Školjka

%V0 An $n \times n, n \geq 2$ chessboard is numbered by the numbers $1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $n.$

%V0 Let $n$ be an even positive integer. Let $A_1, A_2, \ldots, A_{n + 1}$ be sets having $n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $\frac {n}{2}$ zeros?

%V0 Let $a$ and $b$ be two positive integers such that $a \cdot b + 1$ divides $a^{2} + b^{2}$. Show that $\frac {a^{2} + b^{2}}{a \cdot b + 1}$ is a perfect square.

%V0 In a right-angled triangle $ABC$ let $AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ABD, ACD$ intersect the sides $AB, AC$ at the points $K,L$ respectively. If $E$ and $E_1$ dnote the areas of triangles $ABC$ and $AKL$ respectively, show that $$\frac {E}{E_1} \geq 2.$$

%V0 Consider 2 concentric circle radii $R$ and $r$ ($R > r$) with centre $O.$ Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large circle; $B,P,C$ are collinear and $BC$ is perpendicular to $AP.$ i.) For which values of $\angle OPA$ is the sum $BC^2 + CA^2 + AB^2$ extremal? ii.) What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies?

%V0 A function $f$ defined on the positive integers (and taking positive integers values) is given by: $\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2n) = f(n) \\ f(4n + 1) = 2f(2n + 1) - f(n) \\ f(4n + 3) = 3f(2n + 1) - 2f(n)\text{,} \end{matrix}$ for all positive integers $n.$ Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n.$