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Contests
National and Regional Contests
Switzerland Contests
Preliminary Round - Switzerland
2011 Preliminary Round - Switzerland
2011 Preliminary Round - Switzerland
Part of
Preliminary Round - Switzerland
Subcontests
(5)
5
1
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Right angle [Switzerland 2011]
Let
A
B
C
D
ABCD
A
BC
D
an inscribed quadrilateral and
r
r
r
and
s
s
s
the reflections of the straight line through
A
A
A
and
B
B
B
over the inner angle bisectors of angles
∠
C
A
D
\angle{CAD}
∠
C
A
D
and
∠
C
B
D
\angle{CBD}
∠
CB
D
, respectively. Let
P
P
P
the point of intersection of
r
r
r
and
s
s
s
and let
O
O
O
the circumcentre of
A
B
C
D
ABCD
A
BC
D
. Prove that
O
P
⊥
C
D
OP \perp CD
OP
⊥
C
D
.
4
1
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Number of bus routes [Switzerland 2011]
Given is a circular bus route with
n
⩾
2
n\geqslant2
n
⩾
2
bus stops. The route can be frequented in both directions. The way between two stops is called section and one of the bus stops is called Zürich. A bus shall start at Zürich, pass through all the bus stops at least once and drive along exactly
n
+
2
n+2
n
+
2
sections before it returns to Zürich in the end. Assuming that the bus can chance directions at each bus stop, how many possible routes are there?EDIT: Sorry, there was a mistake...corrected now, thanks mavropnevma! :oops:
3
1
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Placing "+" and "-" [Switzerland 2011]
On a blackboard, there are
11
11
11
positive integers. Show that one can choose some (maybe all) of these numbers and place "
+
+
+
" and "
−
-
−
" in between such that the result is divisible by
2011
2011
2011
.
2
1
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Product of divisors [Switzerland 2011]
Find all positive integers
n
n
n
such that
n
3
n^3
n
3
is the product of all divisors of
n
n
n
.
1
1
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Collinearity [Switzerland 2011]
Let
△
A
B
C
\triangle{ABC}
△
A
BC
a triangle with
∠
C
A
B
=
9
0
∘
\angle{CAB}=90^{\circ}
∠
C
A
B
=
9
0
∘
and
L
L
L
a point on the segment
B
C
BC
BC
. The circumcircle of triangle
△
A
B
L
\triangle{ABL}
△
A
B
L
intersects
A
C
AC
A
C
at
M
M
M
and the circumcircle of triangle
△
C
A
L
\triangle{CAL}
△
C
A
L
intersects
A
B
AB
A
B
at
N
N
N
. Show that
L
L
L
,
M
M
M
and
N
N
N
are collinear.