A country has $n\ge2$ cities. Every pair is joined by a direct two-way air route. Licences for these routes are to be given to airlines subject to:
(i) Each route is licensed to exactly one airline.
(ii) Using routes of any one airline, one can travel from any city to every other city.
What is the greatest possible number of airlines?
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$ and midpoint $M$ of $BC$. Let $D$ lie on the ray opposite to $HA$ and satisfy $DM=BC/2$. Let $D'$ be the reflection of $D$ in $BC$, and let $X=AO\cap MD$.
(a) Prove that $AM$ passes through the midpoint of $D'X$.
(b) Define $E,F$ cyclically in the same way as $D$, and $Y,Z$ cyclically in the same way as $X$. Let $S$ be the intersection of the tangents to $(O)$ at $B,C$. Let $G$ be the perpendicular projection onto $AO$ of the midpoint of $AS$. Prove that a point has equal powers with respect to all four circles $(SGO),(BYE),(CFZ),(O)$.
Let a scalene triangle $ABC$ have circumcircle $(O)$ and incircle $(I)$. Let $E=BI\cap AC$ and $F=CI\cap AB$. The circle through $E$ tangent to the line $OB$ at $B$ meets $(O)$ again at $M$; the circle through $F$ tangent to $OC$ at $C$ meets $(O)$ again at $N$. Let $P,Q$ be the second intersections of $ME,NF$, respectively, with $(O)$. Put $K=EF\cap BC$, $G=PQ\cap BC$, and $H=PQ\cap EF$. Prove that the median from $G$ in triangle $GHK$ is perpendicular to $OI$.
A bug starts at coordinate $x=1$ on the real number line. At each jump, from $x=a$ it may move to $x=a+2$ or $x=a/2$. Prove that the number of distinct positions, including the initial position, reachable in at most $n$ jumps is $F_{n+4}-(n+4)$, where $F_0=F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\ge2$.
Let $f:\mathbb R\to(0,+\infty)$ be continuous and satisfy $\lim_{x\to-\infty}f(x)=\lim_{x\to+\infty}f(x)=0$.
(a) Prove that $f$ attains a greatest value on $\mathbb R$.
(b) Prove that there exist sequences $(x_n),(y_n)$ with $x_n\lt y_n$ for every $n\ge1$, converging to the same limit, such that $f(x_n)=f(y_n)$ for every $n$.
An integer sequence $(x_n)$ satisfies $0\le x_0\lt x_1\le100$ and
$$x_{n+2}=7x_{n+1}-x_n+280\qquad(n\ge0).$$
(a) If $x_0=2,x_1=3$, prove that for every natural number $n$, the sum of the positive divisors of $x_nx_{n+1}+x_{n+1}x_{n+2}+x_{n+2}x_{n+3}+2018$ is divisible by $24$.
(b) Find all pairs $(x_0,x_1)$ for which $x_nx_{n+1}+2019$ is a perfect square for infinitely many natural numbers $n$.
For every polynomial $f(x)=a_0+a_1x+\cdots+a_mx^m$, define $\Gamma(f)=a_0^2+a_1^2+\cdots+a_m^2$. Let $P(x)=(x+1)(x+2)\cdots(x+2020)$. Prove that there exist at least $2^{2019}$ pairwise distinct polynomials $Q_k(x)$, all with positive coefficients, satisfying $\deg Q_k=2020$ and $\Gamma(Q_k(x)^n)=\Gamma(P(x)^n)$ for every positive integer $n$.
Triangle $ABC$ has orthocenter $H$ and incenter $I$. On rays $AB,AC,BC,BA,CA,CB$, respectively, choose $A_1,A_2,B_1,B_2,C_1,C_2$ so that
$$AA_1=AA_2=BC,\quad BB_1=BB_2=CA,\quad CC_1=CC_2=AB.$$
The pairs of lines $(B_1B_2,C_1C_2)$, $(C_1C_2,A_1A_2)$, $(A_1A_2,B_1B_2)$ meet at $A',B',C'$, respectively.
(a) Prove that the area of $A'B'C'$ is at most the area of $ABC$.
(b) Let $J$ be the circumcenter of $A'B'C'$. Lines $AJ,BJ,CJ$ meet $BC,CA,AB$ at $R,S,T$, respectively. The circumcircles of $AST,BTR,CRS$ have a common point $K$. Prove that if $ABC$ is scalene, then $IHJK$ is a parallelogram.
Let $f(x)=x^2-\alpha x+1$, where $\alpha\in\mathbb R$.
(a) For $\alpha=\sqrt{15}/2$, express $f$ as the quotient of two polynomials with nonnegative coefficients.
(b) Find all real $\alpha$ for which such a representation exists.
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. Let $K$ be the reflection of $H$ in $BC$. Lines $DE,MP$ meet at $X$, and lines $DF,MN$ meet at $Y$.
(a) Line $XY$ meets the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
(b) Lines $KE,KF$ meet $(O)$ again at $S,T$, respectively, with $S,T\ne K$. Prove that $BS,CT,XY$ are concurrent.
Identical square sheets are divided into a $5\times5$ grid on both faces. Color the sheets using $n$ colors so that each cell has the same color on its two faces. Two colored sheets are considered identical if they can be placed one on top of the other so that all corresponding cells have the same colors. Prove that there are at most
$$\frac18\left(n^{25}+4n^{15}+n^{13}+2n^7\right)$$
pairwise distinct colored sheets.