BALKAN-SL 2019

Let $a, b, c$ be real numbers such that $0 \leq a \leq b \leq c$. Prove that if

$$a+b+c=a b+b c+c a>0,$$

then $\sqrt{b c}(a+1) \geq 2$. When does the equality hold?

Let $a_{i j}, i=1,2, \ldots, m$ and $j=1,2, \ldots, n$, be positive real numbers. Prove that

$$\sum_{i=1}^{m}\left(\sum_{j=1}^{n} \frac{1}{a_{i j}}\right)^{-1} \leq\left(\sum_{j=1}^{n}\left(\sum_{i=1}^{m} a_{i j}\right)^{-1}\right)^{-1} .$$

When does the equality hold?

Let $a, b, c$ be positive real numbers, such that $(a b)^{2}+(b c)^{2}+(c a)^{2}=3$. Prove that

$$\left(a^{2}-a+1\right)\left(b^{2}-b+1\right)\left(c^{2}-c+1\right) \geq 1 .$$

100 couples are invited to a traditional Moldovan dance. The 200 people stand in a line, and then in a step, two of them (not necessarily adjacent) may swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

A town-planner has built an isolated city whose road network consists of 2 N roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.

Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first exit at each roundabout he encounters. It turns out his journey includes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?

Let $A B C D$ be a square of center $O$ and let $M$ be the symmetric of the point $B$ with respect to the point $A$. Let $E$ be the intersection of $C M$ and $B D$, and let $S$ be the intersection of $M O$ and $A E$. Show that $S O$ is the angle bisector of $\angle E S B$.

Let be a triangle $\triangle A B C$ with $m(\angle A B C)=75^{\circ}$ and $m(\angle A C B)=45^{\circ}$. The angle bisector of $\angle C A B$ intersects $C B$ at the point $D$. We consider the point $E \in(A B)$, such that $D E=D C$. Let $P$ be the intersection of the lines $A D$ and $C E$. Prove that $P$ is the midpoint of the segment $A D$.

Let $A B C$ be a scalene and acute triangle, with circumcentre $O$. Let $\omega$ be the circle with centre $A$, tangent to $B C$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $F G \perp A O, \angle B F D=\angle D G C$ and the couples of points $(B, F)$ and $(C, G)$ are in different halfplanes with respect to the line $A D$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $A B C$.

Given an acute triangle $A B C$, let $M$ be the midpoint of $B C$ and $H$ the orthocentre. Let $\Gamma$ be the circle with diameter $H M$, and let $X, Y$ be distinct points on $\Gamma$ such that $A X, A Y$ are tangent to $\Gamma$. Prove that $B X Y C$ is cyclic.

Let $A B C(B C>A C)$ be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $A B$ at the point $D$. The circumcircles of triangles $B C D, O C D$ and $A O B$ intersect the ray $C A$ (beyond $A$ ) at the points $Q, P$ and $K$, respectively, such that $P \in(A K)$ and $K \in(P Q)$. The line $P D$ intersects the circumcircle of triangle $B K Q$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $B Q$. Prove that $T B=T Q$.

Let $A B C$ be an acute triangle, and $A X, A Y$ two isogonal lines. Also, suppose that $K, S$ are the feet of perpendiculars from $B$ to $A X, A Y$, and $T, L$ are the feet of perpendiculars from $C$ to $A X, A Y$ respectively. Prove that $K L$ and $S T$ intersect on $B C$.

Let $A D, B E$, and $C F$ denote the altitudes of triangle $\triangle A B C$. Points $E^{\prime}$ and $F^{\prime}$ are the reflections of $E$ and $F$ over $A D$, respectively. The lines $B F^{\prime}$ and $C E^{\prime}$ intersect at $X$, while the lines $B E^{\prime}$ and $C F^{\prime}$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\triangle A B C$, then the lines $A X, Y H$, and $B C$ are concurrent.

Given an acute triangle $A B C,(c)$ is circumcircle with center $O$ and $H$ the orthocenter of the triangle $A B C$. The line $A O$ intersects $(c)$ at the point $D$. Let $D_{1}, D_{2}$ and $H_{2}, H_{3}$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $A B, A C$ respectively. Let $\left(c_{1}\right)$ be the circumcircle of the triangle $A D_{1} D_{2}$. Suppose that the line $A H$ intersects again $\left(c_{1}\right)$ at the point $U$, the line $H_{2} H_{3}$ intersects the segment $D_{1} D_{2}$ at the point $K_{1}$ and the line $D H_{3}$ intersects the segment $U D_{2}$ at the point $L_{1}$. Prove that one of the intersection points of the circumcircles of the triangles $D_{1} K_{1} H_{2}$ and $U D L_{1}$ lies on the line $K_{1} L_{1}$.

Given semicircle (c) with diameter $A B$ and center $O$. On the (c) we take point $C$ such that the tangent at the $C$ intersects the line $A B$ at the point $E$. The perpendicular line from $C$ to $A B$ intersects the diameter $A B$ at the point $D$. On the (c) we get the points $H, Z$ such that $C D=C H=C Z$. The line $H Z$ intersects the lines $C O, C D, A B$ at the points $S, I, K$ respectively and the parallel line from $I$ to the line $A B$ intersects the lines $C O, C K$ at the points $L, M$ respectively. We consider the circumcircle $(k)$ of the triangle $L M D$, which intersects again the lines $A B, C K$ at the points $P, U$ respectively. Let $\left(e_{1}\right)$, $\left(e_{2}\right),\left(e_{3}\right)$ be the tangents of the $(k)$ at the points $L, M, P$ respectively and $R=\left(e_{1}\right) \cap\left(e_{2}\right)$, $X=\left(e_{2}\right) \cap\left(e_{3}\right), T=\left(e_{1}\right) \cap\left(e_{3}\right)$. Prove that if $Q$ is the center of $(k)$, the lines $R D, T U$, $X S$ pass through the same point, which lies in the line $I Q$.