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3 2 3
···
2010
5 13 21 26 29 76cm 5cm
1cm 10cm2 14cm2 16cm2 20cm2 25cm2
14cm 2cm2 4cm2 6cm2 9cm2
12
2 3 4 6
8 33 16 33 42 45 48
12 18 30 31 33
0 9 10 12 21 24
6
188 16 13 8 9 11 13 15 6 2 3 4 6 12 18 154cm2 16cm2 24cm2 28cm2 32cm2 36cm2 00 : 00 : 00
07 : 57 : 35 12 : 00 : 00
20 : 09 : 07
2 3 4 5 6
2
2
2
× ×· · · × 2010
2 4 6 8
−2−
−3−
T 1cm
2cm
7 9 11 13 15 1234567891011121314 . . . 0
2010
2 4 6 8 ABC C
110◦ b
120◦
D
125◦ 135◦
b a 70
140◦
20
A
a
B
16 18 19 20 22
1 2 3 4 5
1 10 100
P P AB ABCD
ABCD P BC ABCD
120cm
200cm2 A
400cm2
D P
600cm2 800cm2
C
B 2
1000cm
n n 16 18 22 23
3 4
5 6
0 1 2 3 4 96 278
2 3 4 5 6
5
15 8
3
5 7 10 13 20 12cm
36cm2 40cm2 48cm2 54cm2 72cm2 10 0 2 4
89
6 8
4 5 6 7
5 10 12 12 55 60 66 78 2 2010 612 610 591 572
5+5+5+5 54 5
×4×3×2
45
1
16 28 34 36
·· ·· ·· ··· 2 3 4
16
5 6 AC
ABC
AB BC
1 2 3 4 5 22010 + 32010 + 52010 + 72010 8 7 6 4 2 x2 1 2 3 4
−
(x, y) y 2 = 303
x
y
2 3 4 5 6
1/5
12 18 19 20 40 80
1
×1×1
2 3 4 6 8
a b = (a + 1) (b a b a+b
× − 1)
∗
11 12 13 17 31
∗
3 5 = (3 + 1) (5 1) = 16 a b = 24 b a = 30
∗
∗
× − ∗
1/3 2/5 1/2 3/5 5/8
20 30 35 40 45 18
17 18 20
21 25
3 6 7 9
4 5 6 7
15 20 25 27 30 1
100
1
100
1 3 5 7 10 2 2 3 4 6 8
40km 50km 60km 70km 80km
1 2 3
4
19100
1 4 5 7 9
31 33 35 38
1 2
3 4
150◦ 180◦ 210◦ 240◦
23 28 33 48
x4 +x3 +x2 +x 4 =
−
0 0 1 2 3
ABCD A BC
AB O
AOC
◦ ◦ ◦ ◦
C
4 1
×1×1
×3×3
1 3
3
×3
11 = 1 + 3 + 7
ABC
160◦
AB = AC
∠DCA
∠BAC
A 160 B
C
3
×3
D
ABC 2
A
C
B
1
2
3 1
8 2
4 1 8
3
2
2
1 3
49
2
100
2
64
2
121
2
81
2
12
12 11
11
1
9
9
3
8
7
5
5 6
6
0 minutos
1 minuto
4
→ → →
4
×4
1 2
10 3
8
4 7
2
10
2
10
12 11
1
9
3
8
4 7
5 6
2 minutos
1, 4, 9, 16, 25, . . . 1, 2, 3, 4, . . . , 9998, 9999, 10000.
1%
15 %
1.5 %
20 %
10 % α
α
3α
25◦ 30◦ 36◦
√ 3
60◦
O ABC
√ 2 3
A
√ 4 3
B O
21000
3
×3
(x, y) xy
x
− x − 15 = 0?
y
C
1 4
AB + BC + CD ABD
AD CD D
y
A
x
y
x
B
C
5cm
10 15 20
2
2
25
2
2 2
P
P
Q
Q
66
6!
A B
A
B A
B
A
B
8
×8
1234567891011121314 . . . 979899100,
m c 2
1 0 c m
8cm
2
A
F C C F E
B
B F D D F
AGF
F 1/16 1/8 1/4
E D
C
1/2
B
G
A
x(x 1)(x 2)(x
− 3)
−
x a, b b
c
a
≥b≥c
−
4 4
Juan
Mariana
3
5
8
1
3
Pedro
4
7
3
7
8
4
Oscar
4
8
7
1
2
8
6
7
Luis 3
2 cm
4 cm
1
5 2
3
4
6
...
