MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1999 Greece National Olympiad
1
1
Part of
1999 Greece National Olympiad
Problems
(1)
f(x^2)f(y^2) is at least f(xy)^2 for quadratic f
Source: Greece MO 1999
5/27/2011
Let
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=ax^2+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are nonnegative real numbers, not all equal to zero. Prove that
f
(
x
y
)
2
≤
f
(
x
2
)
f
(
y
2
)
f(xy)^2\le f(x^2)f(y^2)
f
(
x
y
)
2
≤
f
(
x
2
)
f
(
y
2
)
for all real numbers
x
,
y
x,y
x
,
y
.
quadratics
inequalities proposed
inequalities