1 3 1 ⋅ 25 − 5 5
1.
2 5
) −3
) 21
) 25
1 3
2.
:
1 5
) 29
2 5
.
: 0
) 45
3. 50 % )5
30
) 600
) 750
) 900
) 15
) 20
) 50
:
C
4.
,
N
CM < CN .
?
M
) AM = CM
) AM = BN
) AM > BN
) AM < BN
N
B
5.
3:7 :8,
,
:
) ) ) ) 15
6.
34
.
. ) 64
) 78
5
7. ) 0,4 8. ΔABC )4
2
) 83
= −0, 2
−
) 2,2
M 12
. )3
N ,
)6
) 91
: )0
ΔMNC
AB
) 1,8
ΔABC
. :
)8
C
A
M
N
B 1
x
9. 2 x( x − 3) − (2 x − 1)( x + 2) )2 ) −3
: ) −9
)9
ΔABC .
10.
C 130 0
∠ABC
:
) 250
) 550
105 0 ,
) 650
1300 A
) 750
B
11.
D
C
,
ABCD
3
1050
?
(
.)
B
A
. 52.252
12. ) 56
:
) 202 a⊥b
13.
cEd ,
) 58
) 125
α :
c
3α
a
α
d b ) 300
) 450
BL ( L ∈ AC ) 14. BL = BC ∠BAC = 540 , : ) 36
2
0
) 64
0
) 600
∠ACB 0
) 68
) 750
C
ΔABC ,
0
) 78
L A
540
B
4
15.
,
.
.2
.1 4 .1
2
. 2,
) S1 < S 2
.
S1 ,
S2
: ) S1 = S2
) S1 > S2
.
16.
, : )0
) S1 =
v
e
,
, f
, )1
)2
C
ABCD 17. e , M ∠MAD : ∠MBA : ∠MCD = 1: 2 : 5 . ∠BMC .
18.
) 600
) 74030 '
) x = −2
( x + 2)
x,
3
)
− a ( x + 1) x 2 − x + 1 + 3ax 2
)1
)3
-
(
) x = −3
:
a
) −2
B
:
) x=0
19.
(
A
3 +9 −2
) x=2
20.
M
) 1050
-
v+ f −e )4
D
) 52030 '
2 S2 3
)0
= 22007 + 22008 + 22009 ?
, .)
3
21.
,
,
. ? ) 10
) 1103
) 1485
) 2037
1 ⎞⎛ 1 ⎞ ⎛ 1 ⎞⎛ 1 ⎞⎛ 1 ⎞⎛ 1 ⎞ ⎛ ⎜1 − ⎟⎜1 + ⎟⎜ 1 − ⎟⎜ 1 + ⎟ ... ⎜ 1 − ⎟⎜ 1 + ⎟ ⎝ 3 ⎠⎝ 4 ⎠⎝ 5 ⎠⎝ 6 ⎠ ⎝ 19 ⎠⎝ 20 ⎠
22.
: )1
20 21
)
) 0,7
) 0,15
23.
.
= ∠BCD = 240
240
240
CD = CM ,
=
∠
=
D
∠MDB
.
.)
(
2310 + 3510 :
24. )1
)2
)3
)4
25.
.
∠
= 200 ,
:
) 700
) 1400
) 400
) 600
26.
27. )
200
14 − 22b =7 , 3
) 10,5
4
∠
21a − 33b
) 14
:
) 21 a ( x + 1) = x − 1
) a=0
) a = −1
) 31,5
? ) a =1
ΔABC (∠ACB = 900 ) CL ( L ∈ AB ) C. ∠ABC : ∠BAC = 2 : 7 .
28. CH ( H ∈ AB ) ∠HCL ,
) 150
) 250
) 200
A
) 100
5 3
) 11
30
30.
:
)0
,
B
L
6 − 5 − 7 = −2
29. )
C
11 7
,
.
20
)
. ,
?
.)
(
25
31.
20
45%
,
. ?
.)
( ABC
32.
∠BAC = 45
0
C
∠ABC = 30 . 0
M M
, ∠MAB = ∠MBA = 150 .
150
A
∠BMC
150
B
.
.)
(
33. ) m>0
34.
m <0 , k <0 n
,
2 x − 1 < 3x + 7 − 3 3 2
(
) x ∈ −∞; − 14 5
: ) mn < km
) m>n
) x ∈ ( −∞; −1)
n>0,
)
) m<k <n
:
(
) x ∈ − 14 ; +∞ 5
)
) x ∈ ( −1; +∞ )
5
3 x + 13 ≥ 4
,
35. )2
)9
2 − x > −3
1, 2
) 11
:
) 13
( H ∈ AB )
36.
ΔABC
(∠
= 900 ) .
AB
M A ∠
B
,
.
M.
= 600 ,
B
B
= 12
∠
,
. (
.)
37.
3 − 2x ≥ 5
) x ∈ ( −∞; −1] ∪ [ 4; +∞ )
) x ∈ ( −∞; −1] ∪ ( 5; +∞ )
BC
38.