7 8
9
10
2011
2014
A
0 AMOR + AMOR AMOR ODIO
1
1000 =
∠
= 150◦
∠ A
150 B
D
O
C
3
1, 2, 3, 4 2011
A
100100
5050 15050 ?
·
12345678910111213 . . . 200920102011 5 7 4
x, y y2
x2 = 2011+
1
3/2
9/4
√ 1+x ≥ 2 x
x
≥0
x2
− 7y = 10
A
m c 1 = r
m 4 c
2011
2100 1
1, 000, 000
25 24
1 cm
6
1 cm
2 cm
4 cm
x2 +
2
x
≥3
x>0
p2 q
−1
p p2
2
− q
p
A B BC E
C D
B C/2 > DE 5
× 10 n n3 + (n + 1)3 +
(n + 2)3
3
2010
3
670 0 b 10 31 28
32 27
26 12 29
13 13
29 x
42
− x = 16 + x 26 = 2x 13 = x
b 5
3 2cm
2cm 4cm 70cm 72cm 37 78cm
11 30
e
12 2 4 c
16cm2
14cm 7 2cm
22 = 4cm a
3 2a + a = 14cm 7a = 14cm a = 2cm
·
A = a2 = 22 = 4cm2
b
3 c
a b
3
c b
b
c a
a
a
a
c b
b c
3
b
8 9 10 27 10 11 12 10 11 12 13 14 15 16
7 8 9 9 10 11 33
24 30
14 + 15 + 16 = 45 a a+1
a+2
(a) + (a + 1) + (a + 2) = 33 3a + 3 = 33 3a = 30 a = 10 10 11 12 13 14 15 16
14 + 15 + 16 = 45 d
3 3 4
12
3
4 2 2
18
3 + 12 + 18 = 33 e
0
2 4
1 3 5 3
6
0 4 3 = 12 c
·
3 5
5 5 1 1 1 1 1+1+1+1+5= 9
b
A 4
b
D
5
154cm2
22 22 154 cm2 22
4 4
· 154 = 28cm 22
2
c 10 20 09 : 59 : 59 19 : 59 : 59 23 : 59 : 59
→ 10 : 00 : 00 → 20 : 00 : 00 → 00 : 00 : 00
b 2
··· ··· ··· ··· 4 2010 = 502 4 + 2
·
2010 2
2
2
× ×· · · × 2010
4
b
4
2
00
318 317
184 186 188 93
···
···
318
2 92
159
91 1 2 a
···
68 12 12 6
12
68
6 6 6
3
3
9 c 31 5 5 1 8
15 22
20
d
1
2
3
2
×2
1
2
3
8+3 = 11 c
9
×1=9
90 2 = 180 100 3 = 300 500 3 = 1500
× × ×
9 + 180 + 300 + 1500 = 1989 2010 3 7
0
− 1989 = 21 700 − 799
7 706
6
ABD
BD BCD
21
3
2010 699 d 2 1 2 3
4
C
b
20
D 1 2
b
a 3
70 A
4
a
B
360◦ 1 + 2 + 3 + 4 + 70
◦ + 20◦ = 360◦
◦ = 360◦ ◦ 1 + 2 + 3 + 4 = 270
1 + 2 + 3 + 4 + 90
DC
BC 1
= 3
AB 2
BC
= 4
= 270◦ ◦ 3 + 4 + 3 + 4 = 270 2 (3 + 4) = 270◦ ◦ 3 + 4 = 135 1 + 2 + 3 + 4
·
ABC 3 4
d
3
11
11
5
3
11 + 3 + 5 = 19 c K h
m
5
m
h
h
m h
K
m h
5
m
h
e
49.999
49.999 h
.002 h (.001)2 + h2 = (49.999)2 h2 = (49.999)2 (.001)2 h2 = 2499.9 h 49.99899999
−
≈
A
≈ 12 · .002 · 49.99899999 ≈ 0.049999
.1
e x
P BC
P AB
2x ABCD
x
A
4x
2x
D
P 2x x
B
C
ABCD
120cm
2(2x) + 2(4x) = 120cm 4x + 8x = 120cm 12x = 120cm 120 x= cm 12 x = 10cm ABCD A = (2x)(4x) = 8x2 = 8(10cm)2 = 800cm2 d n
n n
2
n
n
−1 n
