As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle.

Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

Given a finite graph $G$, let $f(G)$ be the number of triangles in graph $G$, $g(G)$ be the number of edges in graph $G$, find the minimum constant $c$, so that for each graph $G$, there is $f^ 2(G)\le c \cdot g^3(G)$.

A positive integer $m$ is called a beautiful integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many beautiful integers with square factors, and the square factors of different beautiful integers are relative prime.