Let the acute triangle $ABC$ be inscribed in $(O)$. Let $A_1,B_1,C_1$ be the feet of its altitudes from $A,B,C$, respectively, and let $A_2,B_2,C_2$ be their reflections in the midpoints of $BC,CA,AB$, respectively. Let $A_3,B_3,C_3$ be the intersections, other than the named vertices $A,B,C$, of the circumcircles of $AB_2C_2,BC_2A_2,CA_2B_2$, respectively, with $(O)$. Prove that $A_1A_3,B_1B_3,C_1C_3$ are concurrent.
Editorial clarification: the source says “the intersections” without specifying the second intersections. When two circles are tangent at a named vertex, the second intersection must be understood with multiplicity; without this convention the statement is not defined for all acute triangles.
Let $P(x)=rx^3+qx^2+px+1$, where $p,q,r$ are real and $r\gt 0$. Define $(a_n)$ by
$$a_1=1,\qquad a_2=-p,\qquad a_3=p^2-q,$$
$$a_{n+3}=-pa_{n+2}-qa_{n+1}-ra_n\qquad(n\ge0).$$
Prove that if $P$ has exactly one real root and has no repeated root, then $(a_n)$ has infinitely many negative terms.
Editorial clarification: the source prints $n\ge0$ although $a_0$ is not listed. Its equation at $n=0$ forces $a_0=0$ because $r\gt 0$. Equivalently, use the three displayed initial terms and apply the recurrence for $n\ge1$.
Let $a,b$ be positive integers such that none of $a,b,ab$ is a perfect square. Prove that at least one of the equations
$$ax^2-by^2=1,\qquad ax^2-by^2=-1$$
has no solution in positive integers.
Find all real numbers $r$ such that, for every positive $a,b,c$,
$$\left(r+\frac a{b+c}\right)\left(r+\frac b{c+a}\right)\left(r+\frac c{a+b}\right)\ge\left(r+\frac12\right)^3.$$
Let $(O)$ be a circle with diameter $AB$, and let $M$ be any point strictly inside the circle, not on $AB$. The internal angle-bisector ray at $M$ of triangle $AMB$ meets $(O)$ at $N$. The external angle bisector of $\angle AMB$ meets the lines $NA,NB$ at $P,Q$, respectively. The line $MA$ meets the circle with diameter $NQ$ again at $R$, and the line $MB$ meets the circle with diameter $NP$ again at $S$, where $R,S\ne M$. Prove that the median from $N$ of triangle $NRS$ always passes through a fixed point as $M$ varies inside the circle.
Source defect: for some permitted interior points M, a specified second intersection is tangent at M and cannot satisfy R,S different from M. The intended theorem holds wherever the construction is defined; alternatively allow a repeated intersection at tangency. The solutions give an explicit counterexample to the unrestricted construction and prove the corrected interpretation.
A mathematics conference has $6n+4$ mathematicians and exactly $2n+1$ meetings, where $n\ge1$. At each meeting they sit around one four-seat table and $n$ six-seat tables; the seats are equally spaced around each table. Any two mathematicians who sit next to or opposite each other at one meeting must not sit next to or opposite each other at any other meeting.
(a) Prove that a seating arrangement is possible when $n=1$.
(b) Can a seating arrangement be made for every $n\gt 1$?
Solve the system
$$\begin{cases}\dfrac1{\sqrt{1+2x^2}}+\dfrac1{\sqrt{1+2y^2}}=\dfrac2{\sqrt{1+2xy}},\\ \sqrt{x(1-2x)}+\sqrt{y(1-2y)}=\dfrac29.\end{cases}$$
Let the real sequence $(x_n)$ be defined by
$$x_1=\frac12,\qquad x_n=\frac{\sqrt{x_{n-1}^2+4x_{n-1}}+x_{n-1}}2\quad(n\ge2).$$
For each positive integer $n$, put $y_n=\sum_{i=1}^n1/x_i^2$. Prove that $(y_n)$ has a finite limit as $n\to\infty$ and find that limit.
In the plane, let $A,B$ be fixed distinct points. A point $C$ varies so that $\angle ACB=\alpha$, where $0^\circ\lt \alpha\lt 180^\circ$ is fixed. The incircle of triangle $ABC$, with center $I$, touches $AB,BC,CA$ at $D,E,F$, respectively. Lines $AI,BI$ meet line $EF$ at $M,N$, respectively. Prove that:
1. The length $MN$ is constant.
2. The circumcircle of triangle $DMN$ always passes through a fixed point.
Let real numbers $a,b,c$ satisfy that $a^n+b^n+c^n$ is an integer for every positive integer $n$. Prove that there exist integers $p,q,r$ such that $a,b,c$ are the three roots of
$$x^3+px^2+qx+r=0.$$
Let $n$ be a positive integer and $T=\{1,2,\ldots,2n\}$. How many subsets $S$ of $T$ have the property that there are no $a,b\in S$ with $|a-b|\in\{1,n\}$? The empty set is included.