M
,
ABC
∠AMC = 600
) 100
) 120
)2
6
) x ∈ [ 4; +∞ )
) x ∈ ( −∞; −1]
C
1 CM = CB . 3
∠ABC ,
39. x+5 x−4 1 + =a+ 2 3 2
:
M
∠BAC = 1200 . ) 150
A
) 180
B
a
)0
−2 ? ) −1
) −
1 2
OL→
40.
∠
LH ⊥ OB ( H ∈ OB ). =1
=6
∠AML = ∠LPO .
,
. H
, L
. (
i
.)
iB
41.
,
.
, : .” .” .”
:„ :„ :„ ,
)
)
42.
-
)
92,
1
) 21
?
)
16 % .
) 26
.
)
44.
-
:
) 18
)
)
64 % 64 %
30
32 %
?
(
.) D
AC = 12
45.
M
ABCD
,
∠D
= 900 .
CM = 2
(
,
)
) 15
43.
?
,
. .)
7
∠BDC = 300 .
ABCD ,
46.
N
D
O
O
BD
M ΔMBN
AB N.
OM = 3
,
) 12
CD
) 18
.
M
) 19
) 21
30
47.
4
1
0
,
, .
,
?
(
.) D
48.
32 DP
DQ
AB
BC .
∠
= 300 .
Q
2
D, D
DQ,
30
P ) 1,5
)2
) 2,5
)3 AB = 9
ABCD
49.
(
BC = 7
.
∠ABC = ∠ADC
, AC ?
.) 16, 4, 9, 6, 8, 12, 11, 1
50.
, , 3 )3
8
0
-
3. ,
,
.
, ?
) 6
) 9
) 12
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Ɂɚɞɚɱɚ 6. ȼ ɬɚɡɢ ɡɚɞɚɱɚ ɟ ɜɚɠɧɨ ɞɚ ɫɟ ɢɡɩɨɥɡɜɚ ɧɟɪɚɜɟɧɫɬɜɨɬɨ ɧɚ ɬɪɢɴɝɴɥɧɢɤɚ, ɬ.ɟ.ɫɴɨɛɪɚɠɟɧɢɟɬɨ, ɱɟ ɫɛɨɪɴɬ ɧɚ ɞɜɟ ɨɬ ɫɬɪɚɧɢɬɟ ɜ ɬɪɢɴɝɴɥɧɢɤɚ ɟ ɩɨ-ɝɨɥɹɦ ɨɬ ɬɪɟɬɚɬɚ ɫɬɪɚɧɚ. Ɉɬɬɭɤ ɫɥɟɞɜɚ, ɱɟ ɨɫɧɨɜɚɬɚ ɧɚ ɬɪɢɴɝɴɥɧɢɤɚ ɟ 15 ɫɦ. Ɂɚɞɚɱɚ 14. Ⱥɤɨ ∠ABL = x , ɬɨ ∠ACB = ∠BLC = ∠BAC + ∠ABL = 540 + x . Ɍɨɝɚɜɚ, ɤɚɬɨ ɢɡɩɨɥɡɜɚɦɟ, ɱɟ ɫɭɦɚɬɚ ɨɬ ɴɝɥɢɬɟ ɜ ∆ABC ɟ 1800 , ɩɨɥɭɱɚɜɚɦɟ 540 + 2 x + 540 + x = 1800 . Ɉɬɬɭɤ x = 240 ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ ∠ACB = 540 + 240 = 780 .
Ɂɚɞɚɱɚ 17. Ⱥɤɨ ∠MAD = α , ɬɨ ∠MBA = 2α ɢ ∠MCD = 5α . Ɍɨɝɚɜɚ ∠BAM = ∠BMA = 900 − α ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ ∆BAM ɟ ɪɚɜɧɨɛɟɞɪɟɧ ( AB = MB ). Ɉɬɬɭɤ ɡɚɤɥɸɱɚɜɚɦɟ, ɱɟ ∆BMC ɟ ɫɴɳɨ 1800 − (900 − 2α ) ɪɚɜɧɨɛɟɞɪɟɧ ( MB = BC ). Ɍɨɝɚɜɚ ∠BMC = ∠BCM = = 450 + α ɢ ɡɧɚɱɢ: 2 0 0 ∠BCD = 90 = ∠BCM + ∠MCD = 45 + α + 5α α = 7, 50 . ɋɥɟɞɨɜɚɬɟɥɧɨ ∠BMC = 450 + 7,50 = 52,50 = 52030 ' . D
C M
A
B
Ɂɚɞɚɱɚ 18. Ⱥɛɫɨɥɸɬɧɚɬɚ ɫɬɨɣɧɨɫɬ ɟ ɩɨ-ɝɨɥɹɦɚ ɢɥɢ ɪɚɜɧɚ ɧɚ ɧɭɥɚ, ɨɬɤɴɞɟɬɨ ɫɥɟɞɜɚ, ɱɟ ɧɚɣ-ɦɚɥɤɚɬɚ ɫɬɨɣɧɨɫɬ ɧɚ ɢɡɪɚɡɚ ɫɟ ɩɨɥɭɱɚɜɚ ɜ ɫɥɭɱɚɹ, ɤɨɝɚɬɨ ɚɛɫɨɥɸɬɧɚɬɚ ɫɬɨɣɧɨɫɬ ɫɟ ɚɧɭɥɢɪɚ, ɬ.ɟ. ɩɪɢ x = −3 .
