).

d1d2 > 0 e2e1 > 0

A

B


20/326

P

Q

P =Q P

Q


21/326

T

a

b

c

c2 =a2 +b2


22/326

a b

b

a

a

bc

Q

a+b

a

b

Q

c

a

b

= 180

(+) = 180 90= 90

(a+b)2 = 4 ab2

+c2,

a2 + 2ab+b2 = 2ab+c2,

a2

+b2

=c2

,


23/326

N2

N

N2

N

N2

N2

N


24/326

N2

N

N2

N

N2

N

N

p

N= 2p+1

N2 = (2p+ 1)(2p+ 1)

= 4p2 + 2p+ 2p+ 1

= 4p2 + 4p+ 1

= 2(2p2 + 2p) + 1

= 2q+ 1,

q = 2p2 + 2p N2 = 2q+ 1


25/326

x

x+ 1/x2

x

x+ 1/x2

x

x+ 1

x 0

x2 + 1< 2x.

x2 2x+ 1 < 0 (x 1)2


26/326


27/326


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29/326

9


30/326

0 x y z36

xyz= 36 x

y z

x y z xyz

1 1 36 36

1 2 18 36

1 3 12 36

1 4 9 36

1 6 6 362 2 9 36

2 3 6 36

3 3 4 36

x y z x+y+z

1 1 36 38

1 2 18 21

1 3 12 16

1 4 9 14

1 6 6

2 2 9

2 3 6 11

3 3 4 10


31/326

A

B

C

A

B

C

A B

C

A


32/326

A

A

B

C

D

C

D

A

B

A

B

A


33/326

B

A

B

C

D

C

A

B

D


34/326


35/326

1, 2, . . . , 9,

i

j

i

j


36/326

1 2 3

4 5 6

7 8 9


37/326

Q T 3 4 6


38/326

. . .


39/326

1, 2, 3, . . . , 99

2, . . . , 20


40/326


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42/326

x

y

z


43/326


44/326

x

x

2x

x/5

2x+x

5= 121,

10x+x= 605,

11x= 605 x= 605/11 = 55

x

ax+b= 0,


45/326

a= 0 b R x

2x 3 = 0

4x+ 1 = 0

3

2x = 0.

a

b

a+c b+c

a= b = a+c= b+c.

c

x

5x 3 = 6,


46/326

3 x

(5x 3) + 3 = 6 + 3 5x= 9.

a

b

a c b c

a= b = ac= bc. 5x = 9

1/5

x=5x

5 =

9

5,

5x 3 = 6


47/326

x

x

6x

6x+ 12

6x+ 12

3 = 2x+ 4

2x + 4 2x= 4

1 = 2

x+ 2x= 2x+x.

x x= 2x 2x (x x)

1(x x) = 2(x x) (xx)

1 = 2


48/326

ax+b = 0

b

ax+b+ (b) = 0 + (b)ax =b.

a= 0

x

ax

a =

ba

x = ba

.

ax+b= 0

x = ba

.

x

x11 + 11x+ 1x1 = 777,

x


49/326

abc 10

102a+ 10b+c

x11 = 100x+ 11

11x= 110 +x

1x1 = 101 + 10x

100x+ 11 + 110 +x+ 101 + 10x= 777

111x+ 222 = 777

x=777 222

111 = 5.

1

9

1 6

9

1 9 s

45 = 1 + 2 + + 9 = 3s,

s 15 x


50/326

x

1 +x+ 6 = 15 x= 8

8 9 15

9

x

x

x

15

x 4 15 = 60

x

x

1 + 2 + 3 + 4 + + 9 + 3x= 60,

45 + 3x= 60

x= 5


51/326

6.378.000

C

r

C= 2r,

3, 1416

CT

CT = 2rT=2 3, 1415 6.378.000 = 40.072.974


52/326

rT

x

rf

x= rfrT rf CT+ 1 = 40.072.975

40.072.975 = 2rf rf=40.072.975

2 .

rf

6.378.000, 16 x x =

rf rT = 0, 16

=3, 1415926535897932384626433832795 .


53/326

x

y

2x+ 3y= 14.

x+y = 6.

x1 x2

. . . xn

a1x1+a2x2+ +anxn+b= 0,


54/326

a1, a2, . . . , an b

2x 3y= 0

x y

2a b+ c3

= 5

a, b

c.

(r1, r2, . . . , rn)

x1 r1 x2 r2 xn rn

a1r1+a2r2+ +anrn +b= 0.

(3, 2) 2x 3y = 0

2 3 3 2 = 0.

(2, 3) 2x 3y = 0 2 2 3 3 =5= 0. (2, 0, 3) 2a b+ c

3= 5

2 2 0 +33

= 5.

n

x1 x2 . . . xn k

x1, x2, . . . , xn

a11x1+a12x2+ +a1nxn+b1= 0,a21x1+a22x2+ +a2nxn+b2= 0,

ak1x1+ak2x2+ +aknxn+bk = 0,


55/326

aij (1ik, 1jn)

aij

xj

(r1, r2, . . . , rn)

(r1, r2, . . . , rn)

2x+ 3y= 14,

x+y = 6.

x

x= 6 y.


56/326

2(6 y) + 3y = 14,12 2y+ 3y = 14,

y = 2.

y = 2 x= 6 2 = 4

x y z x+y z 1 = 0,x y 1 = 0.

x= y + 1.

