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Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2015 Switzerland Team Selection Test
2015 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(11)
11
1
Hide problems
Cities connected by car or bicycle
In Thailand there are
n
n
n
cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car. Remark : It is not necessary to use the same means of transport for each city
10
1
Hide problems
Parallelogram and angles
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Suppose that there exists a point
P
P
P
in the interior of the parallelogram which is on the perpendicular bisector of
A
B
AB
A
B
and such that
∠
P
B
A
=
∠
A
D
P
\angle PBA = \angle ADP
∠
PB
A
=
∠
A
D
P
Show that
∠
C
P
D
=
2
∠
B
A
P
\angle CPD = 2 \angle BAP
∠
CP
D
=
2∠
B
A
P
9
1
Hide problems
Cost of controlled garden
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. At the center of a circular garden is a guard tower. On the outskirt of the garden there are
n
n
n
garden dwarfs regularly spaced. In the tower are attentive supervisors. Each supervisor controls a portion of the garden delimited by two dwarfs. We say that the supervisor
A
A
A
controls the supervisor
B
B
B
if the region of
B
B
B
is contained in that of
A
A
A
. Among the supervisors there are two groups: the apprentices and the teachers. Each apprentice is controlled by exactly one teachers, and controls no one, while the teachers are not controlled by anyone. The entire garden has the following maintenance costs: - One apprentice costs 1 gold per year. - One teacher costs 2 gold per year. - A garden dwarf costs 2 gold per year. Show that the garden dwarfs cost at least as much as the supervisors.
7
1
Hide problems
Finite set of functions for functional equation
Find all finite and non-empty sets
A
A
A
of functions
f
:
R
↦
R
f: \mathbb{R} \mapsto \mathbb{R}
f
:
R
↦
R
such that for all
f
1
,
f
2
∈
A
f_1, f_2 \in A
f
1
,
f
2
∈
A
, there exists
g
∈
A
g \in A
g
∈
A
such that for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
f
1
(
f
2
(
y
)
−
x
)
+
2
x
=
g
(
x
+
y
)
f_1 \left(f_2 (y)-x\right)+2x=g(x+y)
f
1
(
f
2
(
y
)
−
x
)
+
2
x
=
g
(
x
+
y
)
6
1
Hide problems
Polynomial equation
Find all polynomial function
P
P
P
of real coefficients such that for all
x
∈
R
x \in \mathbb{R}
x
∈
R
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
+
2
)
P(x)P(x+1)=P(x^2+2)
P
(
x
)
P
(
x
+
1
)
=
P
(
x
2
+
2
)
4
1
Hide problems
Equation with relatively prime integers
Find all relatively prime integers
a
,
b
a,b
a
,
b
such that
a
2
+
a
=
b
3
+
b
a^2+a=b^3+b
a
2
+
a
=
b
3
+
b
3
1
Hide problems
Same angle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
>
A
C
AB> AC
A
B
>
A
C
. Let
D
D
D
be a point on
A
B
AB
A
B
such that
D
B
=
D
C
DB = DC
D
B
=
D
C
and
M
M
M
the middle of
A
C
AC
A
C
. The parallel to
B
C
BC
BC
passing through
D
D
D
intersects the line
B
M
BM
BM
in
K
K
K
. Show that
∠
K
C
D
=
∠
D
A
C
\angle KCD = \angle DAC
∠
K
C
D
=
∠
D
A
C
.
1
1
Hide problems
Maximum colored boxes in chessboard
What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?
8
1
Hide problems
n+c divides a^n+b^n+n for certain n
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers such that if
n
n
n
is not divisible by any prime less than
2014
2014
2014
, then
n
+
c
n+c
n
+
c
divides
a
n
+
b
n
+
n
a^n+b^n+n
a
n
+
b
n
+
n
.Proposed by Evan Chen
2
1
Hide problems
Finally a sorta kinda clever-ish inequality
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers greater than or equal to
1
1
1
. Prove that
min
(
10
a
2
−
5
a
+
1
b
2
−
5
b
+
10
,
10
b
2
−
5
b
+
1
c
2
−
5
c
+
10
,
10
c
2
−
5
c
+
1
a
2
−
5
a
+
10
)
≤
a
b
c
.
\min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc.
min
(
b
2
−
5
b
+
10
10
a
2
−
5
a
+
1
,
c
2
−
5
c
+
10
10
b
2
−
5
b
+
1
,
a
2
−
5
a
+
10
10
c
2
−
5
c
+
1
)
≤
ab
c
.
12
1
Hide problems
Numbers with same digits
Given positive integers
m
m
m
and
n
n
n
, prove that there is a positive integer
c
c
c
such that the numbers
c
m
cm
c
m
and
c
n
cn
c
n
have the same number of occurrences of each non-zero digit when written in base ten.