n e abcd b=6 5
d 4 d=0
0
5
4 00 20 40 60
80 c
6
8
c x 96 = 278 x
x
26
38
3 =3 312x = 324
12x = 24 24 x= 12 x=2 a c
n 5
15
5 c + 15 = n
·
8
· − 3) = n
8 (c
3
8(c 8c 8c
− 3) = 5c + 15 − 24 = 5c + 15 − 5c = 15 + 24 3c = 39 39 c= 3 c = 13
d
12cm
l3
l3 l2
l4
12cm
l1
l1
l1
l4
l2
A = l1 l2
·
12cm l1 + l3 = 12cm l3 = 12
−l
l2 + l4 = 12cm
1
2l1 + 2l2 = 2l1 + 2l4 = 2l3 + 24cm
l2 = l4 l2 = 6cm 2l1 + 2 6 = 2l3 + 24 2l1 = 2(12 l1 ) + 24 2l1 = 36 2l1 4l1 = 36 36 l1 = 4 l1 = 9cm
·
−
−
− 12
A = l1 l2 = 9cm 6cm = 54cm2
·
·
d 9 10 9 = 90
8
·
e 5 6
c
4 3 10 4 + 3 + 2 + 1 = 10 12 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 66 d 2 2 1
2010
2
10 10 2 + 1 2 10
· ·
10 2
·
100
2
2000 11
1 (10 10 2 + 1) 10 (10 2) 100 (2) 11
·
· · · · ·
a
612 4 1 3 5 7
5 9 54
b
4
S 4S 1 1+2+3+
16
· · · + 16 = 16(162 + 1) = 8 · 17
4S 4S = 8 17 8 17 S = 4 S = 2 17 S = 34
· · ·
c
3
4
1
2
c
5 BC 3.8+ .6 = 4.4 AB
AC
AB 3
BC BC
4
3.6 d
2 3 5 7
25 = 32 5
5 3
1
4
7 4
2010
4
4 + 9 + 5 + 9 = 27 b
x2 (x + y) (x
2
− y = 303 · − y) = 1 · 303 = −1 · −303 = 3 · 101 = −3 · −101 = 101 · 3 = −101 · −3
x
y
2
x y=1 x + y = 303
−
2x = 304 x = 152
y = 151 x y= x+y =
−
−1 −303 2x =
x=
−152
y=
−151
−304
x y=3 x + y = 101
−
2x = 104 x = 52
y = 49 x y= x+y =
−
−3 −101 2x =
x=
−52
y=
−49
−104
x y = 101 x+y =3
−
2x = 104 x = 52
y=
−49 x y= x+y =
−
−101 −3
x =
−52
2x = 6
y = 49
−104
e
1
2
4
7
3
5
8
6
9
10
1, 2, 3 4, 7, 8 6, 9, 10 1, 8, 9 5
4 c
j
v
c
3 j = v 2 1/5 1 c=j 5
3 1 c 2 5 3 = c 10
v=
10 3
v = c
12 c=
10 12 = 40 3
·
d
C2 C1
F4
F1
F3
F2
1 2
3
2
2 C 1
F 1 F 1 F 2 F 2
F 3 F 3 C 2
F 4 F 4
C2 C1
3
F4
1
F1
F3
2
F2
4 6
4
d a b = 24
∗
a b = 24 (a + 1)(b 1)(b 1) = 24 ab a + b 1 = 24 ab a + b = 25
−
−
∗ − −
b a = 30
∗
b a = 30 (b + 1)(a 1)(a 1) = 30 ab + a b 1 = 30 ab + a b = 31
∗ − − − −
(ab + a
− b) − (ab − a + b) = 31 − 25 2a − 2b = 6 a−b=3 a = b+3
(a + 1)(b 1) = 24 (b + 3 + 1)(b 1) = 24 (b + 4)(b 1) = 24 b2 + 3b 28 = 0 (b + 7)(b 4) = 0
− − − −
−
b=4
a=7
a
a h
h
a
a
a
At AT = 5
· 1 ah 2
5 = ah 2 As
1 As = base altura 2 1 5 = a h 2 2
·
·
c 4 3
4 + 3 + 2 + 1 = 10 2
10
10 2
d
=
10! 10 9 8! = = 45 2!8! 2 8!
