MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2016 Switzerland - Final Round
2016 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
10
1
Hide problems
f(x + yf(x + y)) = y^2 + f(xf(y + 1))
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
:
f
(
x
+
y
f
(
x
+
y
)
)
=
y
2
+
f
(
x
f
(
y
+
1
)
)
.
f(x + yf(x + y)) = y^2 + f(xf(y + 1)).
f
(
x
+
y
f
(
x
+
y
))
=
y
2
+
f
(
x
f
(
y
+
1
))
.
9
1
Hide problems
n -element subset F of {1, . . . , 2n}
Let
n
≥
2
n \ge 2
n
≥
2
be a natural number. For an
n
n
n
-element subset
F
F
F
of
{
1
,
.
.
.
,
2
n
}
\{1, . . . , 2n\}
{
1
,
...
,
2
n
}
we define
m
(
F
)
m(F)
m
(
F
)
as the minimum of all
l
c
m
(
x
,
y
)
lcm \,\, (x, y)
l
c
m
(
x
,
y
)
, where
x
x
x
and
y
y
y
are two distinct elements of
F
F
F
. Find the maximum value of
m
(
F
)
m(F)
m
(
F
)
.
7
1
Hide problems
2n distinct points on a circle, assign |a - b| to a segmentm sum n^2
There are
2
n
2n
2
n
distinct points on a circle. The numbers
1
1
1
through
2
n
2n
2
n
are randomly assigned to this one points distributed. Each point is connected to exactly one other point, so that no of the resulting connecting routes intersect. If a segment connects the numbers
a
a
a
and
b
b
b
, so we assign the value
∣
a
−
b
∣
|a - b|
∣
a
−
b
∣
to the segment . Show that we can choose the routes such that the sum of these values results
n
2
n^2
n
2
.
6
1
Hide problems
a_n = a_{n-1}^2 + 1 , a_k divides a_{\ell }
Let
a
n
a_n
a
n
be a sequence of natural numbers defined by
a
1
=
m
a_1 = m
a
1
=
m
and for
n
>
1
n > 1
n
>
1
. We call apair
(
a
k
,
a
ℓ
)
(a_k, a_{\ell })
(
a
k
,
a
ℓ
)
interesting if (i)
0
<
ℓ
−
k
<
2016
0 < \ell - k < 2016
0
<
ℓ
−
k
<
2016
, (ii)
a
k
a_k
a
k
divides
a
ℓ
a_{\ell }
a
ℓ
. Show that there exists a
m
m
m
such that the sequence
a
n
a_n
a
n
contains no interesting pair.
2
1
Hide problems
sum (ab+ 1)/(a^2 + ca + 1) >= 3/2 for triangle sidelengths
Let
a
,
b
a, b
a
,
b
and
c
c
c
be the sides of a triangle, that is:
a
+
b
>
c
a + b > c
a
+
b
>
c
,
b
+
c
>
a
b + c > a
b
+
c
>
a
and
c
+
a
>
b
c + a > b
c
+
a
>
b
. Show that:
a
b
+
1
a
2
+
c
a
+
1
+
b
c
+
1
b
2
+
a
b
+
1
+
c
a
+
1
c
2
+
b
c
+
1
>
3
2
\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32
a
2
+
c
a
+
1
ab
+
1
+
b
2
+
ab
+
1
b
c
+
1
+
c
2
+
b
c
+
1
c
a
+
1
>
2
3
4
1
Hide problems
2016 different points in plane with at least 45 different distances
There are
2016
2016
2016
different points in the plane. Show that between these points at least
45
45
45
different distances occur.
3
1
Hide problems
p(p+1)+q(q+1)=n(n+1)
Find all primes
p
,
q
p, q
p
,
q
and natural numbers
n
n
n
such that:
p
(
p
+
1
)
+
q
(
q
+
1
)
=
n
(
n
+
1
)
p(p+1)+q(q+1)=n(n+1)
p
(
p
+
1
)
+
q
(
q
+
1
)
=
n
(
n
+
1
)