34.
(Vyetnam -2001) Musbat , ,
x y z
sonlar
{
}
1
1
min
2,
3 ,
3
6,
3
10 2 5
2
2
z
x
y
x z
y
z
≤ <
+
≥
+
≥
shartlarni
qanoatlantirsa,
2
2
2
1
2
3
( , , )
P x y z
x
y
z
=
+
+
ifodaning eng katta qiymatini toping.
35.
(AQSh -2003) Musbat , ,
a b c
sonlar uchun
(
)
(
)
2
2
2
2
2
2
2
2
2
2
2
(2
)
8
2
(
)
2
(
)
2
(
)
a b c
b c a
c a b
a
b c
b
a c
c
a b
+ +
+ +
+ +
+
+
≤
+ +
+
+
+
+
tengsizlikni isbotlang.
36.
(Albaniya -2004) Musbat , ,
a b c
sonlarning ko’paytmasi birga teng bo’lsa,
9
1
1
1
1,2
1
1
1
0,64
0,64
0,64
a
b
c
b
c
a
+
+
≥
+ +
+ +
+ +
tengsizlikni isbotlang.
37.
Musbat , ,
x y z
sonlar
1
xyz
=
shartni qanoatlantirsa,
1
1
1
1
1
1
1
1
1
(1
)
(1
)
(1
)
x y z
x
y
z
x
z
y
x
z
y
⎛
⎞
+ + −
+
+
≥
+
+
⎜
⎟
+
+
+
+
+
+
⎝
⎠
tengsizlikni isbotlang.
38.
Musbat , ,
x y z
sonlar uchun
2
2
2
2
3(
1)(
1)(
1) (
)
1
x
x
y
y
z
z
xyz
xyz
− +
− +
− + ≥
+
+
tengsizlikni isbotlang.
39.
(Belorussiya -1997)
,
1
n N n
∈
>
uchun
1
1 1 1
1
...
2 3 4
n
n
n n
n
−
+ + + + < −
tengsizlikni isbotlang.
40.
Birdan katta haqiqiy , ,
x y z
sonlar
1
1
1
2
x
y
z
+ + =
tenglikni kanoatlantirsa,
1
1
1
x y z
x
y
z
+ + ≥
− +
− +
−
tengsizlikni isbotlang.
41.
(Hindiston -2002) Musbat
, ,
a b c
sonlar
2
2
2
3
a
b
c
abc
+
+
=
tenglikni
qanoatlantirsa,
2 2
2 2
2 2
9
a
b
c
b c
c a
a b
a b c
+
+
≥
+ +
tengsizlikni isbotlang.
42.
(Ukraina -2002) Musbat , ,
x y z
sonlar uchun
10
2
2
2
2
1
1
1
1
28 3
(
)
9
(
)
x y z
x
y
z
xyz x y z
+
+
+
≥
+ +
+ +
t
tengsizlikni isbotlang.
43.
(Ukraina -2001)
1
2
, ,...,
n
a a
a
haqiqiy sonlar
2
2
2
2
2
1
2
1
2
...
,
...
1
n
n
a
a
a
n a
a
a
n
+
+ +
≥
+
+ +
≤
+
shartlarni qanoatlantirsa,
1
1
k
n
a
n
− ≤
≤ +
tengsizlikni isbotlang.
44.
(Ukraina -2001) Musbat , ,
, ,
1
a b c
ва
α β γ α β γ
+ + =
sonlar uchun
2 (
)(
)
a
b
c
ab bc ca
a b c
α
β
γ
αβ βγ γα
+
+
+
+
+
+
+
≤ + +
tengsizlikni isbotlang.
45.
(Ukraina -2000) Musbat ,
a b
sonlar uchun
3
2
2
3
3
1
1
1
1
64
3
3
3(
)
a
ab
a b b
a b
+
+
+
≥
+
tengsizlikni isbotlang.
46.
(Ukraina -1999) Haqiqiy
[ ]
1
2
6
, ,...,
0;1
x x
x
∈
sonlar uchun
3
3
3
1
2
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
4
5
6
1
3
4
5
6
1
2
3
4
5
3
...
5
5
5 5
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
+
+ +
≤
+ + + + +
+ + + + +
+ + + + +
tengsizlikni isbotlang.
