It is known that the sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}$, $(n \ge 2)$, find the general term of $a_n$.

From a point $P$ exterior of circle $\odot O$, we draw tangents $PA$, $PB$ and the secant $PCD$ . The line passing through point $C$ parallel to $PA$ intersects chords $AB$, $AD$ at points $E$, $F$ respectively. Prove that $CE = EF$.

Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$.

Let $a,b,c$ be positive real numbers such that $(a+2b)(b+2c)=9$. Prove that $$\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{b^3+c^3}{2}}\geq 3.$$

Let $\odot O$ be the inscribed circle of $\vartriangle ABC$, with $D$, $E$, $N$ the touchpoints with sides $AB$, $AC$, $BC$ respectively. Extension of $NO$ intersects segment $DE$ at point $K$. Extension of $AK$ intersects segment $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.

Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$ The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.

Let $x,y,z \in [0,1]$ , and $|y-z|\leq \frac{1}{2},|z-x|\leq \frac{1}{2},|x-y|\leq \frac{1}{2}$ . Find the maximum and minimum value of $W=x+y+z-yz-zx-xy$.