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National and Regional Contests
Romania Contests
JBMO TST - Romania
2011 Junior Balkan Team Selection Tests - Romania
2011 Junior Balkan Team Selection Tests - Romania
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JBMO TST - Romania
Subcontests
(5)
5
1
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n persons, each of them speaking at most 3 languages
Consider
n
n
n
persons, each of them speaking at most
3
3
3
languages. From any
3
3
3
persons there are at least two which speak a common language. i) For
n
≤
8
n \le 8
n
≤
8
, exhibit an example in which no language is spoken by more than two persons. ii) For
n
≥
9
n \ge 9
n
≥
9
, prove that there exists a language which is spoken by at least three persons
1
4
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3
4
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4
4
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2
3
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Problem 5.16 (Star Theorem)
Let
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
be a convex pentagon. Suppose rays
A
2
A
3
A_2A_3
A
2
A
3
and
A
5
A
4
A_5A_4
A
5
A
4
meet at the point
X
1
X_1
X
1
. Define
X
2
X_2
X
2
,
X
3
X_3
X
3
,
X
4
X_4
X
4
,
X
5
X_5
X
5
similarly. Prove that
∏
i
=
1
5
X
i
A
i
+
2
=
∏
i
=
1
5
X
i
A
i
+
3
\displaystyle\prod_{i=1}^{5} X_iA_{i+2} = \displaystyle\prod_{i=1}^{5} X_iA_{i+3}
i
=
1
∏
5
X
i
A
i
+
2
=
i
=
1
∏
5
X
i
A
i
+
3
where the indices are taken modulo 5.
1 nxn square divided into n^2 unit squares, real number in each unti square
We consider an
n
×
n
n \times n
n
×
n
(
n
∈
N
,
n
≥
2
n \in N, n \ge 2
n
∈
N
,
n
≥
2
) square divided into
n
2
n^2
n
2
unit squares. Determine all the values of
k
∈
N
k \in N
k
∈
N
for which we can write a real number in each of the unit squares such that the sum of the
n
2
n^2
n
2
numbers is a positive number, while the sum of the numbers from the unit squares of any
k
×
k
k \times k
k
×
k
square is a negative number.
a^2 + b^2 \in A for every a, b \in A with a \ne b
Find all the finite sets
A
A
A
of real positive numbers having at least two elements, with the property that
a
2
+
b
2
∈
A
a^2 + b^2 \in A
a
2
+
b
2
∈
A
for every
a
,
b
∈
A
a, b \in A
a
,
b
∈
A
with
a
≠
b
a \ne b
a
=
b