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Balkan 1985

This document contains 4 problems from the 2nd Balkan Mathematical Olympiad held in 1985 in Sofia, Bulgaria. Problem 1 asks to prove that lines CD and OE are orthogonal if and only if AB=AC for a given triangle configuration. Problem 2 asks to prove that a certain expression is equal to 0 given relationships between sin terms. Problem 3 asks to find a point A on an axis E such that points symmetrically placed around A have different colors. Problem 4 asks to prove that with 1985 persons in groups of 3 where 2 speak a common language, there must be at least 200 persons speaking a common language.

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Mauricio Mallma
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116 views1 page

Balkan 1985

This document contains 4 problems from the 2nd Balkan Mathematical Olympiad held in 1985 in Sofia, Bulgaria. Problem 1 asks to prove that lines CD and OE are orthogonal if and only if AB=AC for a given triangle configuration. Problem 2 asks to prove that a certain expression is equal to 0 given relationships between sin terms. Problem 3 asks to find a point A on an axis E such that points symmetrically placed around A have different colors. Problem 4 asks to prove that with 1985 persons in groups of 3 where 2 speak a common language, there must be at least 200 persons speaking a common language.

Uploaded by

Mauricio Mallma
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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nd

The 2 Balkan Mathematical Olympiad

1985, Sofia, Bulgaria

Problem 1. Let O be the circumcenter of a triangle ABC, D be the midpoint of the side AB
and E be the centroid of the triangle ACD. Prove that the lines CD and OE are orthogonal if and
only if AB=AC.

Problem 2. Let a, b, c, d be real numbers from the interval so that

sin a + sin b + sin c + sin d =1

and

Prove that .

Problem 3. On an axis E (axis is a oriented straight line with a unit of measurement) the set

S of all the points that have the coordinates of form 19a + 85b, where , is considered.
The points from S are coloured in red and the other points of E with integer coordinates are
coloured in green. Find out if there is a point A on E (not necessary with integer coordinates) so
that every two points B and C with integer coordinates, which are situated on E symmetrically
with respect to A, have different colours.

Problem 4. 1985 persons take part at an international meeting. In every group of three
persons there are at least two persons who speak the same language. Knowing that every
person speaks at most five languages, prove that there are at least 200 persons who speak the
same language.

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