MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2005 Switzerland - Final Round
2005 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(8)
6
1
Hide problems
sum (1+a)/(1+a+ab) where abc=1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers with
a
b
c
=
1
abc = 1
ab
c
=
1
. Find all possible values of the expression
1
+
a
1
+
a
+
a
b
+
1
+
b
1
+
b
+
b
c
+
1
+
c
1
+
c
+
c
a
\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}
1
+
a
+
ab
1
+
a
+
1
+
b
+
b
c
1
+
b
+
1
+
c
+
c
a
1
+
c
can take.
4
1
Hide problems
(a+ b)/gcd(a, b) \in M
Determine all sets
M
M
M
of natural numbers such that for every two (not necessarily different) elements
a
,
b
a, b
a
,
b
from
M
M
M
,
a
+
b
g
c
d
(
a
,
b
)
\frac{a + b}{gcd(a, b)}
g
c
d
(
a
,
b
)
a
+
b
lies in
M
M
M
.
3
1
Hide problems
sum ka_k <= {n \choose 2} + sum a_k^k
Prove for all
a
1
,
.
.
.
,
a
n
>
0
a_1, ..., a_n > 0
a
1
,
...
,
a
n
>
0
the following inequality and determine all cases in where the equaloty holds:
∑
k
=
1
n
k
a
k
≤
(
n
2
)
+
∑
k
=
1
n
a
k
k
.
\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.
k
=
1
∑
n
k
a
k
≤
(
2
n
)
+
k
=
1
∑
n
a
k
k
.
10
1
Hide problems
n > 10 teams take part in a football tournament
n
>
10
n > 10
n
>
10
teams take part in a football tournament. Every team plays exactly once against each other. A win gives two points, a draw a point, and a defeat no point. After the tournament it turns out that each team gets exactly half their points in the games against the bottom
10
10
10
teams has won (in particular, each of these
10
10
10
teams has won the made half their points against the
9
9
9
remaining). Determine all possible values of
n
n
n
, and give an example of such a tournament for these values.
9
1
Hide problems
f(yf(x))(x + y) = x^2(f(x) + f(y))
Find all functions
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
such that for all
x
,
y
>
0
x, y > 0
x
,
y
>
0
f
(
y
f
(
x
)
)
(
x
+
y
)
=
x
2
(
f
(
x
)
+
f
(
y
)
)
.
f(yf(x))(x + y) = x^2(f(x) + f(y)).
f
(
y
f
(
x
))
(
x
+
y
)
=
x
2
(
f
(
x
)
+
f
(
y
))
.
7
1
Hide problems
7 x4^n = a^2 + b^2 + c^2 + d^2
Let
n
≥
1
n\ge 1
n
≥
1
be a natural number. Determine all positive integer solutions of the equation
7
⋅
4
n
=
a
2
+
b
2
+
c
2
+
d
2
.
7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.
7
⋅
4
n
=
a
2
+
b
2
+
c
2
+
d
2
.
5
1
Hide problems
area of P_n >= 1/2 for tweaked a convex n-gon, starting with area 1
Tweaking a convex
n
n
n
-gon means the following: choose two adjacent sides
A
B
AB
A
B
and
B
C
BC
BC
and replaces them with the line segment
A
M
AM
A
M
,
M
N
MN
MN
,
N
C
NC
NC
, where
M
∈
A
B
M \in AB
M
∈
A
B
and
N
∈
B
C
N \in BC
N
∈
BC
are arbitrary points inside these segments. In other words, you cut off a corner and get an
(
n
+
1
)
(n+1)
(
n
+
1
)
-corner. Starting from a regular hexagon
P
6
P_6
P
6
with area
1
1
1
, by continuous Tweaks a sequence
P
6
,
P
7
,
P
8
,
.
.
.
P_6,P_7,P_8, ...
P
6
,
P
7
,
P
8
,
...
convex polygons. Show that Area of
P
n
P_n
P
n
for all
n
≥
6
n\ge 6
n
≥
6
greater than
1
2
\frac1 2
2
1
is, regardless of how tweaks takes place.
2
1
Hide problems
2n black and 2n with points on a row
Of
4
n
4n
4
n
points in a row,
2
n
2n
2
n
are colored white and
2
n
2n
2
n
are colored black. Swot tha tthere are
2
n
2n
2
n
consecutive points of which exactly
n
n
n
are white and
n
n
n
are black.