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Undergraduate contests
Vojtěch Jarník IMC
1997 VJIMC
1997 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4-M
2
Hide problems
# of 0's in decimal expansion, summation converges
Find all real numbers
a
>
0
a>0
a
>
0
for which the series
∑
n
=
1
∞
a
f
(
n
)
n
2
\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}
n
=
1
∑
∞
n
2
a
f
(
n
)
is convergent;
f
(
n
)
f(n)
f
(
n
)
denotes the number of
0
0
0
's in the decimal expansion of
f
f
f
.
infinite factorial sum with trig expression
Prove that
∑
n
=
1
∞
n
2
(
7
n
)
!
=
1
7
3
∑
k
=
1
2
∑
j
=
0
6
e
cos
(
2
π
j
/
7
)
⋅
cos
(
2
k
π
j
7
+
sin
2
π
j
7
)
.
\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).
n
=
1
∑
∞
(
7
n
)!
n
2
=
7
3
1
k
=
1
∑
2
j
=
0
∑
6
e
c
o
s
(
2
πj
/7
)
⋅
cos
(
7
2
kπj
+
sin
7
2
πj
)
.
Problem 3
2
Hide problems
f^(k)(0)>0 for k=1,2,...
Let
c
1
,
c
2
,
…
,
c
n
c_1,c_2,\ldots,c_n
c
1
,
c
2
,
…
,
c
n
be real numbers such that
c
1
k
+
c
2
k
+
…
+
c
n
k
>
0
for all
k
=
1
,
2
,
…
c_1^k+c_2^k+\ldots+c_n^k>0\qquad\text{for all }k=1,2,\ldots
c
1
k
+
c
2
k
+
…
+
c
n
k
>
0
for all
k
=
1
,
2
,
…
Let us put
f
(
x
)
=
1
(
1
−
c
1
x
)
(
1
−
c
2
x
)
⋯
(
1
−
c
n
x
)
.
f(x)=\frac1{(1-c_1x)(1-c_2x)\cdots(1-c_nx)}.
f
(
x
)
=
(
1
−
c
1
x
)
(
1
−
c
2
x
)
⋯
(
1
−
c
n
x
)
1
.
z
∈
C
z\in\mathbb C
z
∈
C
Show that
f
(
k
)
(
0
)
>
0
f^{(k)}(0)>0
f
(
k
)
(
0
)
>
0
for all
k
=
1
,
2
,
…
k=1,2,\ldots
k
=
1
,
2
,
…
.
integral inequality in R^3
Let
u
∈
C
2
(
D
‾
)
u\in C^2(\overline D)
u
∈
C
2
(
D
)
,
u
=
0
u=0
u
=
0
on
∂
D
\partial D
∂
D
where
D
D
D
is the open unit ball in
R
3
\mathbb R^3
R
3
. Prove that the following inequality holds for all
ε
>
0
\varepsilon>0
ε
>
0
:
∫
D
∣
∇
u
∣
2
d
V
≤
ε
∫
D
(
Δ
u
)
2
d
V
+
1
4
ε
∫
D
u
2
d
V
.
\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.
∫
D
∣∇
u
∣
2
d
V
≤
ε
∫
D
(
Δ
u
)
2
d
V
+
4
ε
1
∫
D
u
2
d
V
.
(We recall that
∇
u
\nabla u
∇
u
and
Δ
u
\Delta u
Δ
u
are the gradient and Laplacian, respectively.)
Problem 2
2
Hide problems
convex-type sequence is bounded, hence convergent
Let
α
∈
(
0
,
1
]
\alpha\in(0,1]
α
∈
(
0
,
1
]
be a given real number and let a real sequence
{
a
n
}
n
=
1
∞
\{a_n\}^\infty_{n=1}
{
a
n
}
n
=
1
∞
satisfy the inequality
a
n
+
1
≤
α
a
n
+
(
1
−
α
)
a
n
−
1
for
n
=
2
,
3
,
…
a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots
a
n
+
1
≤
α
a
n
+
(
1
−
α
)
a
n
−
1
for
n
=
2
,
3
,
…
Prove that if
{
a
n
}
\{a_n\}
{
a
n
}
is bounded, then it must be convergent.
|f(z)|=1 if |z|=1, find f if holomorphic
Let
f
:
C
→
C
f:\mathbb C\to\mathbb C
f
:
C
→
C
be a holomorphic function with the property that
∣
f
(
z
)
∣
=
1
|f(z)|=1
∣
f
(
z
)
∣
=
1
for all
z
∈
C
z\in\mathbb C
z
∈
C
such that
∣
z
∣
=
1
|z|=1
∣
z
∣
=
1
. Prove that there exists a
θ
∈
R
\theta\in\mathbb R
θ
∈
R
and a
k
∈
{
0
,
1
,
2
,
…
}
k\in\{0,1,2,\ldots\}
k
∈
{
0
,
1
,
2
,
…
}
such that
f
(
z
)
=
e
i
θ
z
k
f(z)=e^{i\theta}z^k
f
(
z
)
=
e
i
θ
z
k
for all
z
∈
C
z\in\mathbb C
z
∈
C
.
Problem 1
2
Hide problems
cover R3 with skew lines
Decide whether it is possible to cover the
3
3
3
-dimensional Euclidean space with lines which are pairwise skew (i.e. not coplanar).
d|a^2+2, find d mod 8
Let
a
a
a
be an odd positive integer. Prove that if
d
d
d
divides
a
2
+
2
a^2+2
a
2
+
2
, then
d
≡
1
(
m
o
d
8
)
d\equiv1\pmod8
d
≡
1
(
mod
8
)
or
d
≡
3
(
m
o
d
8
)
d\equiv3\pmod8
d
≡
3
(
mod
8
)
.