Ɂɚɞɚɱɚ 21. Ⱥɤɨ ɝɨɞɢɧɢɬɟ ɧɚ ɞɜɚɦɚɬɚ ɫɚ ɫɴɨɬɜɟɬɧɨ 10a + b ɢ 10b + a , ɬɨ: (10a + b)2 − (10b + a )2 = (10a + b + 10b + a )(10a + b − 10b − a ) = 9.11(a + b)(a − b) . ɉɨɥɭɱɟɧɨɬɨ ɱɢɫɥɨ ɫɟ ɞɟɥɢ ɧɚ 9 ɢ ɧɚ 11. Ɉɬ ɩɨɫɨɱɟɧɢɬɟ ɨɬɝɨɜɨɪɢ ɫɚɦɨ ɟɞɢɧɢɹɬ ɩɪɢɬɟɠɚɜɚ ɬɨɜɚ ɫɜɨɣɫɬɜɨ. ɋɴɳɟɫɬɜɭɜɚ ɩɪɢɦɟɪ, ɤɨɣɬɨ ɪɟɚɥɢɡɢɪɚ ɫɢɬɭɚɰɢɹɬɚ: Ɇɚɪɬɢɧ ɟ ɧɚ 14 ɝɨɞɢɧɢ, ɚ ɥɟɥɹɬɚ ɟ ɧɚ 41 ɝɨɞɢɧɢ. Ɂɚɞɚɱɚ 22. § 1 ·§ 1 · 5 4 ¨1 + ¸¨1 − ¸ = ⋅ = 1 ; © 4 ¹© 5 ¹ 4 5
§ 1 ·§ 1 · 7 6 ¨1 + ¸ ¨1 − ¸ = ⋅ = 1 ; © 6 ¹© 7 ¹ 6 7
1 ·§ 1 · 19 18 § … ¨1 + ¸ ¨1 − ¸ = ⋅ = 1 ɢ © 18 ¹ © 19 ¹ 18 19 1 · 2.21 7 § 1 ·§ ɫɥɟɞɨɜɚɬɟɥɧɨ ɫɬɨɣɧɨɫɬɬɚ ɧɚ ɩɪɨɢɡɜɟɞɟɧɢɟɬɨ ɟ ɪɚɜɧɚ ¨1 − ¸ ¨1 + ¸ = = = 0, 7 . © 3 ¹ © 20 ¹ 3.20 10
Ɂɚɞɚɱɚ 23. ȿɞɧɚɤɜɨɫɬɬɚ ɧɚ ɬɪɢɴɝɴɥɧɢɰɢɬɟ AMC ɢ BDC ɩɨɡɜɨɥɹɜɚ ɨɩɪɟɞɟɥɹɧɟɬɨ ɧɚ ∠BDC . ɂɦɚɦɟ ∠BDC = ∠AMC == 1800 − (600 + 240 ) = 960 . Ɍɴɣ ɤɚɬɨ ∆MDC ɟ ɪɚɜɧɨɫɬɪɚɧɟɧ, ɬɨ ∠BDM = ∠BDC − 600 = 360 . Ɂɚɞɚɱɚ 25. Ɍɪɢɴɝɴɥɧɢɰɢɬɟ ABO , BOC ɢ AOC ɫɚ ɪɚɜɧɨɛɟɞɪɟɧɢ. Ⱥɤɨ ∠ACO = x ɢ ∠BCO = y , ɬɨ 2 x + 2 y + 2∠BAO = 1800 . Ɉɬɬɭɤ ɧɚɦɢɪɚɦɟ x + y = 700 ɢ ɢɡɩɨɥɡɜɚɦɟ, ɱɟ ∠ACB = x + y . Ɂɚɞɚɱɚ 28. ɉɨɫɥɟɞɨɜɚɬɟɥɧɨ ɧɚɦɢɪɚɦɟ ∠ABC = 200 , ∠LCB = 450 (ɩɨɥɨɜɢɧɚɬɚ ɨɬ ɩɪɚɜɢɹ ɴɝɴɥ) ɢ ∠ALC = 650 (ɜɴɧɲɟɧ ɴɝɴɥ ɡɚ ∆LBC ). Ɉɬɬɭɤ ∠HCL = 900 − 650 = 250 . Ɂɚɞɚɱɚ 30. ɓɟ ɢɡɦɟɪɜɚɦɟ ɜɪɟɦɟɬɨ ɜ ɦɢɧɭɬɢ, ɚ ɩɴɬɹ – ɜ ɦɟɬɪɢ. ɇɟɤɚ ɫɤɨɪɨɫɬɬɚ ɧɚ ɟɫɤɚɥɚɬɨɪɚ ɟ x , ɚ 1 1 x ɬɚɡɢ ɧɚ ɱɨɜɟɤɚ (ɩɪɢ ɩɴɪɜɚɬɚ ɫɢɬɭɚɰɢɹ) ɟ y . Ɍɨɝɚɜɚ ( x + y ) ⋅ = ( x + 3 y ) ⋅ . Ɉɬɬɭɤ y = 2 3 3 x x+ x+ y x+ y 3 = 2 ɦɢɧɭɬɢ = 40 ɫɟɤɭɧɞɢ. ɢ ɩɨɧɟɠɟ ɩɴɬɹɬ ɟ , ɬɨ ɬɴɪɫɟɧɨɬɨ ɜɪɟɦɟ ɟ = 2 2x 2x 3 Ɂɚɞɚɱɚ 32. Ⱥɤɨ ɫɢɦɟɬɪɚɥɚɬɚ ɧɚ ɨɬɫɟɱɤɚɬɚ AB ɩɪɟɫɢɱɚ ɫɬɪɚɧɚɬɚ BC ɜ ɬɨɱɤɚ N , 600 = ∠MNB = ∠MNA = ∠ANC . Ɉɫɜɟɧ ɬɨɜɚ ∠BAN = 300 ɢ ɫ ɩɨɦɨɳɬɚ ɧɚ ɭɫɥɨɜɢɟɬɨ ɡɚɞɚɱɚɬɚ ɡɚɤɥɸɱɚɜɚɦɟ, ɱɟ ∠MAN = ∠CAN . Ɍɨɝɚɜɚ ɬɪɢɴɝɴɥɧɢɰɢɬɟ AMN ɢ ACN ɟɞɧɚɤɜɢ, ɨɬɤɴɞɟɬɨ MN = CN . ȼ ɪɚɜɧɨɛɟɞɪɟɧɢɹ ∆CMN ɢɦɚɦɟ, ɱɟ ∠MNC = 1200 ɫɥɟɞɨɜɚɬɟɥɧɨ ∠MCB = 300 . ɇɚɣ-ɧɚɤɪɚɹ ∠BMC = 1800 − (300 + ∠MBC ) = 1350 .
ɬɨ ɨɬ ɫɚ ɢ
Ɂɚɞɚɱɚ 38. ɇɟɤɚ MC = x . Ɍɨɝɚɜɚ BM = 2 x . Ⱦɚ ɩɨɫɬɪɨɢɦ ɩɟɪɩɟɧɞɢɤɭɥɹɪ BK ɨɬ ɬɨɱɤɚɬɚ B ɤɴɦ AM ( K ∈ AM ) . Ɉɬ ɩɪɚɜɨɴɝɴɥɧɢɹ ɬɪɢɴɝɴɥɧɢɤ MBK , ɤɨɣɬɨ ɟ ɫ ɨɫɬɴɪ ɴɝɴɥ 30° , 1 ɩɨɥɭɱɚɜɚɦɟ MK = BM = x . Ɉɬɬɭɤ ɫɥɟɞɜɚ, ɱɟ ∆MKC ɟ ɪɚɜɧɨɛɟɞɪɟɧ. Ɉɬ ɬɨɡɢ 2 K ɬɪɢɴɝɴɥɧɢɤ C
M A
B
ɧɚɦɢɪɚɦɟ, ɱɟ ∠MKC = 30° ∠BKC = 120° ɢ ∆CBK ɟ ɪɚɜɧɨɛɟɞɪɟɧ ɫ ɛɟɞɪɚ BK = CK . Ⱥɤɨ ɞɨɩɭɫɧɟɦ ɫɟɝɚ, ɱɟ KA < KB = KC , ɬɨ ɨɬ ∆AKC ɢɦɚɦɟ ∠CAK > ∠ACK ∠CAK > 75° . Ɉɬ ɞɪɭɝɚ ɫɬɪɚɧɚ, ɨɬ ∆ABK ɢɦɚɦɟ ∠BAK > ∠ABK ∠BAK > 45° . ɇɨ ɬɨɝɚɜɚ ɳɟ ɫɥɟɞɜɚ, ɱɟ ∠BAC = ∠CAK + ∠BAK > 120° , ɤɨɟɬɨ ɩɪɨɬɢɜɨɪɟɱɢ ɧɚ ɭɫɥɨɜɢɟɬɨ ɧɚ ɡɚɞɚɱɚɬɚ. