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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2023 Switzerland Team Selection Test
2023 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(5)
9
1
Hide problems
Weird graph
Let
G
G
G
be a graph whose vertices are the integers. Assume that any two integers are connected by a finite path in
G
G
G
. For two integers
x
x
x
and
y
y
y
, we denote by
d
(
x
,
y
)
d(x, y)
d
(
x
,
y
)
the length of the shortest path from
x
x
x
to
y
y
y
, where the length of a path is the number of edges in it. Assume that
d
(
x
,
y
)
∣
x
−
y
d(x, y) \mid x-y
d
(
x
,
y
)
∣
x
−
y
for all integers
x
,
y
x, y
x
,
y
and define
S
(
G
)
=
{
d
(
x
,
y
)
∣
x
,
y
∈
Z
}
S(G)=\{d(x, y) | x, y \in \mathbb{Z}\}
S
(
G
)
=
{
d
(
x
,
y
)
∣
x
,
y
∈
Z
}
. Find all possible sets
S
(
G
)
S(G)
S
(
G
)
.
7
1
Hide problems
Polynomials with all roots <1
Find all monic polynomials
P
(
x
)
=
x
2023
+
a
2022
x
2022
+
…
+
a
1
x
+
a
0
P(x)=x^{2023}+a_{2022}x^{2022}+\ldots+a_1x+a_0
P
(
x
)
=
x
2023
+
a
2022
x
2022
+
…
+
a
1
x
+
a
0
with real coefficients such that
a
2022
=
0
a_{2022}=0
a
2022
=
0
,
P
(
1
)
=
1
P(1)=1
P
(
1
)
=
1
and all roots of
P
P
P
are real and less than
1
1
1
.
5
1
Hide problems
Graph with Tokyo flavortext
The Tokyo Metro system is one of the most efficient in the world. There is some odd positive integer
k
k
k
such that each metro line passes through exactly
k
k
k
stations, and each station is serviced by exactly
k
k
k
metro lines. One can get from any station to any otherstation using only one metro line - but this connection is unique. Furthermore, any two metro lines must share exactly one station. David is planning an excursion for the IMO team, and wants to visit a set
S
S
S
of
k
k
k
stations. He remarks that no three of the stationsin
S
S
S
are on a common metro line. Show that there is some station not in
S
S
S
, which is connected to every station in
S
S
S
by a different metro line.
4
1
Hide problems
Similar triangles with a common vertex
Let
A
B
C
ABC
A
BC
and
A
M
N
AMN
A
MN
be two similar, non-overlapping triangles with the same orientation, such that
A
B
=
A
C
AB=AC
A
B
=
A
C
and
A
M
=
A
N
AM=AN
A
M
=
A
N
. Let
O
O
O
be the circumcentre of the triangle
M
A
B
MAB
M
A
B
. Prove that the points
O
,
C
,
N
O, C, N
O
,
C
,
N
and
A
A
A
lie on a circle if and only if the triangle
A
B
C
ABC
A
BC
is equilateral.
2
1
Hide problems
NT with a set
Let
S
S
S
be a non-empty set of positive integers such that for any
n
∈
S
n \in S
n
∈
S
, all positive divisors of
2
n
+
1
2^n+1
2
n
+
1
are also in
S
S
S
. Prove that
S
S
S
contains an integer of the form
(
p
1
p
2
…
p
2023
)
2023
(p_1p_2 \ldots p_{2023})^{2023}
(
p
1
p
2
…
p
2023
)
2023
, where
p
1
,
p
2
,
…
,
p
2023
p_1, p_2, \ldots, p_{2023}
p
1
,
p
2
,
…
,
p
2023
are distinct prime numbers, all greater than
2023
2023
2023
.