RO-NMO 2004

On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$ . The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$ . Prove that:

(a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$ ;

(b) $P,A,Q$ are collinear.

Virginia Tica, Vasile Tica

The sidelengths of a triangle are $a,b,c$ .

(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$ .

(b) Prove that $\sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$ .

Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$ , $\left( OE \right.$ is the bisector of $\measuredangle AOD$ , $E \in \left( AD \right)$ and $EF \| AB$ , $F \in \left( BC \right)$ . Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$ . Prove that:

(a) $EP=QF$ ;

(b) $EF=AD$ .

Claudiu-Stefan Popa

Let $\mathcal U = \left\{ \left( x,y \right) | x,y \in \mathbb Z, \ 0 \leq x,y < 4 \right\}$ .

(a) Prove that we can choose $6$ points from $\mathcal U$ such that there are no isosceles triangles with the vertices among the chosen points.

(b) Prove that no matter how we choose $7$ points from $\mathcal U$ , there are always three which form an isosceles triangle.

Radu Gologan, Dinu Serbanescu

Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$ .

Lucian Dragomir

Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$ , where $0 \leq x \leq y \leq z \leq t$ , has exactly $2$ solutions in $\mathbb Z$ .

Mihai Baluna

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.

(a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$ ;

(b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$ , then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$ .

Mircea Fianu

In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube.

Dinu Serbanescu

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$ .

Cristinel Mortici

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$ , $f(x)=a x^2 + b x + c$ , with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$ , for all $n \geq 4$ .

Laurentiu Panaitopol

Let $H$ be the orthocenter of the acute triangle $ABC$ . Let $BB'$ and $CC'$ be altitudes of the triangle ( $B^{\prime} \in AC$ , $C^{\prime} \in AB$ ). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$ . The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$ . Determine the locus of the midpoint of the segment $[ PQ]$ .

Gheorghe Szolosy

Let $p,q \in \mathbb N^{\ast}$ , $p,q \geq 2$ . We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$ , $i = \overline{1,n}$ , not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$ 's has an element at most, $i=\overline{1,p}$ . Prove that:

(a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$ ;

(b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$ ;

(c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$ .

Dan Schwarz

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$ , for all $x,y \in \mathbb{R}$ .

Prove that if for any real $x$ , the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$ , for all $x \in \mathbb R$ .

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles.

Prove that it is regular.

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$ . Prove that: $$2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} .$$

Radu Gologan

Let $\left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $X_n \subset P_n$ and the projection of the set $X_{n+1}$ on the plane $P_n$ is included in the set $X_n$ , for all $n$ .

Prove that there is a sequence of points $\left( p_n \right)_{n \geq 1}$ such that $p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$ , for all $n$ .

Does the conclusion of the problem remain true if the sets $X_n$ are infinite?

Claudiu Raicu

Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$ . A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$ . $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$ .

Prove that $$FB_1 + FB_2 + \ldots + FB_n > np .$$

Calin Popescu

Let $n \in \mathbb N$ , $n \geq 2$ .

(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that $$\textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor .$$

(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have $$\textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor .$$

Ion Savu

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$ .

(a) Prove that $\textrm{Im} f$ is finite or numerable.

(b) Find all continuous functions which have the property mentioned in the hypothesis.

(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$ , i.e. all $x \in \mathbb Q$ are local, strict minimum points.

(b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$ , where $I$ is a non-degenerate interval.

(c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$ , where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$ .

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have $$n^2 \int_{x}^{x + \frac{1}{n}} f(t) dt = n f(x) + \frac12 .$$

Mihai Piticari

Let $f \in \mathbb Z[X]$ . For an $n \in \mathbb N$ , $n \geq 2$ , we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$ , for all $x \in \mathbb Z$ .

(a) Prove that $f_n$ is well defined.

(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$ , $n \geq 2$ , the function $f_n$ is surjective.

Bogdan Enescu

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that $$\int_0^1 f(x) dx = \int_0^1 x f(x) dx = 1 .$$ Prove that $$\int_0^1 f^2 (x) dx \geq 4 .$$ Ion Rasa

Let $\mathcal K$ be a field of characteristic $p$ , $p \equiv 1 \left( \bmod 4 \right)$ .

(a) Prove that $-1$ is the square of an element from $\mathcal K.$

(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$ , of elements from $\mathcal K$ .

(c) Can $0$ be written in the same way?

Marian Andronache