JBMO-SL 2006

For an acute triangle $ABC$ prove the inequality: $\sum_{cyclic} \frac{m_a^2}{-a^2+b^2+c^2}\ge \frac{9}{4}$ where $m_a,m_b,m_c$ are lengths of corresponding medians.

Let $x,y,z$ be positive real numbers such that $x+2y+3z=\frac{11}{12}$ . Prove the inequality $6(3xy+4xz+2yz)+6x+3y+4z+72xyz\le \frac{107}{18}$ .

Let $n\ge 3$ be a natural number. A set of real numbers $\{x_1,x_2,\ldots,x_n\}$ is called summable if $\sum_{i=1}^n \frac{1}{x_i}=1$ . Prove that for every $n\ge 3$ there always exists a summable set which consists of $n$ elements such that the biggest element is: a) bigger than $2^{2n-2}$ b) smaller than $n^2$

Determine the biggest possible value of $m$ for which the equation $2005x + 2007y = m$ has unique solution in natural numbers.

Determine all pairs $(m,n)$ of natural numbers for which $m^2=nk+2$ where $k=\overline{n1}$ .

EDIT. It has been discovered the correct statement is with $k=\overline{1n}$ .

Prove that for every composite number $n>4$ , numbers $kn$ divides $(n-1)!$ for every integer $k$ such that $1\le k\le \lfloor \sqrt{n-1} \rfloor$ .

Determine all numbers $\overline{abcd}$ such that $\overline{abcd}=11(a+b+c+d)^2$ .

Prove that there do not exist natural numbers $n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

Let $ABCD$ be a trapezoid with $AB\parallel CD,AB>CD$ and $\angle{A} + \angle{B} = 90^\circ$ . Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.

Let $ABCD$ be a trapezoid inscribed in a circle $\mathcal{C}$ with $AB\parallel CD$ , $AB=2CD$ . Let $\{Q\}=AD\cap BC$ and let $P$ be the intersection of tangents to $\mathcal{C}$ at $B$ and $D$ . Calculate the area of the quadrilateral $ABPQ$ in terms of the area of the triangle $PDQ$ .

Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect at $A$ and $B$ . Let $M\in AB$ . A line through $M$ (different from $AB$ ) cuts circles $\mathcal{C}_1$ and $\mathcal{C}_2$ at $Z,D,E,C$ respectively such that $D,E\in ZC$ . Perpendiculars at $B$ to the lines $EB,ZB$ and $AD$ respectively cut circle $\mathcal{C}_2$ in $F,K$ and $N$ . Prove that $KF=NC$ .

Let $ABC$ be an equilateral triangle of center $O$ , and $M\in BC$ . Let $K,L$ be projections of $M$ onto the sides $AB$ and $AC$ respectively. Prove that line $OM$ passes through the midpoint of the segment $KL$ .

Let $A$ be a subset of the set $\{1, 2,\ldots,2006\}$ , consisting of $1004$ elements. Prove that there exist $3$ distinct numbers $a,b,c\in A$ such that $gcd(a,b)$ : a) divides $c$ b) doesn't divide $c$

Let $n\ge 5$ be a positive integer. Prove that the set $\{1,2,\ldots,n\}$ can be partitioned into two non-zero subsets $S_n$ and $P_n$ such that the sum of elements in $S_n$ is equal to the product of elements in $P_n$ .

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$ . The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$ , and argue the existence of such a triangle.