2006 Romanian NMO Problems
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
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