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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2002 Switzerland Team Selection Test
2002 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(9)
10
1
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any partition of the set {m,m + 1,..,k} into 2 classes contains ab,c such a^b=c
Given an integer
m
≥
2
m\ge 2
m
≥
2
, find the smallest integer
k
>
m
k > m
k
>
m
such that for any partition of the set
{
m
,
m
+
1
,
.
.
,
k
}
\{m,m + 1,..,k\}
{
m
,
m
+
1
,
..
,
k
}
into two classes
A
A
A
and
B
B
B
at least one of the classes contains three numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
(not necessarily distinct) such that
a
b
=
c
a^b = c
a
b
=
c
.
9
1
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a^n +1a^n -2 >= n^2 (a +1/a -2 )
For each real number
a
a
a
and integer
n
≥
1
n \ge 1
n
≥
1
prove the inequality
a
n
+
1
a
n
−
2
≥
n
2
(
a
+
1
a
−
2
)
a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)
a
n
+
a
n
1
−
2
≥
n
2
(
a
+
a
1
−
2
)
and find the cases of equality.
6
1
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for any positive integer k exist indices i,jsuch that k =k =x_i -x_j
A sequence
x
1
,
x
2
,
x
3
,
.
.
.
x_1,x_2,x_3,...
x
1
,
x
2
,
x
3
,
...
has the following properties: (a)
1
=
x
1
<
x
2
<
x
3
<
.
.
.
1 = x_1 < x_2 < x_3 < ...
1
=
x
1
<
x
2
<
x
3
<
...
(b)
x
n
+
1
≤
2
n
x_{n+1} \le 2n
x
n
+
1
≤
2
n
for all
n
∈
N
n \in N
n
∈
N
. Prove that for each positive integer
k
k
k
there exist indices
i
i
i
and
j
j
j
such that
k
=
x
i
−
x
j
k =x_i -x_j
k
=
x
i
−
x
j
.
4
1
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Swiss crosses insides a 7x7 square, divided into unit squares
A
7
×
7
7 \times 7
7
×
7
square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side
3
3
3
) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.
3
1
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d_1^2+d_2^2+d_3^2+d_4^2 = n
Let
d
1
,
d
2
,
d
3
,
d
4
d_1,d_2,d_3,d_4
d
1
,
d
2
,
d
3
,
d
4
be the four smallest divisors of a positive integer
n
n
n
(having at least four divisors). Find all
n
n
n
such that
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
=
n
d_1^2+d_2^2+d_3^2+d_4^2 = n
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
=
n
.
2
1
Hide problems
<AOB + < COD = \pi inside parallelogram ABCD => < CBO =< CDO
A point
O
O
O
inside a parallelogram
A
B
C
D
ABCD
A
BC
D
satisfies
∠
A
O
B
+
∠
C
O
D
=
π
\angle AOB + \angle COD = \pi
∠
A
OB
+
∠
CO
D
=
π
. Prove that
∠
C
B
O
=
∠
C
D
O
\angle CBO = \angle CDO
∠
CBO
=
∠
C
D
O
.
1
1
Hide problems
exactly 2002 planes by 24 points
In space are given
24
24
24
points, no three of which are collinear. Suppose that there are exactly
2002
2002
2002
planes determined by three of these points. Prove that there is a plane containing at least six points.
5
1
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functional eq.
Find all
f
:
R
→
R
f: R\rightarrow R
f
:
R
→
R
such that (i) The set
{
f
(
x
)
x
∣
x
∈
R
−
{
0
}
}
\{\frac{f(x)}{x}| x\in R-\{0\}\}
{
x
f
(
x
)
∣
x
∈
R
−
{
0
}}
is finite (ii)
f
(
x
−
1
−
f
(
x
)
)
=
f
(
x
)
−
1
−
x
f(x-1-f(x)) = f(x)-1-x
f
(
x
−
1
−
f
(
x
))
=
f
(
x
)
−
1
−
x
for all
x
x
x
8
1
Hide problems
party acquaintances
In a group of
n
n
n
people, every weekend someone organizes a party in which he invites all of his acquaintances. Those who meet at a party become acquainted. After each of the
n
n
n
people has organized a party, there still are two people not knowing each other. Show that these two will never get to know each other at such a party.