)
©
O’zbekiston Respublikasi Xalq ta’limi vazirligi
3
Masalalar.
1.
Agar
, ,
a b c
musbat sonlar va
,
n k N
∈
bo’lsa, u holda quyidagi
n k
n k
n k
n
n
n
k
k
k
a
b
c
a
b
c
b
c
a
+
+
+
+
+
≥
+
+
tengsizlikni isbotlang.
2.
(Rossiya -2003) Musbat
a,b,c
sonlar
a+b+c=
1 tenglikni qanoatlantirilsa, u
holda
1
1
1
2
2
2
1
1
1
1
1
1
a
b
c
a
b
c
+
+
≥
+
+
−
−
−
+
+
+
tengsizlikni isbotlang.
3.
(Moldova -2005) Musbat , ,
a b c
sonlar
4
4
4
3
a
b
c
+
+
=
tenglikni qanoatlantirsa,
1
1
1
1
4
4
4
ab
ac
bc
+
+
≤
−
−
−
tengsizlikni isbotlang.
4.
(Koreya -2000) Aytaylik, , ,
a b c
, , ,
x y z
haqiqiy sonlar quyidagi
0 ,
0
a b c
x y z
≥ ≥ >
≥ ≥ >
shartlarni qanoatlantirsin, u holda
(
)(
) (
)(
) (
)(
)
2 2
2 2
2 2
3
4
a x
b y
c z
by cz bz cy
cz ax cx az
ax by ay bx
+
+
≥
+
+
+
+
+
+
tengsizlikni isbotlang.
5.
(Yaponiya -2002) Aytaylik,
3
n
≥
va
n N
∈
da musbat
1
2
1
2
, ,..., , , ,....,
n
n
a a
a b b
b
sonlar quyidagi
2
2
2
1
2
1
2
...
1
...
1
n
n
a
a
a
ва
b
b
b
+
+ +
=
+
+ +
=
shartlarni
qanoatlantirsin. U holda
1
1
2
2
2
3
1
(
)
(
) ...
(
) 1
n
n
a b
a
a b
a
a b
a
+
+
+
+ +
+
<
tengsizlikni isbotlang.
4
6.
Musbat , ,
a b c
sonlar
1
a b c
+ + =
shartni qanoatlantirsa, u holda
3
2
ab
bc
ac
ab c
bc a
ac b
+
+
≤
+
+
+
tengsizlikni isbotlang.
7.
(Eron -2005) Musbat , ,
a b c
sonlar uchun
(
)
2
1 1 1
a b
c
a b c
b
c
a
a b c
⎛
⎞
⎛
⎞
+ +
≥
+ +
+ +
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
tengsizlikni isbotlang.
8.
(Vengriya -1996) Musbat ,
a b
sonlarning yig’indisi birga teng bo’lsa
2
2
1
1
1 3
a
b
a
b
+
≥
+
+
tengsizlikni isbotlang.
9.
Haqiqiy musbat , ,
x y z
sonlar
1
xyz
≥
shartni qanoatlantirsa,
5
5
5
5
2
2
5
2
2
5
2
2
1
x
y
z
x
y
z
y
z
x
z
x
y
+
+
≥
+
+
+
+
+
+
tengsizlikni isbotlang.
10.
(XMO, Xalqaro Matematika olimpiadasi -2005) Musbat , ,
x y z
sonlar
1
xyz
≥
shartni qanoatlantirsa,
5
2
5
2
5
2
5
2
2
5
2
2
5
2
2
0
x
x
y
y
z
z
x
y
z
y
z
x
z
x
y
−
−
−
+
+
≥
+
+
+
+
+
+
tengsizlikni isbotlang.
11.
(Bosniya -2002) Agar musbat , ,
x y z
sonlar
2
xyz x y z
= + + +
tenglikni
qanoatlantirsa,
5(
) 18 8(
)
x y z
xy
yz
zx
+ + +
≥
+
+
5
tengsizlik o’rinli bo’lishini isbotlang.
12.
(APMO -2005) Musbat , ,
a b c
sonlar
8
abc
=
shartni qanoatlantirsa, u holda.
(
)
2
2
2
3
3
3
3
3
3
4
3
(1
)(1
)
(1
)(1
)
(1
) 1
a
b
c
a
b
b
c
c
a
+
+
≥
+
+
+
+
+
+
tengsizlikni isbotlang.
13.
(Rossiya -1999) Musbat haqiqiy
x
va
y
sonlar
2
3
3
4
x
y
x
y
+
≥
+
tengsizlikni
qanoatlantirsa, u holda
3
3
2
x
y
+
≤
tengsizlikni isbotlang.
14.
(APMO -2003). Agar , ,
a b c
sonlari uchburchak tomonlarining uzunliklari
bo’lib, 1
a b c
+ + =
shartni qanoatlantirsa,
n N
∈
, 2
n
≥
uchun
2
1
2
n
n
n
n
n
n
n
n
n
n
a
b
b
c
c
a
+
+
+
+
+
< +
tengsizlikni isbotlang.
15.
Musbat
1
2
, ,...,
n
a a
a
sonlar uchun quyidagi
1
n
n
n
n
n
G
g
n
n
A
G
⋅
+
≤ +
tengsizlikni isbotlang. Bu yerda
1
2
1 2
1 2
...
... ,
,
...
n
n
n
n
n
n
n
n
a
a
a
g
a a a A
G
A A A
n
+
+ +
=
=
=
.