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɟ ɞɚ ɫɟ ɞɨɤɚɠɟ, ɱɟ ɧɟ ɟ ɜɴɡɦɨɠɧɨ KA > KB = KC . ɋɥɟɞɨɜɚɬɟɥɧɨ KA = KB = KC ɢ ɤɚɬɨ ɢɡɩɨɥɡɜɚɦɟ ɴɝɥɢɬɟ ɧɚ ∆ABK , ɧɚɦɢɪɚɦɟ ∠ABC = 15° . Ɂɚɞɚɱɚ 40. Ⱥɤɨ LK ⊥ AO ( K ∈ AO ), ɬɨ ɨɬ ɫɜɨɣɫɬɜɨɬɨ ɧɚ ɴɝɥɨɩɨɥɨɜɹɳɚɬɚ ɫɥɟɞɜɚ, ɱɟ LH = LK . Ɍɨɝɚɜɚ ɬɪɢɴɝɴɥɧɢɰɢɬɟ LKM ɢ LHP ɫɚ ɟɞɧɚɤɜɢ. ȿɞɧɚɤɜɢ ɫɚ ɢ ɬɪɢɴɝɴɥɧɢɰɢɬɟ KLO ɢ HLO . ɋɥɟɞɨɜɚɬɟɥɧɨ, ɚɤɨ HP = KM = x , ɬɨ OM + x = OP − x ɢ ɨɬɬɭɤ ɧɚɦɢɪɚɦɟ x = 2 ɫɦ. Ɂɚɞɚɱɚ 41. Ɍɜɴɪɞɟɧɢɹɬɚ ɧɚ ɞɜɚɦɚ ɨɬ ɛɪɚɬɹɬɚ (Ⱥɥɟɤɨ ɢ ȼɟɥɢɧ) ɫɚ ɮɚɤɬɢɱɟɫɤɢ ɟɞɧɢ ɢ ɫɴɳɢ. ɋɥɟɞɨɜɚɬɟɥɧɨ ɬɟ ɫɚ ɢɥɢ ɟɞɧɨɜɪɟɦɟɧɧɨ ɜɟɪɧɢ, ɢɥɢ ɟɞɧɨɜɪɟɦɟɧɧɨ ɧɟɜɟɪɧɢ. Ɍɨɜɚ ɨɡɧɚɱɚɜɚ, ɱɟ Ⱥɥɟɤɨ ɢ ȼɟɥɢɧ ɢɥɢ ɟɞɧɨɜɪɟɦɟɧɧɨ ɤɚɡɜɚɬ ɢɫɬɢɧɚɬɚ, ɢɥɢ ɟɞɧɨɜɪɟɦɟɧɧɨ ɥɴɠɚɬ. Ɉɬ ɭɫɥɨɜɢɟɬɨ ɧɚ ɡɚɞɚɱɚɬɚ ɫɥɟɞɜɚ, ɱɟ Ⱥɥɟɤɨ ɢ ȼɟɥɢɧ ɥɴɠɚɬ. ɋɚɦɨ Ȼɨɪɢɥ ɤɚɡɜɚ ɢɫɬɢɧɚɬɚ ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ Ⱥɥɟɤɨ ɟ ɭɛɢɥ ɥɚɦɹɬɚ. Ɂɚɞɚɱɚ 42. ɋɛɨɪɴɬ ɧɚ ɩɴɪɜɢɬɟ 13 ɟɫɬɟɫɬɜɟɧɢ ɱɢɫɥɚ ɟ ɪɚɜɟɧ ɧɚ 91, ɤɨɟɬɨ ɟ ɫ ɟɞɢɧɢɰɚ ɩɨ-ɦɚɥɤɨ ɨɬ 92. ɋɥɟɞɨɜɚɬɟɥɧɨ ɟɞɢɧɫɬɜɟɧɚɬɚ ɜɴɡɦɨɠɧɨɫɬ ɟ ɩɨɫɥɟɞɧɨɬɨ ɱɢɫɥɨ 13 ɞɚ ɫɟ ɡɚɦɟɧɢ ɫɴɫ ɫɥɟɞɜɚɳɨɬɨ 14. ɂɦɚɦɟ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 14 = 92 , ɨɬɤɴɞɟɬɨ 1 + 14 = 15 . Ɂɚɞɚɱɚ 43. Ⱥɤɨ ɟɞɢɧ ɞɟɧ ɨɬ ɫɟɞɦɢɰɚɬɚ ɩɪɟɡ ɦɟɫɟɰ ɹɧɭɚɪɢ (ɤɚɤɬɨ ɢ ɡɚ ɜɫɟɤɢ ɞɪɭɝ ɦɟɫɟɰ ɫ 31 ɞɧɢ) ɟ ɧɚ ɞɚɬɚ 1, 2 ɢɥɢ 3, ɬɨ ɬɨɡɢ ɞɟɧ ɫɟ ɫɪɟɳɚ ɨɛɳɨ 5 ɩɴɬɢ. Ɍɨɜɚ ɟ ɬɚɤɚ, ɡɚɳɨɬɨ 1 + 4.7 = 29 < 31 , 2 + 4.7 = 30 < 31 ɢ 3 + 4.7 = 31 . ɋɚɦɨ ɜ ɫɥɭɱɚɣ, ɱɟ ɬɨɡɢ ɞɟɧ ɟ ɧɚ ɞɚɬɚ 4, 5, 6 ɢɥɢ 7, ɬɨɣ ɫɟ ɫɪɟɳɚ ɨɛɳɨ 4 ɩɴɬɢ ɜ ɦɟɫɟɰɚ (ɫɟɝɚ a + 4.7 > 31 ɡɚ a = 4, 5, 6 ɢɥɢ 7). Ɉɬ ɞɪɭɝɚ ɫɬɪɚɧɚ ɨɬ ɜɬɨɪɧɢɤ ɞɨ ɫɴɛɨɬɚ ɢɦɚ ɬɪɢ ɞɪɭɝɢ ɞɧɢ ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ ɧɟ ɟ ɜɴɡɦɨɠɧɨ ɜɬɨɪɧɢɤ ɞɚ ɟ ɧɚ ɞɚɬɚ 4, ɚ ɫɴɛɨɬɚ ɞɚ ɟ ɧɚ ɞɚɬɚ 7. ɇɨ ɨɬ ɫɴɛɨɬɚ ɞɚ ɜɬɨɪɧɢɤ ɢɦɚ ɞɜɚ ɞɪɭɝɢ ɞɧɢ ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ ɟɞɢɧɫɬɜɟɧɚɬɚ ɜɴɡɦɨɠɧɨɫɬ ɟ ɫɴɛɨɬɚ ɞɚ ɫɟ ɩɚɞɧɟ ɧɚ ɞɚɬɚ 4, ɚ ɜɬɨɪɧɢɤ – ɫɴɨɬɜɟɬɧɨ ɧɚ ɞɚɬɚ 7. ɋɟɝɚ ɥɟɫɧɨ ɫɟ ɫɴɨɛɪɚɡɹɜɚ, ɱɟ 1 ɹɧɭɚɪɢ ɟ ɜ ɫɪɹɞɚ.
Ɂɚɞɚɱɚ 46. Ɍɪɢɴɝɴɥɧɢɰɢɬɟ AMO ɢ CNO ɫɚ ɟɞɧɚɤɜɢ ɩɨ ɜɬɨɪɢ ɩɪɢɡɧɚɤ. ɋɥɟɞɨɜɚɬɟɥɧɨ MO = NO . Ɍɨɝɚɜɚ ∆MBN ɟ ɪɚɜɧɨɛɟɞɪɟɧ, ɡɚɳɨɬɨ BO ɟ ɟɞɧɨɜɪɟɦɟɧɧɨ ɜɢɫɨɱɢɧɚ ɢ ɦɟɞɢɚɧɚ ɜ ɧɟɝɨ. Ɉɫɜɟɧ ɬɨɜɚ ∠BMN = 900 − ∠ABD = 900 − 300 = 600 . ɉɨ ɬɨɡɢ ɧɚɱɢɧ ɡɚɤɥɸɱɚɜɚɦɟ, ɱɟ ∆MBN ɟ ɪɚɜɧɨɫɬɪɚɧɟɧ ɢ ɩɟɪɢɦɟɬɴɪɴɬ ɦɭ ɟ ɪɚɜɟɧ ɧɚ 6 ɩɴɬɢ ɞɴɥɠɢɧɚɬɚ ɧɚ OM , ɬ.