XK-TST 2015

a)Prove that for every n,natural number exist natural numbers a and b such that $(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$ b)Using first equation prove that for every n exist m such that $(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$

Prove that circle l(0,2) with equation $x^2+y^2=4$ contains infinite points with rational coordinates

It's given system of equations $a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$ $a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$ .......... $a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$ such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it

Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$ . If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$ , with vertices from the set $S$ .

In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular. a)Prove that midpoints M,N,P,Q of AD,AB,BC,CD form a rectangular b)If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO