Difference between revisions of "2006 Romanian NMO Problems"
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==8th Grade== | ==8th Grade== | ||
===Problem 1=== | ===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. | + | We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals. |
+ | |||
+ | [[2006 Romanian NMO Problems/Grade 8/Problem 1|Solution]] | ||
===Problem 2=== | ===Problem 2=== | ||
Let <math>n</math> be a positive integer. Prove that there exists an integer <math>k</math>, <math>k\geq 2</math>, and numbers <math>a_i \in \{ -1, 1 \}</math>, such that <center><math>n = \sum_{1\leq i < j \leq k } a_ia_j</math>.</center> | Let <math>n</math> be a positive integer. Prove that there exists an integer <math>k</math>, <math>k\geq 2</math>, and numbers <math>a_i \in \{ -1, 1 \}</math>, such that <center><math>n = \sum_{1\leq i < j \leq k } a_ia_j</math>.</center> | ||
+ | |||
+ | [[2006 Romanian NMO Problems/Grade 8/Problem 2|Solution]] | ||
+ | ===Problem 3=== | ||
+ | Let <math>ABCDA_1B_1C_1D_1</math> be a cube and <math>P</math> a variable point on the side <math>[AB]</math>. The perpendicular plane on <math>AB</math> which passes through <math>P</math> intersects the line <math>AC'</math> in <math>Q</math>. Let <math>M</math> and <math>N</math> be the midpoints of the segments <math>A'P</math> and <math>BQ</math> respectively. | ||
+ | |||
+ | [[2006 Romanian NMO Problems/Grade 8/Problem 3|Solution]] | ||
+ | a) Prove that the lines <math>MN</math> and <math>BC'</math> are perpendicular if and only if <math>P</math> is the midpoint of <math>AB</math>. | ||
+ | |||
+ | b) Find the minimal value of the angle between the lines <math>MN</math> and <math>BC'</math>. | ||
+ | ===Problem 4=== | ||
+ | Let <math>a,b,c \in \left[ \frac 12, 1 \right]</math>. Prove that <center><math>2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3</math>.</center> | ||
+ | |||
+ | ''selected by Mircea Lascu'' | ||
+ | |||
+ | [[2006 Romanian NMO Problems/Grade 8/Problem 4|Solution]] |
Revision as of 09:32, 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.
Solution
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