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Baltic Way
1999 Baltic Way
4
4
Part of
1999 Baltic Way
Problems
(1)
There exists x_0 and y_0
Source: Baltic Way 1999
12/23/2010
For all positive real numbers
x
x
x
and
y
y
y
let
f
(
x
,
y
)
=
min
(
x
,
y
x
2
+
y
2
)
f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right)
f
(
x
,
y
)
=
min
(
x
,
x
2
+
y
2
y
)
Show that there exist
x
0
x_0
x
0
and
y
0
y_0
y
0
such that
f
(
x
,
y
)
≤
f
(
x
0
,
y
0
)
f(x, y)\le f(x_0, y_0)
f
(
x
,
y
)
≤
f
(
x
0
,
y
0
)
for all positive
x
x
x
and
y
y
y
, and find
f
(
x
0
,
y
0
)
f(x_0,y_0)
f
(
x
0
,
y
0
)
.
quadratics
trigonometry
algebra proposed
algebra