(y+ 1) +y z 1 = 0,2y z= 0,

z= 2y.

x z y

y

t

x

z

t

t

x= t+ 1, y =t, z= 2t


57/326

x

y

z

x+y+ 2z 1 = 0,x+z 2 = 0,y+z 3 = 0.

x= 2 z y= 3 z.

(2 z) + (3 z) + 2z 1 = 04 = 0,


58/326

p

g

2g=p

1.

3(g 1) =p. 2g = 3g 3 1

g = 4 p= 9

A1 A2

V1 V2

A1

A2

A1+A2=

4

,

A1+ 2A2 = 1,


59/326

A1 A2

A1=

4 A2;

4 A2+ 2A2= 1,

4

+A2= 1.

A2= 1 4 A1= 4

1

4

=

2 1

x

y


60/326

y < x

c

15x+ 15y =c.

20

20x= c.

15x+ 15y= 20x 15y = 20x 15x= 5x. 5 3y=x

20 3 60y= 20 3y = 20x= c

x y 3y y = 2y 30

30 2y = 60y =c


61/326

x2 6x 8 = 0.

x2 6x= 8.

9 x2 6x + 9 (x 3)2

(x 3)2 = 9 + 8 = 17.

x 3 = 17 x 3 =17

x1 = 3 +

17 x2= 3

17.

a, b

c

ax2 +bx+c= 0,

a= 0 b, c R x

x

ax2

+bx=c


62/326

a

x2 + ba

x=ca

.

b2

4a2

x+ b2a

2=x2 + 2 b2a

x+

b2a2

= b2

4a2 ca = b2

4ac4a2 .

b24ac

x+

b

2a2

=

b2

4ac

4a2 =

4a2 .

0

0

x+ b

2a =

b2

4ac

2a x+

b

2a =

b2

4ac

2a .

x1 = b

2a+

b2 4ac

2a =

b+

2a


63/326

x2 = b

2a

b2 4ac

4a2 =

b

2a .

> 0

= 0 x1 =x2=b/2a

< 0

2x2 4x+ 2 = 0.

a= 2, b=4 c= 2

=b2 4ac= (4)2 4 2 2 = 0.

x= b2a

=44

= 1

x2 x 1 = 0.

a= 1

b=1

c=1

=b2 4ac= (1)2 4 1 (1) = 5.

x1=b+

2a =

1 +

5

2

x2=b

2a =

1 52

.


64/326

x

x= 1 +

1

1 + 1x

,

x

1 +1x

= x+ 1x

.

1 + 1

1 +1

x

= 1 + x

1 +x=

1 + 2x

1 +x.

x=

1 + 2x

1 +x

x2 +x= 1 + 2xx2 x 1 = 0.

x1=1 +

5

2

x2=1 5

2 .

( + )/


65/326

an

n0

an+2=an+1+an.

an an = x

n

x= 0

an

an

xn

n0

xn+2 xn+1 xn = 0.

xn

xn(x2

x

1) = 0

xn = 0 x2 x1 = 0

x= 0 xn = 0 x2 x 1 = 0

an=

1 +

5

2

n

an=

1 5

2

n.

an bn

an+

bn

an = x

n1 bn = x

n2


66/326

ax2

+ bx + c= 0

a= 0

x1 x2

a, b c

x1 x2 0

x1=b+

2a

x2=b

2a .

x1 x2

x1+x2=b+

2a +

b 2a

=2b

2a = b

a.

x1x2

x1x2=b+

2a

b

2a

=

b+ b 4a2

=b2

4a2

=4ac

4a2 =

c

a.

a= 1

x2 sx+p= 0

+=s =p.


67/326

(x )(x ) =x2 sx+p

ax2 +bx+c= 0

a

= 0 a(x2

sx+p) = 0 s=

b/a

p= c/a x2 sx+p= 0

ax2 +bx+c= a(x2 sx+p) =a(x )(x )

ax2 +bx+c= 0

(x )

m2

a

b

a+b= 12,

ab= 28.

a

b

x2 12x+ 28 = 0

= 122

428 = 32


68/326

a=12 +

32

2 = 6 + 22

b=12 32

2 = 6 2

2.

x2 +bx+ 17 = 0

b

n

m

m= n n + m=b

m=n b

m

n

m n = 17

m = 1 n = 1

1 +b+ 17 = 0

b=18

b

a

P ={1, 2, . . . , n}

(i, j) i

j

1

A1=

(1, 2), (1, 3), . . . , (1, n)

.


69/326

2

1

A2=

(2, 3), (2, 4), . . . , (2, n)

.

(2, 1) (1, 2) 1

2

Ai=

(i, i+ 1), (i, i+ 2), . . . , (i, n)

,

1in 1.

AiAj = i= j

Ai A1 An1

|X|

X

a

|(A1 A2 An1)|=|A1| + |A2| + + |An1|= (n

1) + (n

2) +

+ 2 + 1

=(n 1)n

2 =a.

n2n2a= 0

a


70/326

a < 75

n1 = 12

a = 66

x4 2x2 + 1 = 0.

y

x2

y=x2 0 =y2 2y + 1 =

(y 1)2 y= 1 x2 =y = 1 x= 1

x=

1

ax2k +bxk +c= 0, k N,

y = xk

ay2

+by+c= 0,


71/326

y =

xk =,

x= k

k

0 k

ax2 +bx+c= 0

Ay2 +B = 0,

A

B

a,b,c

y1= B

A y2=

B

A,

B

A0.


