MK-MO 2015

Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$ . Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$ . Prove that: $$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$

All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$ .

Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$ . A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$ . Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$ , respectively. Let $P$ be the intersection of the lines $AM$ and $CD$ . The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$ . The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$ . Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$ . Prove that $MNPQ$ is a parallelogram.

Find all natural numbers $m$ having exactly three prime divisors $p,q,r$ , such that $$p-1\mid m; qr-1 \mid m; q-1 \nmid m; r-1 \nmid m; 3 \nmid q+r.$$