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Contests
National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2001 Switzerland Team Selection Test
2001 Switzerland Team Selection Test
Part of
Switzerland Team Selection Test
Subcontests
(9)
10
1
Hide problems
1000-element subset M of {0,1,...,2001}, has power of 2 or 2 nos with sum 2
Prove that every
1000
1000
1000
-element subset
M
M
M
of the set
{
0
,
1
,
.
.
.
,
2001
}
\{0,1,...,2001\}
{
0
,
1
,
...
,
2001
}
contains either a power of two or two distinct numbers whose sum is a power of two.
9
1
Hide problems
16 secret agents in Geneva
In Geneva there are
16
16
16
secret agents, each of whom is watching one or more other agents. It is known that if agent
A
A
A
is watching agent
B
B
B
, then
B
B
B
is not watching
A
A
A
. Moreover, any
10
10
10
agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any
11
11
11
agents can also be so ordered.
8
1
Hide problems
\frac{68}{n+70},...,\frac{69}{n+71},\frac{70}{n+72},...,\frac{133}{n+135}
Find two smallest natural numbers
n
n
n
for which each of the fractions
68
n
+
70
,
69
n
+
71
,
70
n
+
72
,
.
.
.
,
133
n
+
135
\frac{68}{n+70},\frac{69}{n+71},\frac{70}{n+72},...,\frac{133}{n+135}
n
+
70
68
,
n
+
71
69
,
n
+
72
70
,
...
,
n
+
135
133
is irreducible.
7
1
Hide problems
perpendicular wanted , where O is circumcenter
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumcenter
O
O
O
. The circle
S
S
S
through
A
,
B
A,B
A
,
B
, and
O
O
O
intersects
A
C
AC
A
C
and
B
C
BC
BC
again at points
P
P
P
and
Q
Q
Q
respectively. Prove that
C
O
⊥
P
Q
CO \perp PQ
CO
⊥
PQ
.
6
1
Hide problems
f(x+y) >= f(x)+ f(y), prove f(x) \le 2x for all x \in [0,1]
A function
f
:
[
0
,
1
]
→
R
f : [0,1] \to R
f
:
[
0
,
1
]
→
R
has the following properties: (a)
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for
0
<
x
<
1
0 < x < 1
0
<
x
<
1
, (b)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
, (c)
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
f(x+y) \ge f(x)+ f(y)
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
whenever
x
,
y
,
x
+
y
∈
[
0
,
1
]
x,y,x+y \in [0,1]
x
,
y
,
x
+
y
∈
[
0
,
1
]
. Prove that
f
(
x
)
≤
2
x
f(x) \le 2x
f
(
x
)
≤
2
x
for all
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
.
5
1
Hide problems
sum 1/a_i <= 1+1/2 +1/3 +...+1/9, decimal representation
Let
a
1
<
a
2
<
.
.
.
<
a
n
a_1 < a_2 < ... < a_n
a
1
<
a
2
<
...
<
a
n
be a sequence of natural numbers such that for
i
<
j
i < j
i
<
j
the decimal representation of
a
i
a_i
a
i
does not occur as the leftmost digits of the decimal representation of
a
j
a_j
a
j
. (For example,
137
137
137
and
13729
13729
13729
cannot both occur in the sequence.) Prove that
∑
i
=
1
n
1
a
i
≤
1
+
1
2
+
1
3
+
.
.
.
+
1
9
\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19
∑
i
=
1
n
a
i
1
≤
1
+
2
1
+
3
1
+
...
+
9
1
.
4
1
Hide problems
all representations of n as a sum of its distinct divisors,
For a natural number
n
≥
2
n \ge 2
n
≥
2
, consider all representations of
n
n
n
as a sum of its distinct divisors,
n
=
t
1
+
t
2
+
.
.
.
+
t
k
,
t
i
∣
n
n = t_1 + t_2 + ... + t_k, t_i| n
n
=
t
1
+
t
2
+
...
+
t
k
,
t
i
∣
n
. Two such representations differing only in order of the summands are considered the same (for example,
20
=
10
+
5
+
4
+
1
20 = 10+5+4+1
20
=
10
+
5
+
4
+
1
and
20
=
5
+
1
+
10
+
4
20 = 5+1+10+4
20
=
5
+
1
+
10
+
4
). Let
a
(
n
)
a(n)
a
(
n
)
be the number of different representations of
n
n
n
in this form. Prove or disprove: There exists M such that
a
(
n
)
≤
M
a(n) \le M
a
(
n
)
≤
M
for all
n
≥
2
n \ge 2
n
≥
2
.
2
1
Hide problems
\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}
If
a
,
b
a,b
a
,
b
, and
c
c
c
are the sides of a triangle, prove the inequality
a
+
b
−
c
+
c
+
a
−
b
+
b
+
c
−
a
≤
a
+
b
+
c
\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}
a
+
b
−
c
+
c
+
a
−
b
+
b
+
c
−
a
≤
a
+
b
+
c
. When does equality occur?
1
1
Hide problems
2001 x 2001 trees in a park form a square grid
The
2001
×
2001
2001 \times 2001
2001
×
2001
trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)