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.