MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2004 Switzerland - Final Round
2004 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(8)
5
1
Hide problems
min sum x^2/(ay + bz)(az + by)
Let
a
a
a
and
b
b
b
be fixed positive numbers. Find the smallest possible depending on
a
a
a
and
b
b
b
value of the sum
x
2
(
a
y
+
b
z
)
(
a
z
+
b
y
)
+
y
2
(
a
z
+
b
x
)
(
a
x
+
b
z
)
+
z
2
(
a
x
+
b
y
)
(
a
y
+
b
x
)
,
\frac{x^2}{(ay + bz)(az + by)}+\frac{y^2}{(az + bx)(ax + bz)}+\frac{z^2}{(ax + by)(ay + bx)},
(
a
y
+
b
z
)
(
a
z
+
b
y
)
x
2
+
(
a
z
+
b
x
)
(
a
x
+
b
z
)
y
2
+
(
a
x
+
b
y
)
(
a
y
+
b
x
)
z
2
,
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers.
10
1
Hide problems
L triomino on a nxn chsseboard
Let
n
>
1
n > 1
n
>
1
be an odd natural number. The squares of an
n
×
n
n \times n
n
×
n
chessboard are alternately colored white and black so that the four corner squares are black. An
L
L
L
-triomino is an
L
L
L
-shaped piece that covers exactly three squares of the board. For which values of
n
n
n
is it possible to cover all black squares with
L
L
L
-triominoes, so that no two
L
L
L
-triominos overlap? For these values of
n
n
n
determine the smallest possible number of
L
L
L
-triominoes that are necessary for this.
8
1
Hide problems
replace a,b with gcd(a, b) and lcm(a, b) in a list of numbers on blackboard
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers
a
,
b
a, b
a
,
b
, wipe them out and instead write gcd
(
a
,
b
)
(a, b)
(
a
,
b
)
and lcm
(
a
,
b
)
(a, b)
(
a
,
b
)
. Show that the content of the list no longer changed after a certain point in time.
7
1
Hide problems
< ABC <= 180^o/ m
Given are
m
≥
3
m\ge 3
m
≥
3
points in the plane. Prove that you can always choose three of these points
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
∠
A
B
C
≤
18
0
o
m
.
\angle ABC \le \frac{180^o}{m}.
∠
A
BC
≤
m
18
0
o
.
6
1
Hide problems
1^n + 2^n + 3^n + 4^n ending in k zeros
Determine all
k
k
k
for which there exists a natural number n such that
1
n
+
2
n
+
3
n
+
4
n
1^n + 2^n + 3^n + 4^n
1
n
+
2
n
+
3
n
+
4
n
with exactly
k
k
k
zeros at the end.
3
1
Hide problems
\sqrt{k^2 - pk} is positive integer
Let
p
p
p
be an odd prime number. Find all natural numbers
k
k
k
such that
k
2
−
p
k
\sqrt{k^2 - pk}
k
2
−
p
k
is a positive integer.
4
1
Hide problems
f(xf(x) + f(y)) = y + f(x)^2
Determine 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
holds
f
(
x
f
(
x
)
+
f
(
y
)
)
=
y
+
f
(
x
)
2
f(xf(x) + f(y)) = y + f(x)^2
f
(
x
f
(
x
)
+
f
(
y
))
=
y
+
f
(
x
)
2
2
1
Hide problems
sum of at least 2 of every 3 elements of M lies on M , subset of R
Let
M
M
M
be a finite set of real numbers with the following property: From three different elements of
M
M
M
can always be chosen two whose sum is located in
M
M
M
. How many elements can
M
M
M
have at most?