MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania National Olympiad
1996 Romania National Olympiad
2
Inequality
Inequality
Source: Romania 1996
January 27, 2010
inequalities
inequalities unsolved
Problem Statement
a
,
b
,
c
,
d
∈
[
0
,
1
]
a,b,c,d \in [0,1]
a
,
b
,
c
,
d
∈
[
0
,
1
]
and
x
,
y
,
z
,
t
∈
[
0
,
1
2
]
x,y,z,t \in [0, \frac{1}{2}]
x
,
y
,
z
,
t
∈
[
0
,
2
1
]
and
a
+
b
+
c
+
d
=
x
+
y
+
z
+
t
=
1
a+b+c+d=x+y+z+t=1
a
+
b
+
c
+
d
=
x
+
y
+
z
+
t
=
1
.prove that:
(
i
)
(i)
(
i
)
a
x
+
b
y
+
c
z
+
d
t
ax+by+cz+dt
a
x
+
b
y
+
cz
+
d
t
≥
\geq
≥
m
i
n
(
a
+
b
2
,
b
+
c
2
,
c
+
d
2
,
d
+
a
2
,
a
+
c
2
,
b
+
d
2
)
min( {\frac{a+b}{2} , \frac{b+c}{2} , \frac{c+d}{2} , \frac{d+a}{2} , \frac{a+c}{2} , \frac{b+d}{2} )}
min
(
2
a
+
b
,
2
b
+
c
,
2
c
+
d
,
2
d
+
a
,
2
a
+
c
,
2
b
+
d
)
(
i
i
)
(ii)
(
ii
)
a
x
+
b
y
+
c
z
+
d
t
ax+by+cz+dt
a
x
+
b
y
+
cz
+
d
t
≥
\geq
≥
54
a
b
c
d
54abcd
54
ab
c
d
Back to Problems
View on AoPS