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Contests
Undergraduate contests
Vojtěch Jarník IMC
2010 VJIMC
2010 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 4
2
Hide problems
weaker abundance of a number
For every positive integer
n
n
n
let
σ
(
n
)
\sigma(n)
σ
(
n
)
denote the sum of all its positive divisors. A number
n
n
n
is called weird if
σ
(
n
)
≥
2
n
\sigma(n)\ge2n
σ
(
n
)
≥
2
n
and there exists no representation
n
=
d
1
+
d
2
+
…
+
d
r
,
n=d_1+d_2+\ldots+d_r,
n
=
d
1
+
d
2
+
…
+
d
r
,
where
r
>
1
r>1
r
>
1
and
d
1
,
…
,
d
r
d_1,\ldots,d_r
d
1
,
…
,
d
r
are pairwise distinct positive divisors of
n
n
n
. Prove that there are infinitely many weird numbers.
image as a subset of rectangles
Let
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
be a function satisfying
∣
f
(
x
)
−
f
(
y
)
∣
≤
∣
x
−
y
∣
|f(x)-f(y)|\le|x-y|
∣
f
(
x
)
−
f
(
y
)
∣
≤
∣
x
−
y
∣
for every
x
,
y
∈
[
0
,
1
]
x,y\in[0,1]
x
,
y
∈
[
0
,
1
]
. Show that for every
ε
>
0
\varepsilon>0
ε
>
0
there exists a countable family of rectangles
(
R
i
)
(R_i)
(
R
i
)
of dimensions
a
i
×
b
i
a_i\times b_i
a
i
×
b
i
,
a
i
≤
b
i
a_i\le b_i
a
i
≤
b
i
in the plane such that
{
(
x
,
f
(
x
)
)
:
x
∈
[
0
,
1
]
}
⊂
⋃
i
R
i
and
∑
i
a
i
<
ε
.
\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.
{(
x
,
f
(
x
))
:
x
∈
[
0
,
1
]}
⊂
i
⋃
R
i
and
i
∑
a
i
<
ε
.
(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
Problem 3
2
Hide problems
integral is bounded
Prove that there exist positive constants
c
1
c_1
c
1
and
c
2
c_2
c
2
with the following properties: a) For all real
k
>
1
k>1
k
>
1
,
∣
∫
0
1
1
−
x
2
cos
(
k
x
)
d
x
∣
<
c
1
k
3
/
2
.
\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.
∫
0
1
1
−
x
2
cos
(
k
x
)
d
x
<
k
3/2
c
1
.
b) For all real
k
>
1
k>1
k
>
1
,
∣
∫
0
1
1
−
x
2
sin
(
k
x
)
d
x
∣
<
c
2
k
.
\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.
∫
0
1
1
−
x
2
sin
(
k
x
)
d
x
<
k
c
2
.
invertible: A,A+B,A+2B,...,A+(2n)B implies A+(2n+1)B is invertible
Let
A
A
A
and
B
B
B
be two
n
×
n
n\times n
n
×
n
matrices with integer entries such that all of the matrices
A
,
A
+
B
,
A
+
2
B
,
A
+
3
B
,
…
,
A
+
(
2
n
)
B
A,\enspace A+B,\enspace A+2B,\enspace A+3B,\enspace\ldots,\enspace A+(2n)B
A
,
A
+
B
,
A
+
2
B
,
A
+
3
B
,
…
,
A
+
(
2
n
)
B
are invertible and their inverses have integer entries, too. Show that
A
+
(
2
n
+
1
)
B
A+(2n+1)B
A
+
(
2
n
+
1
)
B
is also invertible and that its inverse has integer entries.
Problem 1
2
Hide problems
convergence sum(1,∞)1/nf(n) for f:N->N
a) Is it true that for every bijection
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
the series
∑
n
=
1
∞
1
n
f
(
n
)
\sum_{n=1}^\infty\frac1{nf(n)}
n
=
1
∑
∞
n
f
(
n
)
1
is convergent? b) Prove that there exists a bijection
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
such that the series
∑
n
=
1
∞
1
n
+
f
(
n
)
\sum_{n=1}^\infty\frac1{n+f(n)}
n
=
1
∑
∞
n
+
f
(
n
)
1
is convergent. (
N
\mathbb N
N
is the set of all positive integers.)
n=ax+by for given a,b
Let
a
a
a
and
b
b
b
be given positive coprime integers. Then for every integer
n
n
n
there exist integers
x
,
y
x,y
x
,
y
such that
n
=
a
x
+
b
y
.
n=ax+by.
n
=
a
x
+
b
y
.
Prove that
n
=
a
b
n=ab
n
=
ab
is the greatest integer for which
x
y
≤
0
xy\le0
x
y
≤
0
in all such representations of
n
n
n
.
Problem 2
2
Hide problems
A,B matrix
If
A
,
B
∈
M
2
(
C
)
A,B\in M_2(C)
A
,
B
∈
M
2
(
C
)
such that
A
B
−
B
A
=
B
2
AB-BA=B^2
A
B
−
B
A
=
B
2
then prove that
A
B
=
B
A
AB=BA
A
B
=
B
A
a_(n-1)-a_n->0 and a_(2n)-2a_n->0
Prove or disprove that if a real sequence
(
a
n
)
(a_n)
(
a
n
)
satisfies
a
n
+
1
−
a
n
→
0
a_{n+1}-a_n\to0
a
n
+
1
−
a
n
→
0
and
a
2
n
−
2
a
n
→
0
a_{2n}-2a_n\to0
a
2
n
−
2
a
n
→
0
as
n
→
∞
n\to\infty
n
→
∞
, then
a
n
→
0
a_n\to0
a
n
→
0
.