16.
(Yaponiya -2005) Musbat , ,
a b c
sonlar yig’indisi birga teng bo’lsa,
3
3
3
1
1
1
1
a
b c b
c a
a b
+ − +
+ − +
+ − ≤
tengsizlikni isbotlang.
17.
(Kolmogorov kubogi, Rossiya -2004) Musbat haqiqiy , , ,
a b c d
sonlar 1
abcd
=
shartni qanoatlantirsa, u holda
6
1
1
1
1
4
1
1
1
1
ab
bc
cd
da
a
b
c
d
+
+
+
+
+
+
+
≥
+
+
+
+
tengsizlikni isbotlang.
18.
(Kolmogorov kubogi, Rossiya -2004) Musbat , ,
a b c
sonlarning yig’indisi
birga teng bo’lsa,
3
(
)
(
1)
(
1)
(
1)
4
a
b
c
ab bc ca
b b
c c
a a
⎛
⎞
+
+
+
+
≥
⎜
⎟
+
+
+
⎝
⎠
tengsizlik o’rinli bo’lishini isbotlang.
19.
Musbat , ,
a b c
sonlar
1
a b c
+ + =
shartni qanoatlantirsa,
2
2
2
2
a
b b
c c
a
b c
c a
a b
+
+
+
+
+
≥
+
+
+
tengsizlikni o’rinli bo’lishini isbotlang
20.
Musbat , ,
x y z
sonlar uchun
1
(
)(
)
(
)(
)
(
)(
)
x
y
z
x
x y x z
y
y x y z
z
z x z y
+
+
≤
+
+
+
+
+
+
+
+
+
tengsizlikni isbotlang.
21.
(Xitoy -2004) Musbat , ,
a b c
sonlar uchun
3 2
2
a
b
c
a b
b c
c a
+
+
≤
+
+
+
tengsizlikni isbotlang.
22.
(Turkiya -1998) Agar 0
a b c
≤ ≤ ≤
shart bajarilsa,
(
3 )(
4 )(
2 ) 60
a
b b
c c
a
abc
+
+
+
≥
tengsizlikni isbotlang.
7
23.
(Buyuk Ipak yo’li Xalqaro olimpiadasi -2006) Musbat , ,
a b c
sonlar uchun
1
abc
=
shart bajarilsa,
2
3
3
3
3
1 1 1
4
3 2
a
b
c
a b c
b
c
a
a b c
⎛
⎞
⎛
⎞
+
+
≤
+ + + + + +
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
tengsizlikni isbotlang.
24.
(Albaniya -2002) Musbat , ,
a b c
sonlar uchun
(
)
2
2
2
2
2
2
3 1 1 1 1
(
)
3 3
a b c
a
b
c
a
b
c
a b c
+ ⎛
⎞
+ + +
+
+
≤
+ +
+
+
⎜
⎟
⎝
⎠
tengsizlikni isbotlang.
25.
(AQSh -2004) Musbat , ,
a b c
sonlar uchun
(
)
(
)
3
5
2
5
2
5
2
3 (
3)(
3)
a
a
b
b
c
c
a b c
−
+
−
+
−
+ ≥
+ +
tengsizlikni isbotlang.
26.
Musbat , ,
a b c
sonlar uchun
2
2
2
2
2
2
3
(
)(
)(
) (
)
a
ab b b
bc c
c
ca a
ab bc ca
+
+
+
+
+
+
≥
+
+
tengsizlikni isbotlang.
27.
(AQSh -2001) Agar musbat , ,
a b c
sonlar
2
2
2
4
a
b
c
abc
+
+
+
=
shartni
qanoatlantirsa, 0
2
ab bc ca abc
<
+
+
−
≤
tengsizlikni isbotlang.
28.
Haqiqiy ,
x y
sonlar
2
2
2
2
0,
(
)
x
xy x
y
x
y
≠
−
=
+
shartlarni qanoatlantirsa,
2
2
4
x
y
+
≥
tengsizlikni isbotlang.
29.
(Vyetnam -2001) Musbat , ,
a b c
sonlar 21
2
8
12
ab
bc
ac
+
+
≤
shartni
qanoatlantirsa,
1
2 3
( , , )
P a b c
a b c
= + +
ifodaning eng kichik qiymatini toping.
8
30.
Musbat , ,
a b c
sonlar uchun
2
2
2
2
4(
)
a
b
c
a b
a b c
b
c
a
a b c
−
+
+
≥ + + +
+ +
tengsizlikni isbotlang.
31. (
Hindiston -2000) Haqiqiy
1
,...,
n
a
a
conlar uchun
1
2
...
n
a
a
a
≤
≤ ≤
va
1
2
...
0
n
a
a
a
+
+ +
=
shartlar bajarilsa,
2
1
1
0
n
n
i
i
na a
a
=
+
≤
∑
tengsizlikni isbotlang.
32.
(Ruminiya -1999) Musbat
1
2
, ,...
n
x x
x
sonlar
1
1
n
i
i
x
=
=
∏
shartni qanoatlantirsa,
1
2
1
1
1
.....
1
1
1
1
n
n
x
n
x
n
x
+
+
+
≤
− +
− +
− +
tengsizlikni isbotlang.
33.
(Polsha -1991) Haqiqiy , ,
x y z
sonlar
2
2
2
2
x
y
z
+
+
=
shartni qanoatlantirsa,
2
x y z xyz
+ + −
≤
tengsizlikni isbotlang.
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