· · ·
x
x 18 3(x 18)
18
18
3(x
−
x
− 18)
−
− 18 x
2x
18 18 3(x 18) + 18 = 2x 3x 3(18) + 18 = 2x x = 18(2) x = 36
−
−
18
b
18
3 3
3 4 5
6 7 c
2 3
x y
z
1 3
1
5
3
5
x x
y 3y
z 5z
25 x + 3y + 5z = 25 10 x + y + z = 10 x + 3y + 5z = 25 (x + y + z ) + 2y + 4z = 25 10 + 2y + 4z = 25 2y + 4z = 15 2 (y + 2z ) = 3 5
·
·
e 3 1
6
10
6 3 6 = 18
·
2 3 3 3=9 10 d
·
27
10
10 10 n a
b n = 10 a + b
·
c c (a2 + b2 ) = n
·
c (a2 + b2 ) = 10 a + b ca2 + cb2 = 10a + b ca2 10a = b cb2 a(ac 10) = b(1 cb)
·
·
− −
b=0 a = 1 10 50 b=0
ac
−
−
a a = 2 20
− 10 = 0
10 a=5
a (ac b
− 10) = 1 − cb
a b
b a 11 11 22 33 44 55 66
77 88 99
b r
A
2 2 r B
r = 23 AB
AB AB
2+r 2 (2 + r) 3 3r = 4 + 2r r=4 r=
c t 4
−t
4
distanciasubida = distanciabajada 15 t = 30 (4 t) 15t = 120 30t 45t = 120 120 t= 45 8 t= 3
·
· − −
15 a 4
·
8 3
= 40km
1
5 2
10 9 8 7 6
4
0 1 2 3 4
5 5 5 5 5
· 10 + 4 · 0 = 50 · 9 + 4 · 1 = 49 · 8 + 4 · 2 = 48 · 7 + 4 · 3 = 47 · 6 + 4 · 4 = 46
d
4 9 9
19100 = (1 + 18)100 = 1 + 9k 9k
k 9 1 + 9s 1 + 9 1 = 10 a
1 +9k 1 + 9k 10
·
1
oi a1 a2
ai
o1
3
a3 o2 2
3 31 39 7 c
7 5 = 35
·
o
a m
s
o m+a s=o s+a m
·
·
· −1
·
o m
· − a · m − o · s + a · s = −1 (o − a) · (m − s) = −1 (o − a) > 0 −1 a
η
α c d
β
b e
a
δ
γ
5 180◦ a + β + η = 180◦ b + α + γ = 180◦ c + β + δ = 180◦ d + η + γ = 180◦ e + α + δ = 180◦
o
−a = 1
540◦ 2(α + β + γ + δ + η) + (a + b + c + d + e) = 900◦ 2(α + β + γ + δ + η) + 540◦ = 900◦ 2(α + β + γ + δ + η) = 900◦ 2(α + β + γ + δ + η) = 360◦ 360 ◦ α + β + γ + δ + η = 2 α + β + γ + δ + η = 180◦
− 540◦
b 6 5
3 5 3 11
11
6 c
× 3 = 33 1 (x
x3 + 2x2 + 3x + 4
− 1) 4
b
−4
−2
−40
−1
−2
2 1
C
D
O
B
A
C
D
O
B
A
AO O
BC DO
AD AOD 30◦
180◦
60◦ ADO ADO
DCO
DOC
DOC
75◦
AOC DOC AOC = AOD
+ DOC = 60◦ + 75◦ = 135◦
ADO ODC DC DOC
AOD
b
6 12 18 2010
6
2010
a 4 9
6 4+ 6+ 1 = 11 11 9 = 99
·
c
4 3 3 = 36
· ·
9 36
18 e
− 18 = 18 4
1232331 2 3
c
67+47 = 114 114 20 = 94
67
− 47 = 20 d
−
1 99 9 19 29 89 99 90 99 10 10 + 10 = 20
9 10
···
d 3 2(12) + 2(36) = 96 cm
36cm 96/3 = 32cm
4
c
32/4 = 8cm
b x 2x
8 9
26x 26
d
x 2x
8x 18x 26x
1
2
3
4
5
6
7
8
9
1
2
3
5
4 7
6 8
9
3 4 4
4
4
5 1
2
4 7
3
6 8
9
4
2
3
6 8
9
8
9
3
9 4
1
3
