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.
Define $x_1=1$ and
$$x_{n+1}=x_n+3\sqrt{x_n}+\frac n{\sqrt{x_n}}\quad(n\ge1).$$
a) Prove $\lim_{n\to\infty}n/x_n=0$.
b) Find $\lim_{n\to\infty}n^2/x_n$.
a) Let real numbers $a,b,c$ satisfy $a^2+b^2+c^2=1$. Prove
$$|a-b|+|b-c|+|c-a|\le2\sqrt2.$$
b) Let real numbers $a_1,\ldots,a_{2019}$ satisfy $\sum_{i=1}^{2019}a_i^2=1$. Find the largest value of
$$S=|a_1-a_2|+|a_2-a_3|+\cdots+|a_{2018}-a_{2019}|+|a_{2019}-a_1|.$$
Define $a_1=5$, $a_2=13$, and $a_{n+1}=5a_n-6a_{n-1}$ for $n\ge2$.
a) Prove that any two consecutive terms are relatively prime.
b) Prove that if $p$ is a prime divisor of $a_{2^k}$, then $2^{k+1}$ divides $p-1$, for every natural number $k$.
Let $ABC$ be an acute scalene triangle inscribed in circle $(O)$, with orthocenter $H$. Let $D,E,F$ be the reflections of $O$ across $BC,CA,AB$, respectively.
a) Let $H_a$ be the reflection of $H$ across $BC$, let $A'$ be the reflection of $A$ across $O$, and let $O_a$ be the circumcenter of $BOC$. Prove that $H_aD$ and $O_aA'$ meet on $(O)$.
b) Choose $X$ so that $AXDA'$ is a parallelogram. Prove that the circumcircles of $AHX$, $ABF$, and $ACE$ have a second common point different from $A$.
Editorial note: the printed assertion in (b) has an allowed exceptional configuration; the solution explains it without changing the source statement.
Consider the real system with parameter $a$:
$$\begin{cases}x-ay=yz,\\ y-az=zx,\\ z-ax=xy.\end{cases}$$
a) Solve the system when $a=0$.
b) Prove that the system has five distinct solutions for every real $a\gt 1$.
Let $ABC$ be an acute scalene triangle. Let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. The circle with diameter $AD$ meets $DE,DF$ again at $M,N$, respectively. Let $P\in AB$ and $Q\in AC$ satisfy $NP\perp AB$, $MQ\perp AC$. Let $(I)$ be the circumcircle of $APQ$.
a) Prove that $(I)$ is tangent to $EF$.
b) Let $T$ be the tangency point, and let $DT$ meet $MN$ at $K$. Let $L$ be the reflection of $A$ across $MN$. Prove that the circumcircle of $DKL$ passes through $EF\cap MN$.
Editorial note: the circle in (b) is undefined for an allowed degenerate configuration; the solution explains this source issue.
Let $n\gt 1$ be an integer. Let $T$ be the set of all ordered triples $(x,y,z)$ of pairwise distinct positive integers with $1\le x,y,z\le2n$. A set $\mathcal A$ of ordered pairs $(u,v)$ is called linked to $T$ if, for every $(x,y,z)\in T$,
$$\{(x,y),(y,z),(z,x)\}\cap\mathcal A\ne\varnothing.$$
a) Find $|T|$.
b) Prove that there exists a set linked to $T$ with exactly $2n(n-1)$ elements.
c) Prove that every set linked to $T$ has at least $2n(n-1)$ elements.