KG-MO 2011

For a given chord $MN$ of a circle discussed the triangle $ABC$ , whose base is the diameter $AB$ of this circle,which do not intersect the $MN$ , and the sides $AC$ and $BC$ pass through the ends of $M$ and $N$ of the chord $MN$ . Prove that the heights of all such triangles $ABC$ drawn from the vertex $C$ to the side $AB$ , intersect at one point.

In a convex $n$ -gon all angles are equal from a certain point, located inside the $n$ -gon, all its sides are seen under equal angles. Can we conclude that this $n$ -gon is regular?

Given equation ${a^5} - {a^3} + a = 2$ , with real $a$ . Prove that $3 < {a^6} < 4$ .

Points $M$ and $N$ are chosen on sides $AB$ and $BC$ ,respectively, in a triangle $ABC$ , such that point $O$ is interserction of lines $CM$ and $AN$ . Given that $AM+AN=CM+CN$ . Prove that $AO+AB=CO+CB$ .

a) Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times.b) Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$ , for natural $n$ . Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$ .

Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$ , where $(n=1,2,3,...)$ . What must be value of $x_1$ , so that $x_{100}$ and $x_{1000}$ becomes equal?