[Solutions] Junior Balkan Mathematical Olympiad 2017

  1. Determine all the sets of six consecutive positive integers such that the product of some two of them added to the product of some other two of them is equal to the product of the remaining two numbers.
  2. Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$. Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$ When does the equality hold?
  3. Let $ABC $ be an acute triangle such that $AB\neq AC$, with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A$, $M$ and $X$ are collinear.
  4. Consider a regular $2n$-gon $ P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We color the sides of $P$ in $3$ different colors (ignore the vertices of $P$, we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once. Moreover ,from every point in the plane external to $P$, points of most $2$ different colors on $P$ can be seen. Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).

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