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Putnam
2018 Putnam
B5
B5
Part of
2018 Putnam
Problems
(1)
Putnam 2018 B5
Source:
12/2/2018
Let
f
=
(
f
1
,
f
2
)
f = (f_1, f_2)
f
=
(
f
1
,
f
2
)
be a function from
R
2
\mathbb{R}^2
R
2
to
R
2
\mathbb{R}^2
R
2
with continuous partial derivatives
∂
f
i
∂
x
j
\tfrac{\partial f_i}{\partial x_j}
∂
x
j
∂
f
i
that are positive everywhere. Suppose that
∂
f
1
∂
x
1
∂
f
2
∂
x
2
−
1
4
(
∂
f
1
∂
x
2
+
∂
f
2
∂
x
1
)
2
>
0
\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0
∂
x
1
∂
f
1
∂
x
2
∂
f
2
−
4
1
(
∂
x
2
∂
f
1
+
∂
x
1
∂
f
2
)
2
>
0
everywhere. Prove that
f
f
f
is one-to-one.
Putnam
Putnam 2018