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Contests
Undergraduate contests
Vojtěch Jarník IMC
1995 VJIMC
1995 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
1
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limit of arctan^n(x)
Let
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
be a sequence such that
x
1
=
25
x_1=25
x
1
=
25
,
x
n
=
arctan
(
x
n
−
1
)
x_n=\operatorname{arctan}(x_{n-1})
x
n
=
arctan
(
x
n
−
1
)
. Prove that this sequence has a limit and find it.
Problem 3
2
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f>f^(-1) if f and f^(-1) are decreasing
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be mutually inverse decreasing functions on the interval
(
0
,
∞
)
(0,\infty)
(
0
,
∞
)
. Can it hold that
f
(
x
)
>
g
(
x
)
f(x)>g(x)
f
(
x
)
>
g
(
x
)
for all
x
∈
(
0
,
∞
)
x\in(0,\infty)
x
∈
(
0
,
∞
)
?
f(x)=g(x)sin+h(x)cos with continuous functions
Let
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
be a continuous function. Do there exist continuous functions
g
:
R
→
R
g:\mathbb R\to\mathbb R
g
:
R
→
R
and
h
:
R
→
R
h:\mathbb R\to\mathbb R
h
:
R
→
R
such that
f
(
x
)
=
g
(
x
)
sin
x
+
h
(
x
)
cos
x
f(x)=g(x)\sin x+h(x)\cos x
f
(
x
)
=
g
(
x
)
sin
x
+
h
(
x
)
cos
x
holds for every
x
∈
R
x\in\mathbb R
x
∈
R
?
Problem 2
2
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even function has extremum at 0
Let
f
(
x
)
f(x)
f
(
x
)
be an even twice differentiable function such that
f
′
′
(
0
)
≠
0
f''(0)\ne0
f
′′
(
0
)
=
0
. Prove that
f
(
x
)
f(x)
f
(
x
)
has a local extremum at
x
=
0
x=0
x
=
0
.
polynomial in z+1/z
Let
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
and
f
k
=
f
2
n
−
k
f_k=f_{2n-k}
f
k
=
f
2
n
−
k
for each
k
k
k
. Prove that
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
f(z)=z^ng(z+z^{-1})
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
, where
g
g
g
is a polynomial of degree
n
n
n
.
Problem 1
2
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solvability of system
Discuss the solvability of the equations \begin{align*}\lambda x+y+z&=a\\x+\lambda y+z&=b\\x+y+\lambda z&=c\end{align*}for all numbers
λ
,
a
,
b
,
c
∈
R
\lambda,a,b,c\in\mathbb R
λ
,
a
,
b
,
c
∈
R
.
hyperbolas orthogonal?
Prove that the systems of hyperbolas \begin{align*}x^2-y^2&=a\\xy&=b\end{align*}are orthogonal.