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IMC
2013 IMC
2
Imc 2013/1/2
Imc 2013/1/2
Source: Imc 2013 Problem 2
August 8, 2013
trigonometry
IMC
college contests
Bad Latex
Problem Statement
Let
f
:
R
→
R
\displaystyle{f:{\cal R} \to {\cal R}}
f
:
R
→
R
be a twice differentiable function. Suppose
f
(
0
)
=
0
\displaystyle{f\left( 0 \right) = 0}
f
(
0
)
=
0
. Prove there exists
ξ
∈
(
−
π
2
,
π
2
)
\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}
ξ
∈
(
−
2
π
,
2
π
)
such that
f
′
′
(
ξ
)
=
f
(
ξ
)
(
1
+
2
tan
2
ξ
)
.
\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.
f
′′
(
ξ
)
=
f
(
ξ
)
(
1
+
2
tan
2
ξ
)
.
Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.
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