(a) Determine the minimal value of

$$\left(x+\frac{1}{y}\right)\left(x+\frac{1}{y}-2018\right)+\left(y+\frac{1}{x}\right)\left(y+\frac{1}{x}-2018\right)$$

where $x$ and $y$ vary over the positive reals. (b) Determine the minimal value of

$$\left(x+\frac{1}{y}\right)\left(x+\frac{1}{y}+2018\right)+\left(y+\frac{1}{x}\right)\left(y+\frac{1}{x}+2018\right)$$

where $x$ and $y$ vary over the positive reals. (Pierre Haas, Luxembourg)

In the land of Heptanomisma, four different coins and three different banknotes are used, and their denominations are seven different (non-zero) natural numbers. The denomination of the smallest banknote is greater than the sum of the denominations of the four different coins. A tourist has exactly one coin of each denomination and exactly one banknote of each denomination, but he cannot afford the book on numismatics he wishes to buy. However, the mathematically inclined shopkeeper offers to sell the book to the tourist at a price of his choosing, provided that he can pay this price in more than one way. (The tourist can pay a price in more than one way if there are two different subsets of his coins and notes, the denominations of which both add up to this price.) (a) Prove that the tourist can purchase the book if the denomination of each banknote is smaller than 49. (b) Show that the tourist may have to leave the shop empty-handed if the denomination of the largest banknote is 49. (Stijn Cambie, Belgium)

Let $A B C$ be a triangle with orthocentre $H$, and let $D, E$, and $F$ denote the respective midpoints of line segments $A B, A C$, and $A H$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively. (a) Show that lines $P E$ and $Q D$ intersect on the circumcircle of triangle $A B C$. (b) Prove that lines $P D$ and $Q E$ intersect on line segment $A H$. (Merlijn Staps, the Netherlands)

An integer $n \geqslant 2$ having exactly $s$ positive divisors $1=d_{1}<d_{2}<\cdots<d_{s}=n$ is said to be good if there exists an integer $k$, with $2 \leqslant k \leqslant s$, such that $d_{k}>1+d_{1}+\cdots+d_{k-1}$. An integer $n \geqslant 2$ is said to be bad if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than 2 , there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers. (Gerhard Woeginger, Luxembourg)