Let positive sequences $(x_n),(y_n)$ be determined by $x_1=1$, $y_1=\sqrt3$ and
$$\begin{cases}x_{n+1}y_{n+1}-x_n=0,\\x_{n+1}^2+y_n=2\end{cases}\qquad(n=1,2,\ldots).$$
Prove that both sequences converge and find their limits.
Let $P(x)=(x^2-7x+6)^{2n}+13$, where $n$ is a positive integer. Prove that $P(x)$ cannot be expressed as the product of $n+1$ nonconstant polynomials with integer coefficients.
A regular polygon has $103$ vertices. Color $79$ vertices red and all remaining vertices blue. Let $A$ be the number of adjacent red-red pairs and $B$ the number of adjacent blue-blue pairs.
(a) Find all possible pairs $(A,B)$.
(b) Find the number of colorings for which $B=14$, where two colorings are identified if one is obtained from the other by a rotation about the circumcenter.
Let an acute triangle $ABC$ be inscribed in a circle $(O)$, with $AB\lt AC$. Let $I$ be the midpoint of the arc $BC$ not containing $A$. Choose $K\ne C$ on segment $AC$ such that $IK=IC$. Line $BK$ meets $(O)$ again at $D\ne B$ and meets line $AI$ at $E$. Line $DI$ meets line $AC$ at $F$.
(a) Prove $EF=BC/2$.
(b) Choose $M$ on line $DI$ so that $CM\parallel AD$. Line $KM$ meets line $BC$ at $N$. The circumcircle of $BKN$ meets $(O)$ again at $P\ne B$. Prove that line $PK$ passes through the midpoint of $AD$.
An acute triangle $ABC$ is inscribed in a fixed circle $(O)$, with $B,C$ fixed and $A$ moving on $(O)$. On rays $AB,AC$, respectively, choose $M,N$ such that $MA=MC$ and $NA=NB$. The circumcircles of $AMN$ and $ABC$ meet again at $P\ne A$. Line $MN$ meets line $BC$ at $Q$.
(a) Prove that $A,P,Q$ are collinear.
(b) Let $D$ be the midpoint of $BC$. The circles centered at $M,N$ and passing through $A$ meet again at $K\ne A$. The line through $A$ perpendicular to $AK$ meets $BC$ at $E$. The circumcircle of $ADE$ meets $(O)$ again at $F\ne A$. Prove that $AF$ passes through a fixed point.
For positive real numbers $x,y,z$, find the greatest value of
$$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}.$$
Find all collections of $2014$ rational numbers, not necessarily distinct, with the following property: after any one number is removed, the remaining $2013$ numbers can be partitioned into three disjoint groups of $671$ numbers each, with equal products in all three groups.