ɟ. ɧɚ 18 ɫɦ. Ɂɚɞɚɱɚ 47. Ɇɚɤɫɢɦɚɥɧɨ ɜɴɡɦɨɠɧɢɹɬ ɪɟɡɭɥɬɚɬ ɟ 30.4 = 120 . ɉɨɧɟɠɟ 29.4 = 116 , ɬɨ ɜɫɟɤɢ ɪɟɡɭɥɬɚɬ, ɤɨɣɬɨ ɟ ɩɨ-ɝɨɥɹɦ ɨɬ 116, ɦɨɠɟ ɞɚ ɫɟ ɩɨɫɬɢɝɧɟ ɫ 30 ɜɹɪɧɨ ɪɟɲɟɧɢ ɡɚɞɚɱɢ. Ɉɬɬɭɤ ɫɥɟɞɜɚ, ɱɟ ɪɟɡɭɥɬɚɬɢɬɟ 119, 118 ɢ 117 ɧɟ ɦɨɝɚɬ ɞɚ ɫɟ ɪɟɚɥɢɡɢɪɚɬ (ɧɹɦɚ ɡɚɞɚɱɚ, ɤɨɹɬɨ ɞɚ ɛɴɞɟ ɨɰɟɧɟɧɚ ɫ −1 ). Ɉɬ ɞɪɭɝɚ ɫɬɪɚɧɚ 28.4 = 112 . Ɍɨɝɚɜɚ ɜɫɟɤɢ ɪɟɡɭɥɬɚɬ, ɤɨɣɬɨ ɟ ɩɨ-ɝɨɥɹɦ ɨɬ 112 ɢ ɟ ɩɨ-ɦɚɥɴɤ ɨɬ 116, ɦɨɠɟ ɞɚ ɫɟ ɩɨɫɬɢɝɧɟ ɫ 29 ɜɹɪɧɨ ɪɟɲɟɧɢ ɡɚɞɚɱɢ. ɋɟɝɚ ɢɦɚ ɟɞɧɚ ɡɚɞɚɱɚ, ɤɨɹɬɨ ɦɨɠɟ ɞɚ ɛɴɞɟ ɨɰɟɧɟɧɚ ɫ −1 . Ɉɬɬɭɤ ɫɥɟɞɜɚ, ɱɟ ɪɟɡɭɥɬɚɬ 115 ɟ ɪɟɚɥɢɡɭɟɦ, ɧɨ ɪɟɡɭɥɬɚɬɢ 114 ɢ 113 ɧɟ ɫɚ. ɉɨ ɩɨɞɨɛɟɧ ɧɚɱɢɧ ɡɚɤɥɸɱɚɜɚɦɟ, ɱɟ ɧɟ ɦɨɠɟ ɞɚ ɫɟ ɪɟɚɥɢɡɢɪɚ ɢ ɪɟɡɭɥɬɚɬ 109. ɋɥɟɞɨɜɚɬɟɥɧɨ ɜɴɡɦɨɠɧɢɬɟ ɪɟɡɭɥɬɚɬɢ ɨɬ 1 ɞɨ 120 ɜɤɥɸɱɢɬɟɥɧɨ ɫɚ ɨɛɳɨ 120 − (3 + 2 + 1) = 114 . ȼɴɡɦɨɠɧɢ ɫɚ ɫɴɳɨ ɜɫɢɱɤɢ ɪɟɡɭɥɬɚɬɢ ɨɬ −30 ɞɨ 0 ɜɤɥɸɱɢɬɟɥɧɨ (ɬɟ ɫɚ ɨɛɳɨ 31). ɉɨɥɭɱɚɜɚɦɟ 114 + 31 = 145 ɢ ɬɨɝɚɜɚ ɭɱɚɫɬɧɢɰɢɬɟ ɬɪɹɛɜɚ ɞɚ ɫɚ ɧɚɣ-ɦɚɥɤɨ 146, ɡɚ ɞɚ ɟ ɫɢɝɭɪɧɨ, ɱɟ ɩɨɧɟ ɞɜɚɦɚ ɨɬ ɬɹɯ ɳɟ ɛɴɞɚɬ ɨɰɟɧɟɧɢ ɫ ɪɚɜɟɧ ɛɪɨɣ ɬɨɱɤɢ. Ɂɚɞɚɱɚ 48. Ⱥɤɨ ɫɬɪɚɧɚɬɚ ɧɚ ɪɨɦɛɚ ɟ x , ɬɨ ɨɬ ∆PAD (ɩɪɚɜɨɴɝɴɥɟɧ ɬɪɢɴɝɴɥɧɢɤ ɫ ɴɝɴɥ 300 ) ɫɥɟɞɜɚ, x x.x ɱɟ ɜɢɫɨɱɢɧɚɬɚ ɧɚ ɪɨɦɛɚ ɟ . Ɍɨɝɚɜɚ ɥɢɰɟɬɨ ɧɚ ɪɨɦɛɚ ɟ = 32 , ɨɬɤɴɞɟɬɨ x = 8 ɫɦ. 2 2 ɇɟɤɚ PH ⊥ DQ ( H ɩɪɢɧɚɞɥɟɠɢ ɧɚ ɩɪɚɜɚɬɚ DQ ). Ɍɪɢɴɝɴɥɧɢɤ HPD ɟ ɩɪɚɜɨɴɝɴɥɟɧ ɫ 1 ɴɝɴɥ 300 . ɋɥɟɞɨɜɚɬɟɥɧɨ ɬɴɪɫɟɧɟɬɨ ɪɚɡɫɬɨɹɧɢɟ ɟ ɨɬ ɜɢɫɨɱɢɧɚɬɚ ɧɚ ɪɨɦɛɚ, ɬ.