72/326

x= u + v

u

v

a(u+v)2 +b(u+v) +c= 0,

au2 + 2auv+av2 +bu+bv+c= 0.

v

av2 + (2au+b)v+au2 +bu+c= 0.

u

(2au+b)v

u=b/2a

av

2

+a

b

2a2 b

b

2a+c= 0av2

+

b2

4a b2

2a+c= 0,

av2 +b2 + 4ac

4a = 0.

v1=

b2 4ac4a2

v2=

b2 4ac

4a2 ,

=b2 4ac0.

u=b/2a x= u+v

x1= b2a

+v1 x2= b2a

+v2,


73/326

x

x

3x100.

99 = 3 33 3 34 = 102> 100

x= 33

x

3x100< 0

x 100/3 3x 100< 0


74/326

ax+b 0,ax+b0, ax+b0,

a, b R a= 0.

S

a

b

a b

a+ c b+ c

c

a b

a

b

ac bc c

a b

ab

ac

bc

c


75/326

ax+b 0.

a >0

ax + b < 0

a

x+b/a b/a

={x R; x >b/a},


76/326

b a

ax + b > 0

={x R; x


77/326

x

3456784

a

b

a= x

(x+ 2) + (x+ 1) b= (x+ 1)2

(x+ 4)

a= x2 + 3x+ 1 b= x2 +x 3 ab

x2 + 3x+ 1x2 +x 3,

x2x+3

2x+ 40.

x R

x 2, x = 3456784

a

b

a > b

3 3

1 + 2 + 3 + + 9 =45

x


78/326

x

x

17 > 15

x

y

1 + 2 +y = 15 y = 12

x

z

z= 15 (x+ 9)16 x1,

x5

x

s

s= 15 (x + 1) = 14 x9 x5 5x5 x= 5

a b

c

P

c

x

y

a > b


79/326

P

= x+y

P

= x+y

a

b

c

Px yB A

C

P

AC

BC

S

a x

b

y

ax

2 +

by

2 =S,

ax= 2S by

x=2S

by

a .

y

x+y =2S by

a +y

=2S by+ay

a

=2S

a +

a

b

a y,


80/326

= +y,

=2S

a =

a ba

.

0 y hb hb

b

a > b 0 y hb + y

+hb

0+hb.

y = 0

P

A

y =hb P B

ax2 + bx + c

ax

2 +bx+c 0,

ax2 +bx+c0, ax2 +bx+c0,

a,b,c R a= 0. a

ax2

+bx+c


81/326

x2 3x + 2> 0

x2

3x + 2 = 0

x2 3x+ 2 = (x 1)(x 2).

(x1)(x2) (x 1) (x 2)

x 1> 0x >1

x 2> 0x >2,

x >2

x 1< 0x


82/326

ax2 +bx+c= a

x2 + ba

x+ ca

=a

x2 +

b

ax+

b2

4a2 b

2

4a2+

c

a

=a

x2 +

b

ax+

b2

4a2

a

b2

4a2 c

a

=a

x+

b

2a2 4a,

=b2 4ac.

=b2 4ac >0

a

a > 0

a

x+

b

2a

2

4a>0.

a > 0 1/a

x+

b

2a

2

4a2 >0.

> 0


83/326

x+ b2a

2 4a2 = x+ b2a2

2a2

=

x+

b+

2a

x+

b 2a

=

x b

2a

x b+

2a

= (x )(x )> 0,

= b

2a

= b+

2a

ax2 + bx + c= 0

(x )(x ) > 0 (x ) (x )

x > x >

<

x >

x < x < x < <

={x R; x < x > },

a < 0

1/a

x+ b2a2

4a2


84/326


85/326

4a>0 a

ax2 +bx+c= a

x+

b

2a

2

4a

4a


86/326

x

x+1

x2,

f(x) =ax2 + bx + c

ax2 +bx+c= a

x+

b

2a

2

4a,

= b2 4ac. f(x) f(x)

x

a > 0

x = b

2a

f( b2a

) =4a

a < 0

x= b

2a

f( b2a

) =4a

a, b

a + b= 1

ab1/4

ab = a(1a) =a2 +a f(a) =a2 +a f(a) 1/4 0 < a < 1

f(a)

f(a) =(a2 a) =(a2 a+ 1/4 1/4) =(a 1/2)2 + 1/4,

1/4

a= 1/2


87/326

ABCD

r

ABCD

xyr

A B

CD

A= 2x 2y= 4xy.

y=

r2 x2,

A= 4x

r2 x2.

ABCD

A2


88/326

A2

A2 = 16x2(r2 x2) = 16r2x2 16x4.

z=x2

A2 =16z2 + 16r2z=16

z r22

2+ 4r4,

A2

z = r

2

2

x= r2

y =

r2 r

2

2 =

r2

.

2r2

=r

2

|2x 5|= 3

|2x 3|= 1 3x


89/326

|3 x| |x+ 1|= 4

|2x 5|= 3.

|a|= a a0,a a


90/326

|2x 3|= 1 3x

(a)

2x 30,2x 3 = 1 3x,

(b)

2x 3< 0,(2x 3) = 1 3x.

2x 3 = 1 3x5x= 4 x= 4/5

2x 30

(2x3) = 13x x=2

2x 3< 0 x=2

1 3x0,(2x 3)2 = (1 3x)2.

(2x 3)2 = (1 3x)2 5x2 + 6x 8 = 0

x1 = 4/5 x2 =2 x1 13x10

x2=2

|3 x| |x+ 1|= 4

|3x| |x+ 1| x1 = 3 x2 =1


91/326


92/326

u= x2

+ 3x

v= (x2

2)3

.

u v= 3,u+v = 1,

u= 2

v=1

x2 + 3x= 2x2 + 3x= 4,

x= 1 x=4

(x2 2)3 =1,

x2

2 =1 x= 1 x=1

x= 1 u= 2

v=1

12.345.679 18 = 222.222.222 12.345.679 27 = 333.333.333

12.345.679 54 = 666.666.666

999.999.999 12.345.679


93/326

250

200

A

B

2/3

2/5

a

3a+68

2a+106

x

x


94/326

x

y

z

xy

yx

zxz

x

y

z

15

2


95/326

8

3

1 2 ax

2 +bx+c a= 0 a, b c

1+22

1+

2

41+ 42

3 x2 7x2c = 0

c

p(x) = 2x4+bx3+cx2+ dx+ e

p(x) =p(1 x)


96/326

a

b

1.998x2 +ax+b = 0

ax2 + 1.998x + b= 0

x2 + y2 + z2 = 3xyz

x,y,z

a2x2

(b2

2ac)x+c2 = 0,

a, b

c

nN

p(n) = 0 n

a, bZ

(ax b)2 + (bx a)2 =x,

2x2

x2 + 1

=x.

[x]

x


97/326

n4

+ 4

n >1

n4 + 4n

n >1

966666555557

966666555558

966666555558

966666555559,

n

n200 100000.

n


98/326

a

ax2

ax+ 12

x

x

x2 + 9

x 3x+ 1

x2 1x2 3x

[x]{x} +x= 2{x} + 10,

[x]

x

[2, 46] = 2

[5, 83] = 5 {x} x {x}= x [x]

p

x2 +ax +b = 0

a1 b1

x2 +a1x+b1 = 0 x2 +a1x+b1 = 0

a2 b2 x

2 + a2x + b2= 0

x2+a2x+b2= 0 a3 b3

x2+a3x+b3= 0


99/326

x2 = 0

x2 = 0


100/326


101/326


102/326


103/326

x,y,z

n

n xn + yn =zn


104/326

Z

N ={1, 2, 3, . . .}

Z={. . . , 3, 2, 1, 0, 1, 2, 3, . . .}

A

N

aA an nA

A

A


105/326

a

b

a

b

q

b = aq a

b b a

a| b

a

b

ab

7 | 21 21 = 73 3 8

M ={3m, m N} ={3, 6, 9, 12, . . .}

a, b c

a|b b|c a|c

a|b a|c a|(b+c) a|(b c)

a

b

a|b 0< ab

a|b b|a a= b a=b

a|b

b|c

q1 q2

b= aq1

c= bq2.


106/326

c= aq1q2 =aq, q=q1q2Z,

a | b a | c

b= aq1, q1Z

c= aq2, q2Z.

b+c= a(q1+q2 rZ

) b c= a(q1 q2 sZ

),

a| b

b= aq

q1.

a

a

b= aqa >0,

a|b

b|a

|a|

|b|

|b|

|a|

|a| |b| |b| |a| |a| |b| |a|

|a|=

|b|

a= b a=

b


107/326

N= 545362 7

545362 7 = 5q 7 =545362 5q 7 5

r1= 31 7 1 = 24,r2= 31 7 2 = 17,r3= 31

7

3 = 10,

r4= 31 7 4 = 3,r5= 31 7 5 =4,r6= 31 7 6 =11,

rq = 317q 1 q 4

q 5

q= 4

b

a


108/326

a

b

a

q r

b= aq+r, 0r < a.

ab

r 0< r < a

b

b a >0

R={b aqZ; b aq0} N {0}

R

ba R

b a > 0 R

r

r = baq 0 q 0 r < a

r=b aqab a(q+ 1)0.

a >0b a(q+ 1)< b aq.

0b a(q+ 1)< b aq,

r=baq

R

r

q

r1 q1

b= aq1+r1, 0r1< a.


109/326

aq+r =aq1+r1

(r r1) = (q1 q)a;

rr1 a a < rr1< a

r r1 a

r r1 = 0 r=r1

q=q1

q

r

b

a

a

b

1 < a b

n

nab


110/326

n = 175 50 + 213 20

n

n

n1

n1 = 1 2 + 0 2 = 2.

n1

n

a


111/326


112/326

nN

an=n(n + 1)

n

r= 0 r= 1

[r = 0] an

an

[r = 1]

an

an

1a

n

n

2

1

n

n= 2m + 1

kZ

n2 1 = (2m+ 1)2 1 = 4m2 + 4m+ 1 1 = 4m2 + 4m.

n2 1 = 4m(m+ 1).

m(m+ 1)

m(m+ 1) = 2q qZ

n2 1 = 4m(m+ 1) = 4 2q= 8q,


113/326

n

n2

n n2

a b c

a2 b2

a2 +b2 12 + 12 = 2

a2 +b2 =c2 c2


114/326

x y

n= xy = 10x + y

(5x+ 3)2 +y = 40

10x+y = 34,

x= 3 y = 4

10

0

1, 2, 3, . . . , 9.

