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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2007 Switzerland - Final Round
2007 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(8)
1
1
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6x6 system, a = max {1/b, 1/c}
Determine all positive real solutions of the following system of equations:
a
=
m
a
x
{
1
b
,
1
c
}
b
=
max
{
1
c
,
1
d
}
c
=
max
{
1
d
,
1
e
}
a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\}
a
=
ma
x
{
b
1
,
c
1
}
b
=
max
{
c
1
,
d
1
}
c
=
max
{
d
1
,
e
1
}
d
=
max
{
1
e
,
1
f
}
e
=
max
{
1
f
,
1
a
}
f
=
max
{
1
a
,
1
b
}
d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}
d
=
max
{
e
1
,
f
1
}
e
=
max
{
f
1
,
a
1
}
f
=
max
{
a
1
,
b
1
}
7
1
Hide problems
sum \sqrt{a+\sqrt{b + \sqrt{c}}} <= 3\sqrt{m+\sqrt{m + \sqrt{m}}}
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be nonnegative real numbers with arithmetic mean
m
=
a
+
b
+
c
3
m =\frac{a+b+c}{3}
m
=
3
a
+
b
+
c
. Provethat
a
+
b
+
c
+
b
+
c
+
a
+
c
+
a
+
b
≤
3
m
+
m
+
m
.
\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.
a
+
b
+
c
+
b
+
c
+
a
+
c
+
a
+
b
≤
3
m
+
m
+
m
.
8
1
Hide problems
subset of 1-2007, one divides another
Let
M
⊂
{
1
,
2
,
3
,
.
.
.
,
2007
}
M\subset \{1, 2, 3, . . . , 2007\}
M
⊂
{
1
,
2
,
3
,
...
,
2007
}
a set with the following property: Among every three numbers one can always choose two from
M
M
M
such that one is divisible by the other. How many numbers can
M
M
M
contain at most?
10
1
Hide problems
stones on infinity grid of equalteral triangles
The plane is divided into equilateral triangles of side length
1
1
1
. Consider a equilateral triangle of side length
n
n
n
whose sides lie on the grid lines. On every grid point on the edge and inside of this triangle lies a stone. In a move, a unit triangle is selected, which has exactly
2
2
2
corners with is covered with a stone. The two stones are removed, and the third corner is turned a new stone was laid. For which
n
n
n
is it possible that after finitely many moves only one stone left?
9
1
Hide problems
(a^3 + 1)/( 2ab^2 + 1) is an integer
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of natural numbers such that
a
3
+
1
2
a
b
2
+
1
\frac{a^3 + 1}{2ab^2 + 1}
2
a
b
2
+
1
a
3
+
1
is an integer.
5
1
Hide problems
f(xf(y)) f(y) = f (xy/(x + y) )
Determine all functions
f
:
R
≥
0
→
R
≥
0
f : R_{\ge 0} \to R_{\ge 0}
f
:
R
≥
0
→
R
≥
0
with the following properties: (a)
f
(
1
)
=
0
f(1) = 0
f
(
1
)
=
0
, (b)
f
(
x
)
>
0
f(x) > 0
f
(
x
)
>
0
for all
x
>
1
x > 1
x
>
1
, (c) For all
x
,
y
≥
0
x, y\ge 0
x
,
y
≥
0
with
x
+
y
>
0
x + y > 0
x
+
y
>
0
holds
f
(
x
f
(
y
)
)
f
(
y
)
=
f
(
x
y
x
+
y
)
f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)
f
(
x
f
(
y
))
f
(
y
)
=
f
(
x
+
y
x
y
)
2
1
Hide problems
a^{2007}+b^{2007}+c^{2007}+2 x 2007abc is divisible by 13
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be three integers such that
a
+
b
+
c
a + b + c
a
+
b
+
c
is divisible by
13
13
13
. Prove that
a
2007
+
b
2007
+
c
2007
+
2
⋅
2007
a
b
c
a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc
a
2007
+
b
2007
+
c
2007
+
2
⋅
2007
ab
c
is divisible by
13
13
13
.
3
1
Hide problems
L-tetromino, infinite grid
The plane is divided into unit squares. Each box should be be colored in one of
n
n
n
colors , so that if four squares can be covered with an
L
L
L
-tetromino, then these squares have four different colors (the
L
L
L
-Tetromino may be rotated and be mirrored). Find the smallest value of
n
n
n
for which this is possible.