RO-NMO 2008

Let $ABC$ be an acute angled triangle with $\angle B > \angle C$ . Let $D$ be the foot of the altitude from $A$ on $BC$ , and let $E$ be the foot of the perpendicular from $D$ on $AC$ . Let $F$ be a point on the segment $(DE)$ . Show that the lines $AF$ and $BF$ are perpendicular if and only if $EF\cdot DC = BD \cdot DE$ .

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

Let $p,q,r$ be 3 prime numbers such that $5\leq p <q<r$ . Knowing that $2p^2-r^2 \geq 49$ and $2q^2-r^2\leq 193$ , find $p,q,r$ .

Let $ABCD$ be a rectangle with center $O$ , $AB\neq BC$ . The perpendicular from $O$ to $BD$ cuts the lines $AB$ and $BC$ in $E$ and $F$ respectively. Let $M,N$ be the midpoints of the segments $CD,AD$ respectively. Prove that $FM \perp EN$ .

A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.

a) We call admissible sequence a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences.

b) For each integer $n\geq 2$ we denote $d_n$ the number of possibilities of completing with even digits an array with $n$ rows and 4 columns, such that

(1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array.

Find the values of $n$ for which the number $\frac {d_{n+1}}{d_n}$ is an integer.

Let $a,b \in [0,1]$ . Prove that $$\frac 1{1+a+b} \leq 1 - \frac {a+b}2 + \frac {ab}3.$$

Let $ABCDA'B'C'D'$ be a cube. On the sides $(A'D')$ , $(A'B')$ and $(A'A)$ we consider the points $M_1$ , $N_1$ and $P_1$ respectively. On the sides $(CB)$ , $(CD)$ and $(CC')$ we consider the points $M_2$ , $N_2$ and $P_2$ respectively. Let $d_1$ be the distance between the lines $M_1N_1$ and $M_2N_2$ , $d_2$ be the distance between the lines $N_1P_1$ and $N_2P_2$ , and $d_3$ be the distance between the lines $P_1M_1$ and $P_2M_2$ . Suppose that the distances $d_1$ , $d_2$ and $d_3$ are pairwise distinct. Prove that the lines $M_1M_2$ , $N_1N_2$ and $P_1P_2$ are concurrent.

Let $ABC$ be a triangle and the points $D\in (BC)$ , $E\in (CA)$ , $F\in (AB)$ such that $$\frac {BD}{DC} = \frac {CE}{EA} = \frac {AF}{FB}.$$ Prove that if the circumcenters of the triangles $DEF$ and $ABC$ coincide then $ABC$ is equilateral.

Let $a,b,c$ be 3 complex numbers such that $$a|bc| + b|ca| + c|ab| = 0.$$ Prove that $$|(a-b)(b-c)(c-a)| \geq 3\sqrt 3 |abc|.$$

Let $A=\{1,2,\ldots, 2008\}$ . We will say that set $X$ is an $r$ -set if $\emptyset \neq X \subset A$ , and $\sum_{x\in X} x \equiv r \pmod 3$ . Let $X_r$ , $r\in\{0,1,2\}$ be the set of $r$ -sets.

Find which one of $X_r$ has the most elements.

We consider the proposition $p(n)$ : $n^2+1$ divides $n!$ , for positive integers $n$ . Prove that there are infinite values of $n$ for which $p(n)$ is true, and infinite values of $n$ for which $p(n)$ is false.

Let $f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $\{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $x$ . Prove that $f$ is nondecreasing.

Let $A$ be a $n\times n$ matrix with complex elements. Prove that $A^{-1} = \overline{A}$ if and only if there exists an invertible matrix $B$ with complex elements such that $A= B^{-1} \cdot \overline{B}$ .

Let $f: \mathbb R \to \mathbb R$ be a function, two times derivable on $\mathbb R$ for which there exist $c\in\mathbb R$ such that $$\frac { f(b)-f(a) }{b-a} \neq f'(c) ,$$ for all $a\neq b \in \mathbb R$ .

Prove that $f''(c)=0$ .

Let $A=(a_{ij})_{1\leq i,j\leq n}$ be a real $n\times n$ matrix, such that $a_{ij} + a_{ji} = 0$ , for all $i,j$ . Prove that for all non-negative real numbers $x,y$ we have $$\det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.$$

Let $a>0$ and $f: [0,\infty) \to [0,a]$ be a continuous function on $(0,\infty)$ and having Darboux property on $[0,\infty)$ . Prove that if $f(0)=0$ and for all nonnegative $x$ we have $$xf(x) \geq \int^x_0 f(t) dt ,$$ then $f$ admits primitives on $[0,\infty)$ .

Let $f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $f'$ on $[0,1]$ . Prove that if $f\left( \frac 12\right) = 0$ , then $$\int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.$$

Let $A$ be a unitary finite ring with $n$ elements, such that the equation $x^n=1$ has a unique solution in $A$ , $x=1$ . Prove that

a) $0$ is the only nilpotent element of $A$ ;

b) there exists an integer $k\geq 2$ , such that the equation $x^k=x$ has $n$ solutions in $A$ .

Let $\mathcal G$ be the set of all finite groups with at least two elements.

a) Prove that if $G\in \mathcal G$ , then the number of morphisms $f: G\to G$ is at most $\sqrt [p]{n^n}$ , where $p$ is the largest prime divisor of $n$ , and $n$ is the number of elements in $G$ .

b) Find all the groups in $\mathcal G$ for which the inequality at point a) is an equality.