Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that $$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$for all $n$

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

Let $I, O$ be the incenter, circumcenter of triangle $ABC$ and $A_1, B_1, C_1 $be arbitrary points on the segments $AI, BI, CI$ respectively. The perpendicular bisectors of $AA_1, BB_1, CC_1$ intersect each other at $X, Y$ and $Z$. Prove that the circumcenter of triangle $XYZ$ coincides with $O$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$

In each of the cells of a $13 \times 13$ board is written an integer such that the integers in adjacent cells differ by $1$. If there are two $2$s and two $24$s on this board, how many $13$s can there be?

Let $n$ be an odd positive integer with $n > 1$ and let $a_1, a_2,... , a_n$ be positive integers such that gcd $(a_1, a_2,... , a_n) = 1$. Let $d$ = gcd $(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)$. Show that the possible values of $d$ are $d = 1, d = 2$

Two positive integers $m$ and $n$ are called similar if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

Let $C$ be a point lies outside the circle $(O)$ and $CS, CT$ are tangent lines of $(O)$. Take two points $A, B$ on $(O)$ with $M$ is the midpoint of the minor arc $AB$ such that $A, B, M$ differ from $S, T$. Suppose that $MS, MT$ cut line $AB$ at $E, F$. Take $X \in OS$ and $Y \in OT$ such that $EX, FY$ are perpendicular to $AB$. Prove that $X Y$ and $C M$ are perpendicular.

In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be?