ɟ. 2 ɫɦ. 2 Ɂɚɞɚɱɚ 49. ɑɟɬɢɪɢɴɝɴɥɧɢɤɴɬ ɨɬ ɭɫɥɨɜɢɟɬɨ ɧɚ ɡɚɞɚɱɚɬɚ ɟ ɭɫɩɨɪɟɞɧɢɤ. Ⱦɨ ɬɨɡɢ ɢɡɜɨɞ ɦɨɠɟ ɞɚ ɫɟ ɫɬɢɝɧɟ ɩɨ ɫɥɟɞɧɢɹ ɧɚɱɢɧ. ɇɟɤɚ O ɟ ɩɪɟɫɟɱɧɚɬɚ ɬɨɱɤɚ ɧɚ ɞɢɚɝɨɧɚɥɢɬɟ. Ɍɨɝɚɜɚ ɫ ɞɨɩɭɫɤɚɧɟ ɧɚ ɩɪɨɬɢɜɧɨɬɨ ɡɚɤɥɸɱɚɜɚɦɟ, ɱɟ BO = OD , ɡɚɳɨɬɨ ɚɤɨ ɬɨɜɚ ɧɟ ɟ ɬɚɤɚ, ɬɨ ɜɴɪɯɭ BD ɦɨɠɟ ɞɚ ɫɟ ɧɚɦɟɪɢ ɬɨɱɤɚ M ɬɚɤɚ, ɱɟ BO = OM . ɇɨ ɬɨɝɚɜɚ ABCM ɟ ɭɫɩɨɪɟɞɧɢɤ (ɞɢɚɝɨɧɚɥɢɬɟ ɜ ɧɟɝɨ ɜɡɚɢɦɧɨ ɫɟ ɪɚɡɩɨɥɨɜɹɜɚɬ) ɢ ɫɥɟɞɨɜɚɬɟɥɧɨ ∠ABC = ∠AMC . Ɂɚɤɥɸɱɚɜɚɦɟ, ɱɟ ∠AMC = ∠ADC , ɤɨɟɬɨ ɥɟɫɧɨ ɫɟ ɜɢɠɞɚ, ɱɟ ɟ ɧɟɜɴɡɦɨɠɧɨ (ɤɚɬɨ ɢɡɩɨɥɡɜɚɦɟ, ɱɟ ɜɴɧɲɧɢɹɬ ɴɝɴɥ ɜ ɬɪɢɴɝɴɥɧɢɤɚ ɟ ɜɢɧɚɝɢ ɩɨ-ɝɨɥɹɦ ɨɬ ɤɨɣ ɞɚ ɟ ɧɟɫɴɫɟɞɟɧ ɧɟɦɭ ɜɴɬɪɟɲɟɧ ɴɝɴɥ). ɋɥɟɞ ɤɚɬɨ ɱɟɬɢɪɢɴɝɴɥɧɢɤɴɬ ABCD ɟ ɭɫɩɨɪɟɞɧɢɤ, ɩɟɪɢɦɟɬɴɪɴɬ ɦɭ ɟ ɪɚɜɟɧ ɧɚ 2( AB + BC ) = 32 ɫɦ. Ɂɚɞɚɱɚ 50. ɋɛɨɪɴɬ ɧɚ ɜɫɢɱɤɢ ɱɢɫɥɚ ɟ 70. Ɉɬ ɭɫɥɨɜɢɟɬɨ ɫɥɟɞɜɚ, ɱɟ ɫɛɨɪɴɬ ɧɚ ɢɡɬɪɢɬɢɬɟ ɱɢɫɥɚ ɫɟ ɞɟɥɢ ɧɚ 4. ɉɨɧɟɠɟ 70 ɫɟ ɞɟɥɢ ɧɚ 2, ɬɨ ɢ ɨɫɬɚɜɚɳɨɬɨ (ɬɨɜɚ, ɤɨɟɬɨ ɧɟ ɟ ɢɡɬɪɢɬɨ) ɱɢɫɥɨ ɬɪɹɛɜɚ ɞɚ ɫɟ ɞɟɥɢ ɧɚ 2. Ɉɬ ɞɪɭɝɚ ɫɬɪɚɧɚ ɨɛɚɱɟ 70 ɧɟ ɫɟ ɞɟɥɢ 4 ɢ ɡɧɚɱɢ ɨɫɬɚɜɚɳɨɬɨ ɱɢɫɥɨ ɫɴɳɨ ɧɟ ɬɪɹɛɜɚ ɞɚ ɫɟ ɞɟɥɢ ɧɚ 4. ȿɞɢɧɫɬɜɟɧɨɬɨ ɱɢɫɥɨ ɢɡɦɟɠɞɭ ɧɚɩɢɫɚɧɢɬɟ, ɤɨɟɬɨ ɫɟ ɞɟɥɢ ɧɚ 2, ɧɨ ɧɟ ɫɟ ɞɟɥɢ ɧɚ 4, ɟ 6. ɋɥɟɞɨɜɚɬɟɥɧɨ ɨɬɝɨɜɨɪɴɬ ɧɚ ɡɚɞɚɱɚɬɚ ɟ 6. ȼɴɡɦɨɠɧɨɫɬɬɚ ɡɚ ɪɟɚɥɢɡɚɰɢɹ ɟ ɟɞɢɧɫɬɜɟɧɚ: ɋɢɦɟɨɧ ɢɡɬɪɢɜɚ ɱɢɫɥɚɬɚ 1, 3, 4 ɢ 8, ɚ Ƚɟɨɪɝɢ – ɫɴɨɬɜɟɬɧɨ 9, 11, 12 ɢ 16.