Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram. Prove the lines $AC$, $KH$, $BD$ are concurrent. (I. Nagel)

Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two). Exhibit (at least) two values, one even and one odd, for such numbers $n>8$. (Pál Erdös & Kurt Mahler)

For a given integer $n\geq 3$, determine the range of values for the expression \[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\] over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$. Do also determine when the extremal values are achieved. (Dan Schwarz)

Given $n$ sets $A_i$, with $| A_i | = n$, prove they may be indexed $A_i = \{a_{i,j} \mid j=1,2,\ldots,n \}$, in such way that the sets $B_j = \{a_{i,j} \mid i=1,2,\ldots,n \}$, $1\leq j\leq n$, also have $| B_j | = n$. (Anonymous)

For positive real numbers $a,b,c,d$, with $abcd = 1$, determine all values taken by the expression \[\frac {1+a+ab} {1+a+ab+abc} + \frac {1+b+bc} {1+b+bc+bcd} +\frac {1+c+cd} {1+c+cd+cda} +\frac {1+d+da} {1+d+da+dab}.\] (Dan Schwarz)

Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$. (U.S.A. - 1989 IMO Shortlist)

The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares). Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color. (Alexander Magazinov)

Let $n\geq 2$ be an integer. Let us call interval a subset $A \subseteq \{1,2,\ldots,n\}$ for which integers $1\leq a < b\leq n$ do exist, such that $A = \{a,a+1,\ldots,b-1,b\}$. Let a family $\mathcal{A}$ of subsets $A_i \subseteq \{1,2,\ldots,n\}$, with $1\leq i \leq N$, be such that for any $1\leq i < j \leq N$ we have $A_i \cap A_j$ being an interval. Prove that $\displaystyle N \leq \left \lfloor n^2/4 \right \rfloor$, and that this bound is sharp. (Dan Schwarz - after an idea by Ron Graham)