345


115/326

345 = 300 + 40 + 5 = 3 102 + 4 10 + 5,

2768

2768 = 2000 + 700 + 60 + 8 = 2 103 + 7 102 + 6 10 + 8.

a= anan1 . . . a1a0

an, an1, . . . , a1

a0

a

a= an10n +an110

n1 +. . .+a110 +a0

a,b,nN

an bn = (a b)(an1 +an2b+ +abn2 +bn1).

0< b < a a b an bn.

b=1

s= 1 +a+a2 + +an1

s

a

as= (a+a2

+ +an1

) +an

=s 1 +an

.


116/326

(a 1)s= as s= an 1,

a

n

1 = (a 1)(an1

+a

n2

+ +a+ 1).

b= 1

bN an bn =bn (a

b)n 1 .

an bn =bn(ab 1) (a

b)n1 + ( a

b)n2 + + ( a

b) + 1

= (a

b)bn1

(ab

)n1 + ( ab

)n2 +

+ ( a

b) + 1

= (a b)(an1 +an2b+ +abn2 +bn1),

a = an . . . a1a0

a

a0

a

a

a0

a= 10(an10n1 +an110

n2 +

+a1) +a0.

10| a 10|a0

a= an10n +an110

n1 + + 10a1+a0=an(10

n 1) +an1(10n1 1) + + (10 1)a1

+an+an1+ +a1+a0.


117/326

10j 1 = 9qj 1 j n

a= 9(anqn+an1qn1+ +a1) +an+an1+ +a1+a0.

9|a 9|(an+an1+ +a1+a0).

N= 13424136 + 1234567890

3

3

3


118/326

a, bN

b >1

r0, r1, . . . , rn 0

ri

b

1

0in

a= rnbn +rn1b

n1 + +r1b+r0.

a

b

a= (rncn1 . . . r1r0)b,

a= bq0+r0, r0< b,

q0 =bq1+r1, r1< b,

q1 =bq2+r2, r2< b,

qj1=bqj+rj, rj < b,

a > q0 > q1 > q2 > > qj1

j = n qn1 < b qj = 0 j n

rj = 0

j n+ 1.


119/326

1jn

a= bq0+r0,bq0=b

2q1+br1,

b2q1=b3q2+b

2r2,

bn1qn2=bnqn+b

n1rn1

bnqn1=bn+10 +bnrn.

a= rnbn +rn1b

n1 + +r1b+r0.

ri

2

g

g

2

a

1a100

a= r626

+r525

+r424

+r323

+r222

+r12 +r01,


120/326

ri {0, 1}, 0 i 1 2n 128 n 7

ri 0 1

1, 22, 23, 24, 25, 26

1g

100g

a

b

a b

d

d

a

b

d|a d|b


121/326

d

a

b

d =

(a, b)

d= (a, b) (a, b) = 1 a b

12 = 62 18 = 63 (12, 18) = 6 (4, 15) = 1 4

15

a b

a

b

(a, b) =b

a= bq+ c, c

= 0

a b

b

c

(a, b) = (b, c)

a

b

b

a

b

b

a

a

b

a b

b

b

(a, b) =b

a

b

c

b

c

b

c

a

a

b


122/326

a

b

b

c

(a, b) = (b, c)

d

mdc

a

b

x0 y0 d= (a, b) =ax0 + by0

ax+by

x

y

Ca,b={ax+by; x, yZ},

x= y = 0

Ca,b

x0 y0

= ax0+by0

Ca,b

| a | b | a

a


123/326

a

q

r

a= q+r 0< r <

r=a q=a q(ax0+by0) =a(1 qx0) +b(qy0)

r

Ca,b

Ca,b

a

b

= d

d= (a, b) a= da1, b= db1

= ax0+by0=d(a1x0+b1y0).

d|

d

d <

d = (a, b)

d= = ax0+by0

d, N

a,b,c Z

d|a

d|b

d|(a, b)

(a, b) ax+ by x y

(a,b) =(a, b)

d|a

d|b

(ad

, bd

) = 1d

(a, b) a

(a, b),

b

(a, b)

= 1.


124/326

(a, c) = (b, c) = 1 (ab,c) = 1

c|ab (b, c) = 1 c|a

(a, b) = ax0+ by0

(a)x+ (b)y =(ax+by) x, yZ.

(a,b) = min

(a)x+ (b)y >0; x, yZ= min

ax+by ; x, yZ

=(a, b).

(a, b) =

d

a

d, d

b

d

= d

a

d,b

d

.

(a, c) = (b, c) = 1

xj yj j = 1, 2

ax1+cy1= 1,

bx2+cy2= 1.

(x1x2

x)ab+ (ax1y2+y1bx2+cy1y2

y)c= 1.


125/326


126/326

b= aq1+r1, 0r1< a,a= r1q2+r2, 0r2< r1,r1=r2q3+r3, 0r3< r2, rn2=rn1qn+rn, 0rn< rn1,rn1=rnqn+1,

rn rn1 (a, b) = rn

rk

{r Z, 0 r < a} a

(a, b) = (a, r1) = (r1, r2) = = (rn1, rn) =rn.


