Difference between revisions of "2006 Romanian NMO Problems"
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===Problem 2=== | ===Problem 2=== | ||
− | Let <math>\displaystyle ABC</math> and <math>\displaystyle DBC</math> be isosceles triangle with the base <math>\displaystyle BC</math>. We know that <math>\displaystyle \ | + | Let <math>\displaystyle ABC</math> and <math>\displaystyle DBC</math> be isosceles triangle with the base <math>\displaystyle BC</math>. We know that <math>\displaystyle \angle ABD = \frac{\pi}{2}</math>. Let <math>\displaystyle M</math> be the midpoint of <math>\displaystyle BC</math>. The points <math>\displaystyle E,F,P</math> are chosen such that <math>\displaystyle E \in (AB)</math>, <math>\displaystyle P \in (MC)</math>, <math>\displaystyle C \in (AF)</math>, and <math>\displaystyle \angle BDE = \angle ADP = \angle CDF</math>. Prove that <math>\displaystyle P</math> is the midpoint of <math>\displaystyle EF</math> and <math>\displaystyle DP \perp EF</math>. |
+ | |||
===Problem 3=== | ===Problem 3=== | ||
We have a quadrilateral <math>ABCD</math> inscribed in a circle of radius <math>r</math>, for which there is a point <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>. | We have a quadrilateral <math>ABCD</math> inscribed in a circle of radius <math>r</math>, for which there is a point <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>. |
Revision as of 10:47, 27 July 2006
Contents
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
![$n = \sum_{1\leq i < j \leq k } a_ia_j$](http://latex.artofproblemsolving.com/c/3/f/c3f8748bf68501f3212d2a2fabd83d374f793ea2.png)
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
![$2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3$](http://latex.artofproblemsolving.com/9/f/3/9f3b9cb0473c32de0b51a50dcdee707eb7fc00fd.png)
selected by Mircea Lascu
Grade 9
Problem 1
Find the maximal value of
![$\left( x^3+1 \right) \left( y^3 + 1\right)$](http://latex.artofproblemsolving.com/5/c/9/5c9771df93a5ab9e4e2c14fbf59265a888c74397.png)
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)