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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2004 Switzerland Team Selection Test
2004 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(9)
1
1
Hide problems
E(a)is the number of elements of $X_1\cup ... \cup X_n, \sum_{a\in S} E(a)
Let
S
S
S
be the set of all n-tuples
(
X
1
,
.
.
.
,
X
n
)
(X_1,...,X_n)
(
X
1
,
...
,
X
n
)
of subsets of the set
{
1
,
2
,
.
.
,
1000
}
\{1,2,..,1000\}
{
1
,
2
,
..
,
1000
}
, not necessarily different and not necessarily nonempty. For
a
=
(
X
1
,
.
.
.
,
X
n
)
a = (X_1,...,X_n)
a
=
(
X
1
,
...
,
X
n
)
denote by
E
(
a
)
E(a)
E
(
a
)
the number of elements of
X
1
∪
.
.
.
∪
X
n
X_1\cup ... \cup X_n
X
1
∪
...
∪
X
n
. Find an explicit formula for the sum
∑
a
∈
S
E
(
a
)
\sum_{a\in S} E(a)
∑
a
∈
S
E
(
a
)
2
1
Hide problems
max n such 4^{995} +4^{1500} +4^n is perfect square
Find the largest natural number
n
n
n
for which
4
995
+
4
1500
+
4
n
4^{995} +4^{1500} +4^n
4
995
+
4
1500
+
4
n
is a square.
7
1
Hide problems
a =\sqrt{45-\sqrt{21-a}}, b =\sqrt{45+\sqrt{21-b}}, c =\sqrt{45-\sqrt{21+c}},
The real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy the equations:
{
a
=
45
−
21
−
a
b
=
45
+
21
−
b
c
=
45
−
21
+
c
d
=
45
+
21
+
d
\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}
⎩
⎨
⎧
a
=
45
−
21
−
a
b
=
45
+
21
−
b
c
=
45
−
21
+
c
d
=
45
+
21
+
d
Prove that
a
b
c
d
=
2004
abcd = 2004
ab
c
d
=
2004
.
10
1
Hide problems
sum of areas of triangles ABZ, AYC, XBC equals area of ABC.
In an acute-angled triangle
A
B
C
ABC
A
BC
the altitudes
A
U
,
B
V
,
C
W
AU,BV,CW
A
U
,
B
V
,
C
W
intersect at
H
H
H
. Points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
, different from
H
H
H
, are taken on segments
A
U
,
B
V
AU,BV
A
U
,
B
V
, and
C
W
CW
C
W
, respectively. (a) Prove that if
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
and
H
H
H
lie on a circle, then the sum of the areas of triangles
A
B
Z
,
A
Y
C
,
X
B
C
ABZ, AYC, XBC
A
BZ
,
A
Y
C
,
XBC
equals the area of
A
B
C
ABC
A
BC
. (b) Prove the converse of (a).
11
1
Hide problems
f((x+y)/(x-y) )= (f(x)+ f(y))/(f(x)- f(y)) , injective
Find all injective functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all real
x
≠
y
x \ne y
x
=
y
,
f
(
x
+
y
x
−
y
)
=
f
(
x
)
+
f
(
y
)
f
(
x
)
−
f
(
y
)
f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}
f
(
x
−
y
x
+
y
)
=
f
(
x
)
−
f
(
y
)
f
(
x
)
+
f
(
y
)
12
1
Hide problems
naturals in form (a+b+c)^2/(abc) where a,b,c \in N
Find all natural numbers which can be written in the form
(
a
+
b
+
c
)
2
a
b
c
\frac{(a+b+c)^2}{abc}
ab
c
(
a
+
b
+
c
)
2
, where
a
,
b
,
c
∈
N
a,b,c \in N
a
,
b
,
c
∈
N
.
5
1
Hide problems
building bricks
A brick has the shape of a cube of size
2
2
2
with one corner unit cube removed. Given a cube of side
2
n
2^{n}
2
n
divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
9
1
Hide problems
subtraction of subsets
Let
A
1
,
.
.
.
,
A
n
A_{1}, ..., A_{n}
A
1
,
...
,
A
n
be different subsets of an
n
n
n
-element set
X
X
X
. Show that there exists
x
∈
X
x\in X
x
∈
X
such that the sets
A
1
−
{
x
}
,
A
2
−
{
x
}
,
.
.
.
,
A
n
−
{
x
}
A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}
A
1
−
{
x
}
,
A
2
−
{
x
}
,
...
,
A
n
−
{
x
}
are all different.
4
1
Hide problems
Swiss IMO TST 2004
Second Test, May 16 Let
a
a
a
,
b
b
b
, and
c
c
c
be positive real numbers such that
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that
a
b
a
5
+
b
5
+
a
b
+
b
c
b
5
+
c
5
+
b
c
+
c
a
c
5
+
a
5
+
c
a
≤
1
\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1
a
5
+
b
5
+
ab
ab
+
b
5
+
c
5
+
b
c
b
c
+
c
5
+
a
5
+
c
a
c
a
≤
1
. When does equality hold?