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8/13/2019 MA21 - Resoluo de Problemas.pdf
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c
a
b
c2 =a2 +b2
a a
a= 1/2 (1/2)2 =
1/4 1/2
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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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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
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P
Q
P =Q P
Q
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T
a
b
c
c2 =a2 +b2
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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
,
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N2
N
N2
N
N2
N2
N
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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
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x
x+ 1/x2
x
x+ 1/x2
x
x+ 1
x 0
x2 + 1< 2x.
x2 2x+ 1 < 0 (x 1)2
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9
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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
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A
B
C
A
B
C
A B
C
A
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A
A
B
C
D
C
D
A
B
A
B
A
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B
A
B
C
D
C
A
B
D
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1, 2, . . . , 9,
i
j
i
j
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1 2 3
4 5 6
7 8 9
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Q T 3 4 6
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. . .
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1, 2, 3, . . . , 99
2, . . . , 20
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x
y
z
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x
x
2x
x/5
2x+x
5= 121,
10x+x= 605,
11x= 605 x= 605/11 = 55
x
ax+b= 0,
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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,
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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
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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
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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
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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
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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
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6.378.000
C
r
C= 2r,
3, 1416
CT
CT = 2rT=2 3, 1415 6.378.000 = 40.072.974
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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 .
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x
y
2x+ 3y= 14.
x+y = 6.
x1 x2
. . . xn
a1x1+a2x2+ +anxn+b= 0,
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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,
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aij (1ik, 1jn)
aij
xj
(r1, r2, . . . , rn)
(r1, r2, . . . , rn)
2x+ 3y= 14,
x+y = 6.
x
x= 6 y.
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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
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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,
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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,
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A1 A2
A1=
4 A2;
4 A2+ 2A2= 1,
4
+A2= 1.
A2= 1 4 A1= 4
1
4
=
2 1
x
y
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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
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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
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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
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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
.
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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 .
( + )/
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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
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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.
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(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
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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)
.
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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
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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,
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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.
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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,
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x
x
3x100.
99 = 3 33 3 34 = 102> 100
x= 33
x
3x100< 0
x 100/3 3x 100< 0
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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
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ax+b 0.
a >0
ax + b < 0
a
x+b/a b/a
={x R; x >b/a},
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b a
ax + b > 0
={x R; x
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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
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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
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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,
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= +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
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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
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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
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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
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4a>0 a
ax2 +bx+c= a
x+
b
2a
2
4a
4a
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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
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ABCD
r
ABCD
xyr
A B
CD
A= 2x 2y= 4xy.
y=
r2 x2,
A= 4x
r2 x2.
ABCD
A2
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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
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|3 x| |x+ 1|= 4
|2x 5|= 3.
|a|= a a0,a a
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|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
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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
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250
200
A
B
2/3
2/5
a
3a+68
2a+106
x
x
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x
y
z
xy
yx
zxz
x
y
z
15
2
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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)
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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
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n4
+ 4
n >1
n4 + 4n
n >1
966666555557
966666555558
966666555558
966666555559,
n
n200 100000.
n
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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
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x2 = 0
x2 = 0
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x,y,z
n
n xn + yn =zn
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Z
N ={1, 2, 3, . . .}
Z={. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
A
N
aA an nA
A
A
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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.
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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
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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
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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.
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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
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n = 175 50 + 213 20
n
n
n1
n1 = 1 2 + 0 2 = 2.
n1
n
a
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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,
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n
n2
n n2
a b c
a2 b2
a2 +b2 12 + 12 = 2
a2 +b2 =c2 c2
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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
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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
.
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(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.
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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
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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.
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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,
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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
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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
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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
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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.
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(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.
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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.
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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
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a
b
a
b
m
m
a
b
a|m
b|m
m
a b m= (a, b)
m= [a, b]
a,b,c Z
Z
c
a
b
[a, b]|c
[a,b] =[a, b]
|ab|= [a, b] (a, b)
c [a, b]
c= [a, b]q+r, 0r
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T= [15, 18] =15 18
3
= 90.
n1= 6 n2= 5
AB =m CD =n
A B
C
m
n
mn
m
n
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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
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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
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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).
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(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.
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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.
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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,
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N
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, . . . , n
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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).
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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
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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
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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)
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(x1, y1)
(x2, y2)
ax+by =c
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N +
N
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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
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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 +
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k N
N k
N k+ 1 N
k= 1
16 =
3 5 + 1
N= 3
k= 5
Aj j = 1, 2, . . . , 6
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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
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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
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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
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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
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/
1/2
1
2/2
3
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A B
C
D
E
DBE 3
DBE 3/2
DBE
3/2< 3
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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
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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
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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
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5
a
r =p/q
|a
r
|< 1/q2
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A
|A|
n
A1, A2, . . . , An1
An
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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
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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.
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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|,
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S2=
1i1
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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.
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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
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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
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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
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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.
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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
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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
}.
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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
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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!
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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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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
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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
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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
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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
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103
104
p 10p
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