6 7
8
6
7
8
6 1
8
9
4+1+1=6 d 140 140
28
28
=5
28 28 1 14 2 7 4
· · ·
5
7
2 14 2 + 14 + 5 = 21 3 11 10 9 8+2+1 7+3+1 6+4+1 6+3+2 5+3+2+1 4 1 + 2 + 3 + 4 = 10 5+3+2+1 5 180◦ AB ABC BAC
−
BC A 160◦ = 20◦
180◦
DCA AC ACB 20◦
− 2(20◦) = 180◦ −
180◦ 140◦ = 140◦
2
90◦
4 3 + 2 = 14
·
6
5
6 = 4 + 1 + 1 4 2 + 12 + 12 = 18 6 = 3 + 2 + 1 3 2 + 22 + 12 = 14 6 = 2 + 2 + 2 2 2 + 22 + 22 = 12 6
12 d
O A
C
B
1 4
·
1
AOBC 1 = 14 cm2 AOBC
4
ABC
· 1 2
c
1 4
1 = cm2 8 7
1 2
3
1 +1 +1 +1 +1 +1 +1 2 +1 +1 +1 +1 +1 2 +2 +1 +1 +1 2+2+2+1 8
3 +1 +1 +1 +1 3+ 3+ 1 3+ 2+ 1+ 1 3+2+2 b
3 2(24) + 2(36) = 96 cm 96 = 32 3 s
36cm 4
4s = 32 s=8 s2 = 82 = 64cm2 b 5 12 12
12
5
60
60 = 12 5 e
·
2 c 1 10000 1002 = 10000 1 10000 a
100 10000
10000
·
100 100 = 1
a b 180◦
a + b + 3α = 180◦
2a + 2b + α = 180◦ 2
− α2 = 90◦ 6α − α = 180◦
3α
5α = 180◦ α = 36◦
c ABO
r
h
√
1 3 = rh 2
ABC
h
2r
1 2r h = 2 A= 2
· ·
1 rh 2
=2
a
1000 < 1024 103 < 210 (103 )100 < 21000 10300 < 21000 10300
301 23 < 101 (23 )333 < 10333 2 2999 < 2 10333 21000 < 2 10333
×
2
× 10
333
334
× ×
d
·
√
3
1
2
3
4
5
6
7
8
9
1 5 9 3
5 3
=
4 3
5 4 3! = 10 3! 2!
· · ·
3
4
4 3! = =4 3! 1!
· ·
10 + 4 + 8 = 22 15951
5 9 15
6
9
16061 16061
1
− 15951 = 110
0
5
xy x = 15 x(y 1) = 15
− −
15 15 = 1 15 = 15 1 =3 5 =5 3
· · ·
·
8 r1
C 1 r2 r1 = 4 r2 = 1 r3 =
C 2 Asombreada = AC = πr 12
−A +A −A − πr + πr − πr 1 4 −1 + − 4
1
=π =
C 2
C 3
2 2
2 3
3855 π 256
2
4
C 4
2 4
2
2
1
1 16
2
r4 =
1 16
1 2 3 4 5 2 1 3 1 4 3 5 2 4 2 5 5 + 5 = 10
x
x+5 x + 5 = 2x 2x 5 = 2x 5=x
−x d
5 a
p
a + 2 p 2 p + 3m 3m = 15
m
a + 2 p 2 p = 3(5
− m) m
m=0 2 p = 15 a
−a
m=1 2 p = 12 a
−a
8
0
15
m=2 2 p = 9 a a
−
m=3 2 p = 6 a
7
0
5
−a
m=4 2 p = 3 a a
−
4
2
12
0
9
0
6
0
3
m=5 p = 0 a = 0 8 + 7 + 5 + 4 + 2 + 1 = 27 d ABD 23.75 BD + AD = 23.
ADB BD
BC
AB + BAD AB
BD BC AB = BD = AB + BD + AD = AB + AB + AD = 23. 23.75
AD
22 c
−
AB + BC + CD = AB + AB + CD = CD = 1.75 AD = CD + 1.75
4 1
a1
4
4a1 = 1 5 5a2 = 1
a2
4a1 = 5a2 4 a1 = a2 5
4 1 5
a1 = 20 %a1
10cm2 10 + 10 = 20cm2
c
d
P
Q
P Q
32 + a2 = 52 a2 = 25 a2 = 16 a=4
−9 d
2a = 8cm
1
6 2
3
5 3
1
2
· · 6 1
5 2
3 3
6 5! 5 4 3! = 1 = 60 5! 1! 3! 2!
