BALKAN-SL 2014

$\boxed{\text{A1}}$ Let $a,b,c$ be positive reals numbers such that $a+b+c=1$ .Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

$\boxed{A3}$ The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$ Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

$\boxed{A4}$ Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that $$(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3$$ Prove that $$(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3$$

$\boxed{A5}$ Let $n\in{N},n>2$ ,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$ Prove that $$(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3$$

$\boxed{A6}$ The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$ Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$

$\boxed{A7}$ Prove that for all $x,y,z>0$ with $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ and $0\leq a,b,c<1$ the following inequality holds $$\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}$$

The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?

Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$ , respectively such that $\angle BAM = \angle MNC$ . Suppose that the lines $MN$ and $AB$ intersects at $P$ . Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$

Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$ ,from $B$ to $FD$ and from $C$ to $DE$ are concurrent.Prove that if $ABC$ is perpendicular to $DEF$ ,then $DEF$ is perpendicular to $ABC$

Let $\triangle ABC$ be an isosceles. $(AB=AC)$ .Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$ , $E\in DC$ and $2\angle DAE = \angle BAC$ .Prove that we can construct a triangle $XYZ$ such that $XY=BD$ , $YZ=DE$ and $ZX=EC$ .Find $\angle BAC + \angle YXZ$ .

Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$ . We consider the excenters $A_1$ , $B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$ ,of the triangle $A_1B_1C_1$ . By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.

Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$ . A circle with center $B$ and radius $BE$ ,where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$ . If the line ,perpendicular to $BD$ at $E$ ,intersects $CD$ at $M$ ,prove that $KM\perp DL$ .

In $\triangle ABC$ with $AB=AC$ , $M$ is the midpoint of $BC$ , $H$ is the projection of $M$ onto $AB$ and $D$ is arbitrary point on the side $AC$ .Let $E$ be the intersection point of the parallel line through $B$ to $HD$ with the parallel line through $C$ to $AB$ .Prove that $DM$ is the bisector of $\angle ADE$ .

Let $I$ be the incenter of $\triangle ABC$ and let $H_a$ , $H_b$ , and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$ , and $\triangle AIB$ , respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$ . If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$ , prove that the line $H_aT$ is perpendicular to $BC$

$\boxed{N1}$ Let $n$ be a positive integer, $g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ of the form $6k-1,$ where $k$ is a nonnegative integer.Find all positive integers $n$ such that $g(n)$ and $h(n)$ have different parity.

$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if $$x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}$$ for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$

$\boxed{N3}$ Prove that there exist infinitely many non isosceles triangles with rational side lengths $,$ rational lentghs of altitudes and $,$ perimeter equal to $3.$

$\boxed{N5}$ Let $a,b,c,p,q,r$ be positive integers such that $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.$ Prove that $a=b=c$ or $p=q=r.$