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[Solutions] Romanian Masters Mathematics 2016

  1. Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).
  2. Given positive integers $m$ and $n \ge m$, determine the largest number of dominoes ($1\times2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2n$ columns consisting of cells ($1 \times 1$ squares) so that
    • each domino covers exactly two adjacent cells of the board;
    • no two dominoes overlap;
    • no two form a $2 \times 2$ square; and
    • the bottom row of the board is completely covered by $n$ dominoes.
  3. A "cubic sequence" is a sequence of integers given by $$a_n =n^3 + bn^2 + cn + d,$$ where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.
    a) Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$.
    b) Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part a).
  4. Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $$x^{2016}+y> 1-\frac{1}{100}.$$
  5. A hexagon convex $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$.
    a) Prove that $R\geq r_1+r_2+r_3$
    b) If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic
  6. A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $$|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|,$$ and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

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MOlympiad: [Solutions] Romanian Masters Mathematics 2016
[Solutions] Romanian Masters Mathematics 2016
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