· · · · · · ·
b 6 0 1 2 3 4
5 mod 11
0 mod 11 6 mod 11
5 + 6 mod 11 = 0 mod 11 7 mod 11 7 + 4 mod 11 = 0 mod 11 8 mod 11 8 + 3 mod 11 = 0 mod 11 9 mod 11 9 + 2 mod 11 = 0 mod 11 10 mod 11 10 + 1 mod 11 = 0 mod 11 6 d
8
A B 4
4
8
8 4
b
8 7 6 5 4! = = 70 4! 4 3 2 1
· · · · · · · ·
4
32
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
19
20
17
25
18
26
1
64
32 c
1
100
16 25 36 49 64 81
6162 5253 6364 9495 4647 1819 6+6 = 12
90◦ α 102 82 = 100 64 = 36 = 62
−
6+ h
h
−
α
α
α
α
4 h = 10 8
4 h=8 10 16 h= 5
·
6+ AF c1 c2
AGF
c1 = 16
16 5
cm
AGF EF c2 A1
A2 A1 = 162 = 256 A2
16c2 = c1
A2 = 1cm2 y=x
−
A1 = 256cm2
3 2
− − − − y+
3 2
y+
1 2
1 2
y
y
3 2
9 4
y2
=
1 4
y2
u = y2
− − u
9 4
u
1 4
= u2
−
5 9 u+ = 2 16
− − 5 4
u
2
1
−1 b=c
− s(s
a)(s
− b)(s − c) =
(b + a/2)(b
a2 2
a
≥b
√ b≥ 2
2
=1
2
MA b =
− a/2)(a/2)
− a/2)(a/2) a b − 4 2
2
(b + a/2)(b
+ 2
b=2
8
a2
=1 2 4 = 2 a a2 4 2 b = + 2 = 4 a
a2 2
+ 2
8
a2
− MG
≥ ·
a2 8 = 2 a2 a2 2
√
=
4=2
8
a2
a=2
2011 110 210
211
121 101 1288
2
201
112
120
4 2
2
2
2 4
2
2
4
4
4
102
P = 4(4)+ 2(5) = 16+ 10 = 26cm
2,3, 4 5,6, 7 8,9, 10 11,12, 13 3 2 2012 = 670(3) + 2 3
2
2011 2012
1
2013
1
2
A
3
3
6
2014
A
3 A=3 3 O=9
R 6
3 O
×A
9 3R = 9 A=2 O
R = 3 I = 7 3
2
2M93 + 2M93 2M93 9 D79 3 O=8
× M + 2 ≥ 30
R = 6 I = 5 2
2
1
2M86 + 2M86 2M86 8 D58 M = 9
3
= 8358
× M + 2 = 29 M = 7 3 × M + 2 = 23
2
3
4
1
6
5
D D=3
9
1
6 1
123 124 125 126 134 135 136 145 146 156 2 234 235 236 245 246 256 3 345 346 356 4 456 5
6
10 + 6 + 3 + 1 = 20 1
101
999 1 999
102
− 1 = 99
−1=9 1 (102 10 (10
× ×
3 99 + 90 + 3 = 192 ADB 30◦
− 1) = 99 − 1) = 90 1000
AB = AD
ADC ABD
ABD
ADB 30◦ 180◦
BAD 180◦
− (30◦ + 30◦) = 180◦ − 60◦ = 120◦
6
2 1
· 4 2
3 1
4 3 2! 3 2! = = 18 2! 2 2!
· 4 1
3 2
· · · · ·
=
4 3! 3 2! = 12 3! 2!
