MathDB
Problems
Contests
International Contests
IMO Longlists
1971 IMO Longlists
3
3
Part of
1971 IMO Longlists
Problems
(1)
For any positive reals x, y, z it holds-[IMO LongList 1971]
Source:
1/1/2011
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers,
0
<
a
≤
b
≤
c
0 < a \leq b \leq c
0
<
a
≤
b
≤
c
. Prove that for any positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
the following inequality holds:
(
a
x
+
b
y
+
c
z
)
(
x
a
+
y
b
+
z
c
)
≤
(
x
+
y
+
z
)
2
⋅
(
a
+
c
)
2
4
a
c
.
(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.
(
a
x
+
b
y
+
cz
)
(
a
x
+
b
y
+
c
z
)
≤
(
x
+
y
+
z
)
2
⋅
4
a
c
(
a
+
c
)
2
.
inequalities
inequalities proposed