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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2014 Switzerland - Final Round
2014 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
4
1
Hide problems
a x b colors on infinite grid
The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of
a
⋅
b
a \cdot b
a
⋅
b
colors, so that each rectangle of size
a
×
b
a \times b
a
×
b
or
b
×
a
b \times a
b
×
a
, placed appropriately in the checkered plane, always contains a unit square with each color ?
7
1
Hide problems
n cities on a round lake
There are
n
≥
4
n \ge 4
n
≥
4
cities on a round lake, between which
n
−
4
n -4
n
−
4
people travel and one green drivers operate. Each ferry connects two non-adjacent cities, and itself do not cross two driving routes so that collisions can be avoided. In order to better adapt the transport routes to the needs of the passengers, the following change can be done: A new route can be assigned to any driver. The routes of the remaining drives must not cross and also must not be changed at the same time. Let Santa Marta and Cape Town be two non-adjacent cities. Show that you have finitely many route changes so that the Green Driver will operate between Santa Marta and Cape Town after these changes.Note: At no time may two trips between the same cities or one drive between two neighboring cities.[hide=original wording]An einem runden See liegen
n
>
=
4
n >= 4
n
>=
4
Stadte, zwischen denen
n
−
4
n - 4
n
−
4
Personenfahren und eine grune Autofahre verkehren. Jede Fahre verbindet zwei nicht benachbarte Stadte, wobei sich keine zwei Fahrenrouten uberkreuzen, damit Kollisionen vermieden werden konnen. Um die Transportrouten besser den Bedurfnissen der Passagiere anzupassen, kann folgende Anderung vorgenommen werden: Einer beliebigen Fahre kann eine neue Route zugeordnet werden. Dabei durfen die Routen der restlichen Fahren nicht uberkreuzt und auch nicht gleichzeitig verandert werden. Seien Santa Marta und Kapstadt zwei nicht benachbarte Stadte. Zeige, dass man endlich viele Routenanderungen vornehmen kann, sodass die grune Autofahre nach diesen Anderungen zwischen Santa Marta und Kapstadt verkehrt. Bemerkung: Zu keinem Zeitpunkt durfen zwei Fahren zwischen denselben Stadten oder eine Fahre zwischen zwei benachbarten Stadten verkehren.
9
1
Hide problems
a_n=0 or according to parity of number of divisors > 2014
The sequence of integers
a
1
,
a
2
,
,
,
a_1, a_2, ,,
a
1
,
a
2
,,,
is defined as follows:
a
n
=
{
0
i
f
n
h
a
s
a
n
e
v
e
n
n
u
m
b
e
r
o
f
d
i
v
i
s
o
r
s
g
r
e
a
t
e
r
t
h
a
n
2014
1
i
f
n
h
a
s
a
n
o
d
d
n
u
m
b
e
r
o
f
d
i
v
i
s
o
r
s
g
r
e
a
t
e
r
t
h
a
n
2014
a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}
a
n
=
{
0
i
f
n
ha
s
an
e
v
e
n
n
u
mb
er
o
f
d
i
v
i
sors
g
re
a
t
er
t
han
2014
1
i
f
n
ha
s
an
o
dd
n
u
mb
er
o
f
d
i
v
i
sors
g
re
a
t
er
t
han
2014
Show that the sequence
a
n
a_n
a
n
never becomes periodic.
5
1
Hide problems
n | a_n if sum_{d | n} a_d = 2^n
Let
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
a sequence of integers such that for every
n
∈
N
n \in N
n
∈
N
we have:
∑
d
∣
n
a
d
=
2
n
.
\sum_{d | n} a_d = 2^n.
d
∣
n
∑
a
d
=
2
n
.
Show for every
n
∈
N
n \in N
n
∈
N
that
n
n
n
divides
a
n
a_n
a
n
.Remark: For
n
=
6
n = 6
n
=
6
the equation is
a
1
+
a
2
+
a
3
+
a
6
=
2
6
.
a_1 + a_2 + a_3 + a_6 = 2^6.
a
1
+
a
2
+
a
3
+
a
6
=
2
6
.
6
1
Hide problems
An inequality
Let
a
,
b
,
c
∈
R
≥
0
a,b,c\in \mathbb{R}_{\ge 0}
a
,
b
,
c
∈
R
≥
0
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove the inequality :
3
−
b
a
+
1
+
a
+
1
b
+
1
+
b
+
1
c
+
1
≥
4
\frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4
a
+
1
3
−
b
+
b
+
1
a
+
1
+
c
+
1
b
+
1
≥
4
3
1
Hide problems
A functional equation
Find all such functions
f
:
R
→
R
f :\mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
the following holds :
f
(
x
2
)
+
f
(
x
y
)
=
f
(
x
)
f
(
y
)
+
y
f
(
x
)
+
x
f
(
x
+
y
)
f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)
f
(
x
2
)
+
f
(
x
y
)
=
f
(
x
)
f
(
y
)
+
y
f
(
x
)
+
x
f
(
x
+
y
)
2
1
Hide problems
Perfect Square
Let
a
,
b
∈
N
a,b\in\mathbb{N}
a
,
b
∈
N
such that :
a
b
(
a
−
b
)
∣
a
3
+
b
3
+
a
b
ab(a-b)\mid a^3+b^3+ab
ab
(
a
−
b
)
∣
a
3
+
b
3
+
ab
Then show that
lcm
(
a
,
b
)
\operatorname{lcm}(a,b)
lcm
(
a
,
b
)
is a perfect square.