Difference between revisions of "2006 Romanian NMO Problems"
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<math>\displaystyle 2n</math> students <math>\displaystyle (n \geq 5)</math> participated at table tennis contest, which took <math>\displaystyle 4</math> days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: | <math>\displaystyle 2n</math> students <math>\displaystyle (n \geq 5)</math> participated at table tennis contest, which took <math>\displaystyle 4</math> days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: | ||
− | + | * there is only one winner; | |
− | + | * there are <math>\displaystyle 3</math> students on the second place; | |
− | + | * no student lost all <math>\displaystyle 4</math> matches. | |
How many students won only a single match and how many won exactly <math>\displaystyle 2</math> matches? (In the above conditions) | How many students won only a single match and how many won exactly <math>\displaystyle 2</math> matches? (In the above conditions) | ||
[[2006 Romanian NMO Problems/Grade 7/Problem 4 | Solution]] | [[2006 Romanian NMO Problems/Grade 7/Problem 4 | Solution]] |
Revision as of 09:50, 27 July 2006
Contents
[hide]7th Grade
Problem 1
Let be a triangle and the points
and
on the sides
respectively
, such that
. Let
be a point on the line
. Prove that the lines
and
are perpendicular if and only if
is the interior angle bisector of
.
Problem 2
A square of side is formed from
unit squares, each colored in red, yellow or green. Find minimal
, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
Problem 3
In the acute-angle triangle we have
. The points
and
are the feet of the altitudes from
and
, and
is the orthocenter of the triangle. We consider the points
and
on the segments
and
such that
. Prove that
a) ;
b) .
Problem 4
Let be a set of positive integers with at least 2 elements. It is given that for any numbers
,
we have
, where by
we have denoted the least common multiple of
and
. Prove that the set
has exactly two elements.
Marius Gherghu, Slatina
8th Grade
Problem 1
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
Problem 2
Let be a positive integer. Prove that there exists an integer
,
, and numbers
, such that

Problem 3
Let be a cube and
a variable point on the side
. The perpendicular plane on
which passes through
intersects the line
in
. Let
and
be the midpoints of the segments
and
respectively.
a) Prove that the lines and
are perpendicular if and only if
is the midpoint of
.
b) Find the minimal value of the angle between the lines and
.
Problem 4
Let . Prove that

selected by Mircea Lascu
Grade 9
Problem 1
Find the maximal value of

where ,
.
Dan Schwarz
Problem 2
Let and
be isosceles triangle with the base
. We know that
. Let
be the midpoint of
. The points
are chosen such that
,
,
, and
. Prove that
is the midpoint of
and
.
Problem 3
We have a quadrilateral inscribed in a circle of radius
, for which there is a point
on
such that
.
(a) Prove that there are points which fulfill the above conditions.
(b) Prove that .
Virgil Nicula
Problem 4
students
participated at table tennis contest, which took
days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are
students on the second place;
- no student lost all
matches.
How many students won only a single match and how many won exactly matches? (In the above conditions)