47.
(Bosniya -2002) Musbat , ,
a b c
sonlari
2
2
2
1
a
b
c
+
+
=
shartni qanoatlantir-sa,
2
2
2
3
1 2
1 2
1 2
5
a
b
c
bc
ac
ab
+
+
≥
+
+
+
tengsizlikni isbotlang.
48.
(Yugoslaviya -2002) Musbat
1 2
2001
.....
x x
x
sonlar uchun
2
2
2
2
2
2
3
1
1
3
3
3
...
, 2
2001
2
3
(
1)
i
i
x
x
x
x
x
i
i
−
≥
+
+
+ +
≤ ≤
−
11
shart bajarilsa,
2001
2
1
2
1
1,999
...
i
i
i
x
x
x
x
=
−
>
+
+ +
∑
tengsizlikni isbotlang.
49.
(Hindiston -2002) Musbat , ,
a b c
sonlar uchun
a b
c
c a
a b
b c
b
c
a
c b
a c b a
+
+
+
+ + ≥
+
+
+
+
+
ekanligini ko’rsating.
50.
(XMO -1998) Agar
1 (
1,2,... )
i
y
i
n
≥
=
bo’lsa,
1
2
1 2
1
1
1
....
1
1
1
1
...
n
n
n
n
y
y
y
y y
y
+
+
+
≥
+
+
+
+
tengsizlikni isbotlang.
51.
(Hindiston -2004) Ixtiyoriy
1
(0; ]
2
i
x
∈
sonlar uchun (i=1, 2,…,
n
)
1
1
1
1
(1
)
(1
)
n
n
i
i
i
i
n
n
n
n
i
i
i
i
x
x
x
x
=
=
=
=
Π
Π −
≤
⎛
⎞
⎛
⎞
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∑
∑
tengsizlikni isbotlang.
52.
(XMO -2001)
1
2
, ,.....
n
x x
x
haqiqiy sonlar uchun
1
2
2
2
2
2
2
2
1
1
2
1
2
....
1
1
1
...
n
n
x
x
x
n
x
x
x
x
x
x
+
+
+
<
+
+
+
+
+
+ +
tengsizlikni isbotlang.
53.
(XMO -2004) Musbat , ,
a b c
sonlar
1
ab bc ca
+
+
=
tenglikni qanoatlantirsa,
3
3
3
1
1
1
1
6
6
6
b
c
a
a
b
c
abc
+
+
+
+
+
≤
tengsizlikni isbotlang.
12
54.
(XMO -2004) Musbat , , , , ,
x y z a b c
sonlar
1
ab bc ca
+
+
=
shartni
qanoatlantirsa
3
3
3
2
3
(
)
3
abc x y z
ax
by
cz
+ +
≤ +
+
+
tengsizlikni isbotlang.
55.
(Moldova -2001)
1
2
,
,...,
n
a a
a
haqiqiy musbat sonlar uchun
1
1
1
1
1
1
1
1
n
n
i
i
i
i
n
a
a
=
=
−
≥
+
∑
∑
tengsizlikni isbotlang.
56
. (AQSh -1992)
0
1
2
, , ,....,
n
a a a
a
musbat haqiqiy sonlari
2
1
1
i
i
i
a
a
a
−
+
⋅
≤
(
)
1,2,...,
1
i
n
=
−
shartni qanoatlantirsa
0
1
1
2
1
0
1
1
0
1
...
....
...
...
1
1
n
n
n
n
a
a
a a
a
a
a
a
a
a
a
a
n
n
n
n
−
−
+ + +
+
+
+
+ + +
+ + +
⋅
≥
⋅
+
−
tengsizlikni isbotlang.
57
. (Polsha -1996) , ,
, , ,
a b c
x y z
nomanfiy sonlar bo’lib,
1
1, 0
, ,
2
x y z a b c
x y z
+ + = + + =
≤
≤
shartlarni qanoatlantirsa,
8
ax by cz
abc
+
+
≥
tengsizlikni isbotlang.
58
. (XMO -1998) Yig’indisi birdan kichik bo’lgan
1
2
, ,....,
n
x x
x
musbat sonlar
uchun
(
)
1
1 2
1
2
1
2
1
2
.... (1
...
