# 2006 Romanian NMO Problems

## 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)

## 10th Grade

### Problem 1

Let be a set composed of elements and let be its power set. Find all functions that have the properties

(a) , for ;

(b) , for all , where .

### Problem 2

Prove that for all and we have

### Problem 3

Prove that among the elements of the sequence are an infinity of even numbers and an infinity of odd numbers.

### Problem 4

Let , . Determine sets , , from the plane, pairwise disjoint, such that:

(a) for every circle from the plane and for every we have ;

(b) for all lines from the plane and every , the projection of on is not .

## 11th Grade

### Problem 1

Let be a matrix with complex elements and let be the classical adjoint of . Prove that if there exists a positive integer such that then .

*Marian Ionescu, Pitesti*

### Problem 2

We define a *pseudo-inverse* of a matrix a matrix which fulfills the relations
a) Prove that any square matrix has at least a pseudo-inverse.

b) For which matrix is the pseudo-inverse unique?

*Marius Cavachi*

### Problem 3

We have in the plane the system of points and , which have different centers of mass. Prove that there is a point such that

### Problem 4

Let be a function such that for any the sequence is increasing.

a) If the function is also continuous on is it true that is increasing?

b) The same question if the function is continuous on .

## 12th Grade

### Problem 1

Let be a finite field. Prove that the following statements are equivalent:

(a) ;

(b) for all with , is reducible.

### Problem 2

Prove that where if and .

*Dorin Andrica, Mihai Piticari*

### Problem 3

Let be a finite group of elements and be the smallest prime factor of . If has only a subgroup with elements, then prove that is in the center of .

*Note.* The center of is the set .

### Problem 4

Let be a continuous function such that Prove that there is such that

*Cezar Lupu, Tudorel Lupu*