Find all positive integers $ n$ such that $ n^4-4n^3+22n^2-36n+18$ is a perfect square.

Let $ O$ be the circumcenter of acute triangle $ ABC$. Point $ P$ is in the interior of triangle $ AOB$. Let $ D,E,F$ be the projections of $ P$ on the sides $ BC,CA,AB$, respectively. Prove that the parallelogram consisting of $ FE$ and $ FD$ as its adjacent sides lies inside triangle $ ABC$.

In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 + px^3 + qx^2 + rx + s = 0$. Find the minimum of square areas.

Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n + 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n + 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} = A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.

Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.

Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}+a_{2}+\cdots+a_{n}\ge n^2;$ $ (2): a_{1}^2+a_{2}^2+\cdots+a_{n}^2\le n^3+1.$

Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2-x-1=0$. Let $ a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$, $ n=1, 2, \cdots$. (1) Prove that for any positive integer $ n$, we have $ a_{n+2}=a_{n+1}+a_n$. (2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n-2na^n$ for any positive integer $ n$.

Assume that $ S=(a_1, a_2, \cdots, a_n)$ consists of $ 0$ and $ 1$ and is the longest sequence of number, which satisfies the following condition: Every two sections of successive $ 5$ terms in the sequence of numbers $ S$ are different, i.e., for arbitrary $ 1\le i<j\le n-4$, $ (a_i, a_{i+1}, a_{i+2}, a_{i+3}, a_{i+4})$ and $ (a_j, a_{j+1}, a_{j+2}, a_{j+3}, a_{j+4})$ are different. Prove that the first four terms and the last four terms in the sequence are the same.