PL-OM 2016

Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$ .

Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$ . Assume $\angle BAD+ \angle ADC <\pi$ . Let $M, \ N$ be points of tangency of $\omega$ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$ . Prove that $ID$ bisects the segment $KN$ .

Let $a, \ b \in \mathbb{Z_{+}}$ . Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$ , $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$ . Show that $f(a, b)=f(b, a)$ .

Let $k, n$ be odd positve integers greater than $1$ . Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$ , then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$ .

There are given two positive real number $a<b$ . Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$ . $a< \frac{p}{q} < \frac{r}{s} < b$ . $2.$ $p^2+q^2=r^2+s^2$ .

Let $I$ be an incenter of $\triangle ABC$ . Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$ . Let $P$ be a reflection of $I$ with the respect to $KL$ . Prove that $BP \perp CP$ .