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Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2020 Switzerland Team Selection Test
2020 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(5)
1
1
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easy chessboard combo
Let
n
≥
2
n \geq 2
n
≥
2
be an integer. Consider an
n
×
n
n\times n
n
×
n
chessboard with the usual chessboard colouring. A move consists of choosing a
1
×
1
1\times 1
1
×
1
square and switching the colour of all squares in its row and column (including the chosen square itself). For which
n
n
n
is it possible to get a monochrome chessboard after a finite sequence of moves?
2
1
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Cute nt from Switzerland
Find all positive integers
n
n
n
such that there exists an infinite set
A
A
A
of positive integers with the following property: For all pairwise distinct numbers
a
1
,
a
2
,
…
,
a
n
∈
A
a_1, a_2, \ldots , a_n \in A
a
1
,
a
2
,
…
,
a
n
∈
A
, the numbers
a
1
+
a
2
+
…
+
a
n
and
a
1
⋅
a
2
⋅
…
⋅
a
n
a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n
a
1
+
a
2
+
…
+
a
n
and
a
1
⋅
a
2
⋅
…
⋅
a
n
are coprime.
3
1
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Trivial p3 geo from Swiss TST
Let
k
k
k
be a circle with centre
O
O
O
. Let
A
B
AB
A
B
be a chord of this circle with midpoint
M
≠
O
M\neq O
M
=
O
. The tangents of
k
k
k
at the points
A
A
A
and
B
B
B
intersect at
T
T
T
. A line goes through
T
T
T
and intersects
k
k
k
in
C
C
C
and
D
D
D
with
C
T
<
D
T
CT < DT
CT
<
D
T
and
B
C
=
B
M
BC = BM
BC
=
BM
. Prove that the circumcentre of the triangle
A
D
M
ADM
A
D
M
is the reflection of
O
O
O
across the line
A
D
AD
A
D
.
12
1
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Tournament S.S.D
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers such that
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
prove that: (
a
2
a
+
b
+
b
2
b
+
c
+
c
2
c
+
d
+
d
2
d
+
a
)
5
≥
5
5
(
a
c
27
)
2
\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a})^5 \geq 5^5(\frac{ac}{27})^2
a
+
b
a
2
+
b
+
c
b
2
+
c
+
d
c
2
+
d
+
a
d
2
)
5
≥
5
5
(
27
a
c
)
2
4
1
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Russia 2001
Find all odd positive integers
n
>
1
n > 1
n
>
1
such that if
a
a
a
and
b
b
b
are relatively prime divisors of
n
n
n
, then a\plus{}b\minus{}1 divides
n
n
n
.