· · · 30
14253
1
A
1
2
3
4
4 4
3 2 4 3 2 1 = 24
· · ·
100100 = (1002 )50 = 1000050 5050 15050 = (50 150)50 = 750050 10000 > 7500
·
·
100100 > 5050 15050
·
1
p p p2 p3 p4
5
2011
p4 p4
5
p0 = 1
x2 (x + y ) (x
2
− y = 2011 · − y) = 1 · 2011 = 2011 · 1 = (−1) · (−2011) = (−2011) · (−1) 2011
1 x + y = 1,
x
− y = 2011 2x = 2012
x = 1006 y = 2
−1005 x + y = 2011, 2011,
x
−y =1 2x = 2012
x = 1006 y = 1005 3 x+y = x=
−1006
−1,
x
y = 1005
− y = −2011 2x = −2012
4
x+y = x=
−1006
y=
−1005
−2011, 2011,
x
− y = −1 2x = −2012 a
a h
a 2
2
h = a
2
h = a
√
− a 2
3
2
a=1
√ 3 2
+
√ 3 3 2 2
2
+
= a2
3
9
2
4
√ √ 9 3 3 =
4 2
− +
a2 4
3a2
=
√ 3 3
4
+
√ 9 3
2 4 8 4 3+6 3+9 3 = 8 19 = 3 8
√
√
√
√
√ 0 ≤ ( x − 1)
2
√ 2 x ≤ x+1
√ = x−2 x+1
x2 = 10 + 7y 7y = 3 + 7 + 7y 7y = 3 + 7(y 7(y + 1) x 0
6
x2 = 3 mod 7
1
12
13
14
15
8
9
10
11
4
5
6
7
A
1
2
3
15 1 15 13
15 15 14 13
· · ····
A
14 1 = 15!
M
B O
4
1
C
BC = BM = 4 ACB
AMO
AM AO MO = = AC AB CB AM AO 1 = = A0 + 1 AM + 4 4
4AM = AO + 1 1 AM = (AO + 1) 4
4AO = AM + 4 AM + 4 AO = 4
1 AM + 4 +1 4 4 16AM = AM + 4 + 4 8 AM = 15 AM =
AMO 2099 3
1
·
2
20 3 2 4 = 2 + 2,
9 = 2 + 7,
8 15
20 0
=
4 15
2011 2100 2011 4 9 16
16 = 2 + 14
14 1
999999 105 1
1
−
1000000
104 1
−
103
−1
102 10
1 10 100 1000 10000
× (10 × (10 × (10 × (10 × (10
6 1000000 99900 + 99000 + 90000 + 6 = 488895
5 4 3 2 1
−1
− 1) = 99999 − 1) = 99990 − 1) = 99900 − 1) = 99000 − 1) = 90000 99999 + 99990 + l
A = l2 24
1cm
l1
A = l2 = 24 + l12
−1
24 = l l > l1
2 1
−l
= (l
24 = 1 =2 =3 =4
l
−l
1
= 7,
− l )(l + l ) 1
1
· 24 · 12 ·8 ·6
l + l1 = 12
l
l=7
−l
1
= 4,
l + l1 = 6
l=5 49cm2 25cm2 6 4
6
42 = 16 62 = 36 4
10k + 4
(10k + 4)2 = 100k 2 + 80k + 16 = 100k2 + (8k + 1)10 + 6 8k + 1 6
10k + 6
(10k + 6)2 = 100k 2 + 120k + 36 = 100k2 + (12k + 3)10 + 6 12k + 3 a
a h
a 2
2
h=a
√
3
2
h = a h
− a
2
2
2
= a2
−
a2 4
=
3a2 4
h’
a
√
3
h’
2
a 2
√ − 2
a 3 2
a 2
3a2 4
a h = √ 2
2
= (h )2
a2 = (h )2 4 2a2 = (h )2 4
−
a=1 2
4
1 2 4 7 + + = cm. 2 2 2 2
√ √ √ √
x>0
x+2 > 0 0
2
3
≤ (x − 1) (x + 2) = x − 3x + 2
x>0
x
0
≤ x − 3 + x2 2
x2 +
2
x +
1
x
3 x2 +
2
x
≥3
+
1
x
2 x
≥3
≥ · 3
x2
1 1 x x
·
1 1
−1 −1
22
−1
0
1
11
56
5
9 105 56 =
6
· −
884, 375 p2
− 1 = ( p − 1)( p+1)
p
2 p
3
−
4 1, p , p + 1
p
3
2 3 4 = 24
· ·
1)
2
− (q −
(a) 24 1) = p2
p2
−1
q 2
−1
( p2
2
− q
10
B
C
AB
AC
−
C C’ E A
D B’
B
AB C 1:2 B C > DE
ABC B C = BC/2
A AB
AC
A E
→D→
→ B → C A
5 5
×
1
n=1 n n3 + (n + 1)3 + (n + 2)3 9 (n+1)3 +(n+2)3 +(n+3)3 = n3 +(n+1) 3 +(n+2) 3 +9n2 +27n+27.
