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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1991 Vietnam National Olympiad
1
Vietnam
Vietnam
Source: Vietnam MO 1991, #1, first day
March 22, 2006
function
algebra unsolved
algebra
Problem Statement
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
satisfying:
f
(
x
y
)
+
f
(
x
z
)
2
−
f
(
x
)
f
(
y
z
)
≥
1
4
\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}
2
f
(
x
y
)
+
f
(
x
z
)
−
f
(
x
)
f
(
yz
)
≥
4
1
for all
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
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