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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2016 Kurschak Competition
2016 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Composition is square implies polynomial is square
If
p
,
q
∈
R
[
x
]
p,q\in\mathbb{R}[x]
p
,
q
∈
R
[
x
]
satisfy
p
(
p
(
x
)
)
=
q
(
x
)
2
p(p(x))=q(x)^2
p
(
p
(
x
))
=
q
(
x
)
2
, does it follow that
p
(
x
)
=
r
(
x
)
2
p(x)=r(x)^2
p
(
x
)
=
r
(
x
)
2
for some
r
∈
R
[
x
]
r\in\mathbb{R}[x]
r
∈
R
[
x
]
?
2
1
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Two-colored transitive digraph has independent cover
Prove that for any finite set
A
A
A
of positive integers, there exists a subset
B
B
B
of
A
A
A
satisfying the following conditions: [*]if
b
1
,
b
2
∈
B
b_1,b_2\in B
b
1
,
b
2
∈
B
are distinct, then neither
b
1
b_1
b
1
and
b
2
b_2
b
2
nor
b
1
+
1
b_1+1
b
1
+
1
and
b
2
+
1
b_2+1
b
2
+
1
are multiples of each other, and [*] for any
a
∈
A
a\in A
a
∈
A
, we can find a
b
∈
B
b\in B
b
∈
B
such that
a
a
a
divides
b
b
b
or
b
+
1
b+1
b
+
1
divides
a
+
1
a+1
a
+
1
.
1
1
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k-element subsets with strict relation
Let
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
be integers. At most how many
k
k
k
-element subsets can we select from
{
1
,
2
,
…
,
n
}
\{1,2,\dots,n\}
{
1
,
2
,
…
,
n
}
such that for any two selected subsets, one of the subsets consists of the
k
k
k
smallest elements of their union?