127/326

rn =rn2 rn1qnrn1 =rn3 rn2qn1

rn =rn2 (rn3 rn2qn1)qn.

rn rn2 rn3

rn2 =rn4 rn3qn2 rn

rn3

rn4

x, yZ

d= rn =xr1+yr2.

r1 = b aq1 r2 = a r1q2 = a(1 +q1q2) bq2

1176 = 471 2 + 234,

471 = 234 2 + 3,234 = 78 3,

(471, 1176) = 3

2n+ 8

4n+ 15

n


128/326


130/326


131/326


132/326

T= [15, 18] =15 18

3

= 90.

n1= 6 n2= 5

AB =m CD =n

A B

C

m

n

mn

m

n


133/326

A

m

BC

m

AD

m

BC

pm

p N BC AD

AB

DC

qn

qN

B C

D

x

x = pm = qn x

m

n

x

m

n

x= [m, n]

A


134/326

ax+by =c,

a,b,cZ

a= 0 b= 0

(x, y)

(x, y)

(1, 2) (1, 1) 3x 2y= 1

x

y

3x 2y = 1


135/326

ax+by =c, a, b, cZ,

a= 0

b= 0,

d| c

d = (a, b)

(x0, y0)

(x, y)

x= x0+tbd

y =y0 tad , tZ.

(x0, y0)

ax0 +by0 = c. d = (a, b)

q1 q2 dq1 = a dq2 = b

dq1x0+dq2y0=d(q1x0+q2y0) =c,

d|c

d| c

c = qd

q

x0 y0 ax0+by0 = d

q

ax0q+by0q=c,

(x1, y1) x1 = x0q y1 = y0q

(x, y) (x0, y0)

ax0+ by0=c = ax + by ax0+ by0 =ax + by

a(x x0) =b(y0 y)

d

a

d(x

x0) =

b

d(y

0 y).


136/326

(ad

, bd

) = 1 ad| (y0 y) bd| (x x0)

t

x= x0+tbd

y =y0 tad .

t

x

y

12x+ 33y= 27.

(12, 33) = 3

3| 27

4x+ 11y= 9,

x0 = 5 y0 =1

x= 5 + 11t y= 4t

1, t

Z,

(12, 33)

33 = 12 2 + 9,12 = 9 1 + 3,

9 = 3 3 + 0.


137/326


138/326


139/326

n

1

n

n

n n

p

n p q

1< q < p

q|p p|n,

q|

n

p

p

n

n = pq q p

p

n= pqp2,

n

p

p1, p2, p3, . . . , pk.


140/326

n= p1p2p3 pk+ 1

q

q

pi, 1 i k

n

p

p|ab = p|a p|b

a, bZ

p

pb

(p, b) = 1

p|a

p

p= d1d2, 1< d1< p, 1< d2< p.

p|d1 p|d2

pd1, pd2,


141/326

N

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, . . . , n


142/326


143/326


144/326


145/326

M11 = 2

11

1 = 2.047

2.047 = 23 89

n= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 257,

Mn n

257 Mn

n

Mn M67 M257

a, b R

a4 + 4b4 = (a2 + 2b2 + 2ab)(a2 + 2b2 2ab).

a4 + 4b4 =a4 + 4a2b2 + 4b4 4a2b2= (a2 + 2b2)2 4a2b2= (a2 + 2b2 + 2ab)(a2 + 2b2 2ab).


146/326

qn=n

4 + 4n nN.

n

n= 2m

m1 n4 + 4n = (2m)4 + 42m = 16m4 + 24m,

= 28m4 + 24m1.

n4 + 4n 2 n > 1

n4 + 4n

n n= 2m+ 1

m

1

n4 + 4n = (2m+ 1)4 + 42m+1 = (2m+ 1)4 + 4 42m= (2m+ 1)4 + 4 24m = (2m+ 1)4 + 4 (2m)4.

a = 2m+ 1 b = 2m

520 + 230

520 + 230 = 554 + 22 228 = 554 + 4 274 ,

a= 55

b= 27 520 + 230


147/326


148/326


149/326

pi

p11 p22 p

33 pmm =n = q11 q22 q33 qss

pi qj pi=

qj pi

qj

pis qjs

m=s

p11 p22 p

33 pmm =p11 p22 p33 pmm .

1

1

1< 1

p22 p33 pmm =p111 p22 p33 pmm ,

1 1 > 0 p1

pj j > 1 1 = 1

i=

i

i= 1, . . . , n


150/326


151/326


152/326


153/326


154/326


155/326

n = 1 +

1

2+

1

3+ +1

n

n >1

n=13

+15

+ + 12n+ 1

n >0

p(n) = amn

m +am1nm1 + +a0

m1 nN p(n)

n

aN =|p(a)|> 1

p(k+a) kZ.

An n

s

An

2s An

s

n

(a + b)n

n

2s 1

(x0, y0)

ax by = 1 (0, 0)

(b, a)

(x0, y0)


156/326

(x1, y1)

(x2, y2)

ax+by =c


157/326

N +

N


158/326


159/326

0, 1, 2, 3, . . . , 600 k

k 1000 > 602 = 601 + 1

n

n

0, 1, 2, . . . , n 1

0 n1

0 n1

n1

7 8 =

7 + 1


160/326

n

S

S/n

S/n

53

= 5!

3!2!= 10

42 = 4!

