RO-NMO 2006

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 $\angle MPC$ .

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).

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) $CH=DE$ .

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

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

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

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 $BC'$ .

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

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

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

• there is only one winner;

• there are $3$ students on the second place;

• no student lost all $4$ matches.

How many students won only a single match and how many won exactly $2$ matches? (In the above conditions)

Let $M$ be a set composed of $n$ elements and let $\mathcal P (M)$ be its power set. Find all functions $f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties

(a) $f(A) \neq 0$ , for $A \neq \phi$ ;

(b) $f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$ , for all $A,B \in \mathcal P (M)$ , where $A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$ .

Prove that for all $a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $n \in \mathbb N^\ast$ we have $$\frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} .$$

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

Let $n \in \mathbb N$ , $n \geq 2$ . Determine $n$ sets $A_i$ , $1 \leq i \leq n$ , from the plane, pairwise disjoint, such that:

(a) for every circle $\mathcal C$ from the plane and for every $i \in \left\{ 1,2,\ldots,n \right\}$ we have $A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$ ;

(b) for all lines $d$ from the plane and every $i \in \left\{ 1,2,\ldots,n \right\}$ , the projection of $A_i$ on $d$ is not $d$ .

We define a pseudo-inverse $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations $$A = ABA \text{ and } B=BAB.$$ a) Prove that any square matrix has at least a pseudo-inverse.

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

Marius Cavachi

Let $\mathcal K$ be a finite field. Prove that the following statements are equivalent:

(a) $1+1=0$ ;

(b) for all $f \in \mathcal K \left[ X \right]$ with $\textrm{deg} f \geq 1$ , $f \left( X^2 \right)$ is reducible.