Let $n\gt 2$ be an integer and $a_1\lt a_2\lt \cdots\lt a_n$ positive integers. Let $X\subseteq\{1,\ldots,n\}$ minimize $\left|\sum_{i\notin X}a_i-\sum_{i\in X}a_i\right|$. Prove that there exist positive integers $b_1\lt b_2\lt \cdots\lt b_n$ such that $\sum_{i\notin X}b_i=\sum_{i\in X}b_i$.
Let $ABC$ be an acute scalene triangle inscribed in a circle with centre $O$. Its incircle, with centre $I$, touches $BC,CA,AB$ at $D,E,F$, respectively. Let $K,M,N$ be the midpoints of $BC,CA,AB$, respectively.
(a) Prove that the lines through $D,E,F$ parallel to $IK,IM,IN$, respectively, are concurrent.
(b) Let $T,P,Q$ be the midpoints of the major arcs $BC,CA,AB$, respectively, of the circumcircle. Prove that the lines through $D,E,F$ parallel to $IT,IP,IQ$, respectively, are concurrent.
A football tournament has $4n$ teams, where $n$ is a positive integer. Each round divides all teams into $2n$ pairs, which play against each other. No two teams may play more than once. Find the least positive integer $a$ for which it is possible to arrange $a$ rounds so that every subsequent partition of the teams into $2n$ pairs includes at least one pair that has already played.
Every cell of a $(2n+1)\times(2n+1)$ board is black or white, where $n$ is a positive integer. In each row and column, mark all cells of the less frequent color. Let $a,b$ be the total numbers of black and white cells, and let $c$ be the number of cells marked at least once. Prove $c\ge\min(a,b)/2$.
Let $ABC$ be an acute scalene triangle inscribed in a circle with centre $O$. The altitudes $AD,BE,CF$ meet at the orthocentre $H$. Let $G$ be the reflection of $O$ in $BC$. Draw diameters $EK,FL$ of the circles $(GHE),(GHF)$, respectively.
(a) Suppose $AK,AL$ meet $DE,DF$ at $U,V$, respectively. Prove that $UV\parallel EF$.
(b) The tangents to $(O)$ at $B,C$ meet at $S$. Let $T=DS\cap HG$. Let $M,N$ be the perpendicular projections of $H$ onto the lines $TE,TF$, respectively. Prove that $M,N,E,F$ are concyclic.
Editorial caveat: the source omits a necessary construction condition in (b). Even for an acute scalene triangle, $DS$ and $HG$ can be parallel, so $T$ need not exist. The conclusion below is proved when this intersection is finite; a fully checked counterexample to its automatic existence is given after the proof. The source statement has not been silently amended.