2!2! = 6 Ej

Tj j = 1, 2 . . . 10

E1+E2+ +E10= 6 245 = 1470;

Tj Ej 147010 = 147

N k +

N

k +


161/326

k N

N k

N k+ 1 N

k= 1

16 =

3 5 + 1

N= 3

k= 5

Aj j = 1, 2, . . . , 6


162/326

A=

{A1, A2, . . . , A6

}

A1

5 = 22 + 1

A1A3 A1A4 A1A6

A3A4 A3A6 A4A6

A1

A1

A2A3

A4

A5 A6

A1A2A5 A1 A2 A5

A1A4A6 A1 A4

A6


163/326

n

m

A={m+ 1, m+ 2, . . . , m+n} n

n

n

A

n

B ={1, 2, . . . , n 1} n 1

m+i m+j

1i < j n m+i n

m+j n m+j (m+i) n n > j

i

1 n

n

n

A

m

n

m+ 1, m+ 2, . . . , m+n n n

m= 123456789010n+1

m + 1, m + 2, . . . , m + n

n


164/326

n

n

n=

n+ 1

1, 11, 111, 1111, . . . , 111 1 n+1

n

0, 1, 2, 3, . . . , n 1,

n

n

n

n+

{ }

n+ 1

{1, 2, . . . , 2n} n

Cj ={2j 1, 2j}, 1jn.

n+ 1 n

Cj, 1jn, Cj


165/326

d= (a, b) a

b x y

ax+by =d.

m=a/d n=b/d a

b

mx+ny = 1

m

n

ax+by =d

(a, b) = 1 A={a, 2a , . . . , b a}.

A 1

b

b

A b 1 b

ia

ja

b > j > i 1

b

(j i)a

b (a, b) = 1 b

j i >0 b > j i

a

1

b ax ax 1 b

ax 1 =by


166/326

/

1/2

1

2/2

3


167/326

A B

C

D

E

DBE 3

DBE 3/2

DBE

3/2< 3


168/326


169/326


170/326

22 33 = 162 + 1 2 2

x

y

x y +xy

xy

1+xy

tan( ) = tan tan 1 + tan tan

.

x1, x2,

, x7

(2 , 2 ) R xi, 1i7, i (2 , 2 ) tan(i) =xi (2 , 2 )

6

i

i1

i2


171/326

i1 i2

2

2

6

i1i2

0i2 i1

6.

tan(0)tan(i2 i1)tan(

6).

0 xi2 xi11 +xi2xi1

13

.

C

C

N

p: ININ


172/326

IN={1, 2, . . . , N } P(p) = (1 p(1))(2

p(2))

(N

p(N))

8

8

A={1, 2, 3, . . . , 10}.

x

x, 2x, 3x, . . . , 101x


173/326

5

a

r =p/q

|a

r

|< 1/q2


174/326


175/326


176/326


177/326


178/326

A

|A|

n

A1, A2, . . . , An1

An


179/326

A1 A2 An:= (a1, a2, . . . , an); aiAi, i= 1, 2, . . . , n, (a1, a2, . . . , an) n

A1 A2 A1 A2=

|A1 A2|=|A1| + |A2|.

A1

A2 . . . An Ai

Aj =

,

i

=j

|A1 A2 An|=|A1| + |A2| + + |An|.

4

2


180/326

f1 f2 f3 f4

t1 t2

(i, j) 0i4 0j2

fi tj i= 0 j = 0

A1= (1, 0), (2, 0), (3, 0), (4, 0) A2= (0, 1), (0, 2).

|A1 A2| =|A1| + |A2| = 6

66

P ={1, 2, . . . , n} (i, j)

i

j

1

A1 ={(1, 2), (1, 3), . . . , (1, n)}. 2

1

A2 ={(2, 3), (2, 4), . . . , (2, n)}. (2, 1) (1, 2) 1

2

Ai={(i, i+ 1), (i, i+ 2), . . . , (i, n)}, 1in.


181/326


182/326

A1, A2 A3

|A1 A2 A3|=|A1| + |A2| + |A3| |A1 A2| + |A1 A3| + |A2 A3|+

|A1

A2

A3

|.

|A1 (A2 A3)|=|A1| + |A2 A3| |A1 (A2 A3)|,

|A1 A2 A3|=|A1| + |A2 A3| |(A1 A2) (A1 A3)|.

|A1A2A3|=|A1|+ |A2|+ |A3| |A2A3| |(A1A2) (A1A3)|.

|(A1

A2)

(A1

A3)

|=

|A1

A2

|+

|A1

A3

| |(A1

A2)

(A1

A3).

|A1 A2 A3|=|A1| + |A2| + |A3| |A1 A2| + |A1 A3| + |A2 A3|+ |A1 A2 A3|,


183/326


184/326


185/326

S2=

1i1


186/326

A

Ac

U = A

Ac

|U|=

|A

|+

|Ac

|

|Ac|=|U | |A|.

A1 U A2 U A1A2 (A1A2)c U= (A1A2)(A1A2)c

|U|=

|A1

A2

|+

|(A1

A2)

c

|;

|(A1 A2)c|=|U| (|A1| + |A2|) + |A1A2|.

A1 U, A2 U A3 U

|(A1

A2

A3)

c

|=

|U| (

|A1

|+

|A2

|+

|A3

|)

+ (|A1A2| + |A1A3| + |A2A3|) |A1A2A3|.

S0 =|U|

Ai U i =1, 2, . . . , n

ni=1

Aic =S0 S1 S2+S3 S4 + (1)n1Sn

=S0 S1+S2 S3+S4 + (1)nSn,

n

i=1Ai

c

=|Ac1Ac2 Acn|=n

j=0(1)jSj.


187/326


188/326


189/326

A1 A2 . . . An

|A1 A2 An|=|A1| |A2| |An|.

