2006 Romanian NMO Problems

7th Grade

Problem 1

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$ , such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$ . Let $P$ be a point on the line $AC$ . Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$ .

Solution

Problem 2

A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$ , 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).

Solution

Problem 3

In the acute-angle triangle $ABC$ we have $\angle ACB = 45^\circ$ . The points $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ , and $H$ is the orthocenter of the triangle. We consider the points $D$ and $E$ on the segments $AA_1$ and $BC$ such that $A_1D = A_1E = A_1B_1$ . Prove that

a) $A_1B_1 = \sqrt{ \frac{A_1B^2+A_1C^2}{2} }$ ;

b) $CH=DE$ .

Solution

Problem 4

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$ , $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$ , where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$ . Prove that the set $A$ has exactly two elements.

Marius Gherghu, Slatina

Solution

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.

Solution

Problem 2

Let $n$ be a positive integer. Prove that there exists an integer $k$ , $k\geq 2$ , and numbers $a_i \in \{ -1, 1 \}$ , such that

$n = \sum_{1\leq i < j \leq k } a_ia_j$ .

Solution

Problem 3

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$ . The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$ . Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.

a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$ .

b) Find the minimal value of the angle between the lines $MN$ and $BC'$ .

Solution

Problem 4

Let $a,b,c \in \left[ \frac 12, 1 \right]$ . Prove that

$2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3$ .

selected by Mircea Lascu

Solution

Grade 9

Problem 1

Find the maximal value of

$\left( x^3+1 \right) \left( y^3 + 1\right)$ ,

where $x,y \in \mathbb R$ , $x+y=1$ .

Dan Schwarz

Solution

Problem 2

Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$ . We know that $\displaystyle \angle ABD = \frac{\pi}{2}$ . Let $\displaystyle M$ be the midpoint of $\displaystyle BC$ . The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$ , $\displaystyle P \in (MC)$ , $\displaystyle C \in (AF)$ , and $\displaystyle \angle BDE = \angle ADP = \angle CDF$ . Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$ .

Solution

Problem 3

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$ , for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$ .

(a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.

(b) Prove that $PD=r$ .

Virgil Nicula

Solution

Problem 4

$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ 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: