Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$is constant with respect to the point $P$.

Let $a_1, a_2, a_3$ be three distinct real numbers. Define $$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\ b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\ b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$When does equality hold?

Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.

Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ of a triangle $ABC$ whose incentre is $I$. Suppose $EF$, extended, meets the circumcircle of $ABC$ in $M,N$. Show that the circumradius of $MIN$ is twice that of $ABC$.

There are $N$ points in the plane such that the total number of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.

Define the distance between two $5$-digit numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ to be the largest integer $j$ such that $a_j \ne b_j$ . (Example: the distance between $16523$ and $16452$ is $5$.) Suppose all $5$-digit numbers are written in a line in some order. What is the minimal possible sum of the distances of adjacent numbers in that written order?

Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.

Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.

Let $a$ and $b$ be two positive real numbers such that $a^{2007} = a + 1$ and $b^{4014} = b + 3a$. Determine whether $a > b$ or $b > a$.

Let $A_1,A_2,...,A_n$ be $n$ finite subsets of a set $X, n \ge 2$, such that (i) $|A_i| \ge 2, 1 \le i \le n$, (ii) $ |A_i \cap A_j | \ne 1, j \le i < j \le n$. Prove that the elements of $A_1 \cup A_2 \cup ... \cup A_n$ may be colored with $2$ colors so that no $A_i$ is colored by the same color.

Let $P$ be an interior point of triangle $ABC$ such that $\angle BPC = \angle CPA =\angle APB = 120^o$. Prove that the Euler lines of triangles $APB,BPC,CPA$ are concurrent.

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.