Ai

4

2

f1 f2 f3 f4

t1 t2

A1={f1, f2, f3, f4} A2={t1, t2}.

A1

A2

A1 A2= (fi, tj); 1i4 e 1j2, (fi, tj)

fi tj |A1 A2|=|A1| |A2|= 8

9

5


190/326

b1 b2 b3 b4 b5 b6 b7 b8 b9

c1 c2 c3 c4 c5

B ={b1, b2, b3, b4, b5, b6, b7, b8, b9} C={c1, c2, c3, c4, c5}.

B

C

BC

|B C|= 9 5 = 45.

2

10

910 = 90 10

720


191/326

720 = 24 32 51.

A={ 720 2k k Z+},B =

{ 720 3m m

Z+

},

C={ 720 5n n Z+}.

0k4

k >4

25

0k4

A= 20, 21, 22, 23, 24, 0m2 0n1

B =

30, 31, 32

C=

50, 51

.

720

AB C 720 |AB C|

|AB C|=|A| |B| |C| 720 532 = 30

|A|= 5 |B|= 3 |C|= 2

720

20

A


192/326


193/326

720 3

2

1 = 6 A {2

1, 23

} = 3 B {31

} = 2 C {31} = 1 {1, 4, 9, 16, 36, 144} 720

n

n= pk11 pk22 pkrr ,

pi ki Z+

n

d(n)

d(n) = (k1+ 1)(k2+ 1) . . . (kr+ 1).

A1={ n pm11 m Z+},

Ai ={ n pmii t Z+}.

mipi mi > pi

pki+1i

mipi

Ai =p0i , p

1i , p

2i , . . . , p

kii

,

i= 1, 2, 3, . . . , ki.

Ai =ki+ 1.


194/326


195/326

1a

10

7 5

2

A={a1, . . . , a10}, C={c1, . . . , c7} G ={g1, . . . , g5},

|A|= 10 |C|= 7 |G|= 5

A

C

|A C|=|A| |C|= 70

A

G

|A G| =|A| | G | = 50


196/326

C

G

|C

G|=

|C

| | G |= 35

18 12 5

3 2

15 12 = 180

310 = 30

180 + 30 = 210

5

,

U={ 5 , };A={ U };A ={ U };A=

{

U

}.


197/326


198/326

S3=|AAA|= 0. 5

,

AcAcA

c

|AcA

cA

c

|=S0

S1+S2

S3

=243 96 + 3 0=150.

n! n

n! =n (n 1) (n 2) 2 1

n >0

0! = 1

n

n

n

n

Pn

n


199/326

A=

{a1, a2, a3

}

1 = (a1, a2, a3),

2 = (a1, a3, a2),

3 = (a2, a1, a3),

4 = (a2, a3, a1),

5 = (a3, a1, a2),

6 = (a3, a2, a1).

n1

n

O={o1, o2, . . . , on} Pn=n!

n= 1

Pn Pn1 n2

Pn=nPn1.

i

Ai

n 1 {o1, . . . , oi1, oi+1, . . . , on} |Ai| =

Pn1 i = 1, 2, . . . , n

n

oi

Ai n

1

Pn=|A1| + |A2| + + |An|= nPn1.

n 2

n 1

Pn1= (n 1)Pn2,


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Pn=n(n

1)Pn2.

Pn=n(n 1)(n 2) 3 2 1 =n!,

4

pa, pb, pc, pd

{1, 2, 3, 4} (pc, pa, pb, pd) (3, 1, 2, 4)

4! = 24

k

k

1, 2, . . . , k

(a1, a2, . . . , ak) ai {1, 2, . . . , k}

i

ai

(2, 1, 3, . . . , k)

i i i3

k

k

{1, 2, 3, . . . , k} k!


201/326


202/326

p1 n1

Apn=|A1| + |A2| + + |An|= nAp1n1.

n2

n 1

Ap1n1= (n 1)Ap2n2,

Apn=n(n 1)Ap2n2.

Apn=n(n

1)(n

2)

(n

(p

2))A

p(p1)n(p1)

=n(n 1)(n 2) (n p+ 2)A1np+1.

A1np+1 = n p+ 1

Apn=n(n 1)(n 2) (n p+ 2)(n p+ 1)

=n(n

1)(n

2)

(n

p+ 2)(n

p+ 1)

(n

p)

1

(n p) 1=

n!

(n p)! ,


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204/326

2, 3, 4, 5, 7 9

5

400

O={2, 3, 4, 5, 7, 9}

O

A36 =6!

3!= 120.

2

O{2}

A2|O{2}| = A

25 =

5!3!

= 20

2 A2|O{4}|=

A25 = 5!3! = 20 4

20 + 20 = 40


205/326

120 40 = 80

5

5

O {5

}

5

A2|O{5}|=A25=

5!3!

= 20

400

1 / O

2

O {2} A2|O{3}| = A25 = 5!3! = 20 3

20+ 20 = 40

400

p

n


206/326

n

p

0 < p n n

p

np p n

n 1

p n

O = {o1, o2, . . . , on} n

p

= n!

p!(np)!.

p

p O

p

p

p ! p

p

p O p !

p


207/326

p !np =Apn=

n!

(n p)!,

n

p

=

n!

p!(n p)! .

n

i

a

n i b

i

a (ni) b

n

i

a ni

i

a

n

5

52 = 5 !2!3! = 10

6


208/326


209/326

103

104

p 10p

Documents

MA21 - Resolução de Problemas.pdf