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Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2020 Romanian Master of Mathematics
2020 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(4)
4
1
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Function preserves sum-free-ness
Let
N
\mathbb N
N
be the set of all positive integers. A subset
A
A
A
of
N
\mathbb N
N
is sum-free if, whenever
x
x
x
and
y
y
y
are (not necessarily distinct) members of
A
A
A
, their sum
x
+
y
x+y
x
+
y
does not belong to
A
A
A
. Determine all surjective functions
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
such that, for each sum-free subset
A
A
A
of
N
\mathbb N
N
, the image
{
f
(
a
)
:
a
∈
A
}
\{f(a):a\in A\}
{
f
(
a
)
:
a
∈
A
}
is also sum-free.Note: a function
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
is surjective if, for every positive integer
n
n
n
, there exists a positive integer
m
m
m
such that
f
(
m
)
=
n
f(m)=n
f
(
m
)
=
n
.
5
1
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Lattice Polygons
A lattice point in the Cartesian plane is a point whose coordinates are both integers. A lattice polygon is a polygon all of whose vertices are lattice points. Let
Γ
\Gamma
Γ
be a convex lattice polygon. Prove that
Γ
\Gamma
Γ
is contained in a convex lattice polygon
Ω
\Omega
Ω
such that the vertices of
Γ
\Gamma
Γ
all lie on the boundary of
Ω
\Omega
Ω
, and exactly one vertex of
Ω
\Omega
Ω
is not a vertex of
Γ
\Gamma
Γ
.
3
1
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Odd Airports
Let
n
≥
3
n\ge 3
n
≥
3
be an integer. In a country there are
n
n
n
airports and
n
n
n
airlines operating two-way flights. For each airline, there is an odd integer
m
≥
3
m\ge 3
m
≥
3
, and
m
m
m
distinct airports
c
1
,
…
,
c
m
c_1, \dots, c_m
c
1
,
…
,
c
m
, where the flights offered by the airline are exactly those between the following pairs of airports:
c
1
c_1
c
1
and
c
2
c_2
c
2
;
c
2
c_2
c
2
and
c
3
c_3
c
3
;
…
\dots
…
;
c
m
−
1
c_{m-1}
c
m
−
1
and
c
m
c_m
c
m
;
c
m
c_m
c
m
and
c
1
c_1
c
1
. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
2
1
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Lots of Zeroes
Let
N
≥
2
N \geq 2
N
≥
2
be an integer, and let
a
\mathbf a
a
=
(
a
1
,
…
,
a
N
)
= (a_1, \ldots, a_N)
=
(
a
1
,
…
,
a
N
)
and
b
\mathbf b
b
=
(
b
1
,
…
b
N
)
= (b_1, \ldots b_N)
=
(
b
1
,
…
b
N
)
be sequences of non-negative integers. For each integer
i
∉
{
1
,
…
,
N
}
i \not \in \{1, \ldots, N\}
i
∈
{
1
,
…
,
N
}
, let
a
i
=
a
k
a_i = a_k
a
i
=
a
k
and
b
i
=
b
k
b_i = b_k
b
i
=
b
k
, where
k
∈
{
1
,
…
,
N
}
k \in \{1, \ldots, N\}
k
∈
{
1
,
…
,
N
}
is the integer such that
i
−
k
i-k
i
−
k
is divisible by
n
n
n
. We say
a
\mathbf a
a
is
b
\mathbf b
b
-harmonic if each
a
i
a_i
a
i
equals the following arithmetic mean:
a
i
=
1
2
b
i
+
1
∑
s
=
−
b
i
b
i
a
i
+
s
.
a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.
a
i
=
2
b
i
+
1
1
s
=
−
b
i
∑
b
i
a
i
+
s
.
Suppose that neither
a
\mathbf a
a
nor
b
\mathbf b
b
is a constant sequence, and that both
a
\mathbf a
a
is
b
\mathbf b
b
-harmonic and
b
\mathbf b
b
is
a
\mathbf a
a
-harmonic. Prove that at least
N
+
1
N+1
N
+
1
of the numbers
a
1
,
…
,
a
N
,
b
1
,
…
,
b
N
a_1, \ldots, a_N,b_1, \ldots, b_N
a
1
,
…
,
a
N
,
b
1
,
…
,
b
N
are zero.