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Problems
Contests
National and Regional Contests
Estonia Contests
Estonia Team Selection Test
1997 Estonia Team Selection Test
1997 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(3)
3
2
Hide problems
Combinatorics
There are
n
n
n
boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of
n
n
n
can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than
n
n
n
dances?
It is known that for every integer $n > 1$ there is a prime
It is known that for every integer
n
>
1
n > 1
n
>
1
there is a prime number among the numbers
n
+
1
,
n
+
2
,
.
.
.
,
2
n
−
1.
n+1,n+2,...,2n-1.
n
+
1
,
n
+
2
,
...
,
2
n
−
1.
Determine all positive integers
n
n
n
with the following property: Every integer
m
>
1
m > 1
m
>
1
less than
n
n
n
and coprime to
n
n
n
is prime.
2
2
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Inequality
Prove that for all positive real numbers
a
1
,
a
2
,
⋯
a
n
a_1,a_2,\cdots a_n
a
1
,
a
2
,
⋯
a
n
1
1
1
+
a
1
+
1
1
+
a
2
+
⋯
+
1
1
+
a
n
−
1
1
a
1
+
1
a
2
+
⋯
+
1
a
n
≥
1
n
\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}
1
+
a
1
1
+
1
+
a
2
1
+
⋯
+
1
+
a
n
1
1
−
a
1
1
+
a
2
1
+
⋯
+
a
n
1
1
≥
n
1
When does the inequality hold?
Geometry
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. On each of the sides
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to
C
D
,
D
A
,
A
B
,
B
C
,
CD,DA,AB,BC,
C
D
,
D
A
,
A
B
,
BC
,
respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
1
2
Hide problems
Geometry . Concurrency
In a triangle
A
B
C
ABC
A
BC
points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively,and
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
are the midpoints of the altitudes from
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively. Show that the lines
A
1
A
2
,
B
1
B
2
,
C
1
,
C
2
A_1A_2,B_1B_2,C_1,C_2
A
1
A
2
,
B
1
B
2
,
C
1
,
C
2
are concurrent.
$(a)$ Is it possible to partition the segment $[0,1]$ into t
(
a
)
(a)
(
a
)
Is it possible to partition the segment
[
0
,
1
]
[0,1]
[
0
,
1
]
into two sets
A
A
A
and
B
B
B
and to define a continuous function
f
f
f
such that for every
x
∈
A
f
(
x
)
x\in A \ f(x)
x
∈
A
f
(
x
)
is in
B
B
B
, and for every
x
∈
B
f
(
x
)
x\in B \ f(x)
x
∈
B
f
(
x
)
is in
A
A
A
?
(
b
)
(b)
(
b
)
The same question with
[
0
,
1
]
[0,1]
[
0
,
1
]
replaced by
[
0
,
1
)
.
[0,1).
[
0
,
1
)
.