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Vojtěch Jarník IMC
2000 VJIMC
2000 VJIMC
Part of
Vojtěch Jarník IMC
Subcontests
(4)
Problem 2
2
Hide problems
n^(0.5*d(n)) is injective
Let
f
:
N
→
R
f:\mathbb N\to\mathbb R
f
:
N
→
R
be given by
f
(
n
)
=
n
1
2
τ
(
n
)
f(n)=n^{\frac12\tau(n)}
f
(
n
)
=
n
2
1
τ
(
n
)
for
n
∈
N
=
{
1
,
2
,
…
}
n\in\mathbb N=\{1,2,\ldots\}
n
∈
N
=
{
1
,
2
,
…
}
where
τ
(
n
)
\tau(n)
τ
(
n
)
is the number of divisors of
n
n
n
. Show that
f
f
f
is an injection.
writing words around circle, two letters
If we write the sequence
AAABABBB
\text{AAABABBB}
AAABABBB
along the perimeter of a circle, then every word of the length
3
3
3
consisting of letters
A
A
A
and
B
B
B
(i.e.
AAA
\text{AAA}
AAA
,
AAB
\text{AAB}
AAB
,
ABA
\text{ABA}
ABA
,
BAB
\text{BAB}
BAB
,
ABB
\text{ABB}
ABB
,
BBB
\text{BBB}
BBB
,
BBA
\text{BBA}
BBA
,
BAA
\text{BAA}
BAA
) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a
k
k
k
-element alphabet along the perimeter of a circle in such a way that every word of the length
l
l
l
(i.e. an ordered
l
l
l
-tuple of letters) occurs exactly once on the perimeter.
Problem 4
2
Hide problems
2-coloring 2n-gon, sequence of side lengths are equal
Let us choose arbitrarily
n
n
n
vertices of a regular
2
n
2n
2
n
-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.
existence of balls satisfying Lebesgue inequality
Let
B
\mathcal B
B
be a family of open balls in
R
n
\mathbb R^n
R
n
and
c
<
λ
(
⋃
B
)
c<\lambda\left(\bigcup\mathcal B\right)
c
<
λ
(
⋃
B
)
where
λ
\lambda
λ
is the
n
n
n
-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls
{
U
i
}
i
=
1
k
⊆
B
\{U_i\}^k_{i=1}\subseteq\mathcal B
{
U
i
}
i
=
1
k
⊆
B
such that
∑
j
=
1
k
λ
(
U
j
)
>
c
3
n
.
\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.
j
=
1
∑
k
λ
(
U
j
)
>
3
n
c
.
Problem 1
2
Hide problems
product (k^2+1)=4 mod p from k=1 to k=p if p=4n-1
Let
p
p
p
be a prime of the form
p
=
4
n
−
1
p=4n-1
p
=
4
n
−
1
where
n
n
n
is a positive integer. Prove that
∏
k
=
1
p
(
k
2
+
1
)
≡
4
(
m
o
d
p
)
.
\prod_{k=1}^p(k^2+1)\equiv4\pmod p.
k
=
1
∏
p
(
k
2
+
1
)
≡
4
(
mod
p
)
.
uncountable subset family, pairwise intersections are finite
Is there a countable set
Y
Y
Y
and an uncountable family
F
\mathcal F
F
of its subsets such that for every two distinct
A
,
B
∈
F
A,B\in\mathcal F
A
,
B
∈
F
, their intersection
A
∩
B
A\cap B
A
∩
B
is finite?
Problem 3
2
Hide problems
limits equal for bounded sequence of reals
Let
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
be a bounded sequence of reals. Is it true that the fact
lim
N
→
∞
1
N
∑
n
=
1
N
a
n
=
b
and
lim
N
→
∞
1
log
N
∑
n
=
1
N
a
n
n
=
c
\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c
N
→
∞
lim
N
1
n
=
1
∑
N
a
n
=
b
and
N
→
∞
lim
lo
g
N
1
n
=
1
∑
N
n
a
n
=
c
implies
b
=
c
b=c
b
=
c
?
Czech Republic 2000
Prove that if m,n are nonnegative integers and 0<=x<=1 then
(
1
−
x
n
)
m
+
(
1
−
(
1
−
x
)
m
)
n
≥
1
(1-x^n)^m + (1-(1-x)^m)^n \ge 1
(
1
−
x
n
)
m
+
(
1
−
(
1
−
x
)
m
)
n
≥
1