)
...
(1
)(1
)...(1
)
n
n
n
n
n
n x x
x
x
x
x
x
x
x
x
x
x
+
− −
− −
≤
+
+ +
−
−
−
tengsizlikni isbotlang.
13
59.
0
, ,
1
a b c
≤
≤
sonlar uchun
2
1
1
1
a
b
c
bc
ac
ab
+
+
≤
+
+
+
tengsizlikni isbotlang.
60.
a, b, c, d
[ ]
1;2
∈
sonlar uchun
4(
)
a b
c d
a c
b c
d a
b d
+
+
+
+
≤
+
+
+
tengsizlikni isbotlang.
61.
(Vyetnam -2002)
a, b, c
uchburchak tomonlari va 0
1
t
≤ ≤
uchun
a
b
b c ta
a c tb
+
+ −
+ −
2 1
c
t
a b tc
+
≥
+
+ −
tengsizlikni isbotlang.
62.
Haqiqiy
a, b, c
sonlar
a+b+c
=0 shartni qanoatlantirsa,
2 2
2 2
2 2
3 6
a b
b c
c a
abc
+
+
+ ≥
tengsizlikni isbotlang.
63.
(O’zbekiston -2001)
x
1
, x
2
, …, x
2002
musbat sonlar
2002
2
1
1
1
1
i
i
x
=
=
+
∑
shartni
qanoatlantirsa,
x
1
x
2
…x
2002
≥
2001
1001
tengsizlikni isbotlang.
64.
(Belorussiya -1999) Agar musbat
a, b, c
sonlar
a
2
+b
2
+c
2
=3 tenglikni
qanoatlantirsa,
1
1
1
3
1
1
1
2
ab
bc
ac
+
+
≥
+
+
+
tengsizlik o’rinli bo’lishini isbotlang.
65.
Agar
a
≥
b
≥
c
>0 va , ,
x y z
R
∈
bo’lsa,
2
2
(
)
(
)(
)
(
)
4
x
y
z
a c
ax by cz
x y z
a
b
c
ac
+
+
+
+ +
≤
+ +
tengsizlikni isbotlang.
66.
(Singapur -2004) Agar
a, b, c
> 0 bo’lsa, u holda
(
)
1
2
2
2
4
ab
bc
ac
a b c
a b
c b c
a
a c
b
+
+
≤
+ +
+ +
+ +
+ +
tengsizlikni isbotlang.
14
67.
(Polsha -2006) Agar musbat
a, b, c
sonlar
ab+bc+ca=abc
tenglikni
qanoatlantirsa, u holda
4
4
4
4
4
4
3
3
3
3
3
3
1
(
)
(
)
(
)
a
b
b
c
c
a
ab a
b
bc b
c
ac a
c
+
+
+
+
+
≥
+
+
+
tengsizlikni
isbotlang.
68.
(Sankt-Peterburg -2006) Agar
a, b, c, d
musbat sonlar
a
2
+b
2
+c
2
+d
2
=1
tenglikni qanoatlantirsa, u holda
1
18
a b c d
abcd
+ + + +
≥
tengsizlikni isbotlang.
69.
(APMO , Osiyo va Tinch Okean qirg’og’i davlatlari olimpiadasi -2004) Agar
x,
y, z
musbat sonlar bo’lsa,
2
2
2
(
2)(
2)(
2) 9(
)
x
y
z
xy yz zx
+
+
+
≥
+
+
tengsizlik o’rinli bo’lishini ko’rsating.
70.
(Xitoy -2004) Agar natural
1
2
...
n
a
a
a
≤
≤ ≤
sonlar ketma-ketligi
1
1
1
n
i
i
a
=
<
∑
shartni qanoatlantirsa
2
2
2
2
2
1
1
1
1
1
2
2
2
n
i
i
a
x
a
a
x
=
⎛
⎞
≤
⎜
⎟
+
−
+
⎝
⎠
∑
tengsizlikni isbotlang.
(
x
∈
R
).
71.
(Rossiya -2002) Musbat , ,
x y z
sonlar
3
x y z
+ + =
shartni qanoatlantirsa,
x
y
z
xy yz zx
+
+
≥
+
+
tengsizlikni isbotlang.
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