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Stage 10 · Geometry

8 problems · Hardest shortlist tier · mathsolympiadprep.com

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1. Let $ABC$ be a triangle and $N$ be a point that differs from $A$, $B$ and $C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. We define $B_c$, $C_b$, $A_c$ and $C_a$ similarly. Let $m_a$ be the line passing through $N$ and perpendicular to $B_cC_b$. Define $m_b$, $m_c$ similarly.

   a) Assume that $N$ is the orthocenter of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through each bisector of angles $\angle BNC$, $\angle CNA$ and $\angle ANB$ are coincident.

   b) Assume that $N$ is the nine-point center of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through the lines $BC$, $CA$ and $AB$ concur.

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2. Let $\mathcal{V}$ be a finite set of points in the plane. We say that $\mathcal{V}$ is balanced if for any two distinct points $A, B \in \mathcal{V}$, there exists a point $C \in \mathcal{V}$ such that $AC = BC$. We say that $\mathcal{V}$ is center-free if for any distinct points $A, B, C \in \mathcal{V}$, there does not exist a point $P \in \mathcal{V}$ such that $PA = PB = PC$.

   a. Show that for all $n \geqslant 3$, there exists a balanced set consisting of $n$ points.

   b. For which $n \geqslant 3$ does there exist a balanced, center-free set consisting of $n$ points?

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3. Let $A B C D E$ be a convex pentagon with $C D = D E$ and $\angle E D C \neq 2 \cdot \angle A D B$. Suppose that a point $P$ is located in the interior of the pentagon such that $A P = A E$ and $B P = B C$. Prove that $P$ lies on the diagonal $C E$ if and only if $\operatorname{area}(B C D) + \operatorname{area}(A D E) = \operatorname{area}(A B D) + \operatorname{area}(A B P)$.

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4. Suppose there are beetles on a chessboard consisting of $2012 \times 2012$ unit squares. Each unit square can accommodate at most one beetle. At a moment, all beetles fly and land on the chessboard again. For a beetle, we call the vector from its flying unit to its landing unit the beetle's "displacement vector". We call the sum of all beetle's "displacement vectors" the "total displacement vectors".
   Find the maximum length of "total displacement vector" considering the number of beetles and all possible positions of flying and landing. (posed by Qu Zhenhua)

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5. In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary.
   Find the largest $n$ for which there exist $n$ boxes $B_{1}, \ldots, B_{n}$ such that $B_{i}$ and $B_{j}$ intersect if and only if $i \not \equiv j \pm 1(\bmod n)$.

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6. There are $2017$ mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn. Find all possible numbers of tangent segments when he stops drawing.

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7. For an integer $m \geq 1$, we consider partitions of a $2^{m} \times 2^{m}$ chessboard into rectangles consisting of cells of the chessboard, in which each of the $2^{m}$ cells along one diagonal forms a separate rectangle of side length 1. Determine the smallest possible sum of rectangle perimeters in such a partition.

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8. Determine the smallest positive real number $k$ with the following property.
   Let $ABCD$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $AB, BC$, $CD$ and $DA$, respectively. Consider the areas of triangles $AA_{1}D_{1}, BB_{1}A_{1}, CC_{1}B_{1}$, and $DD_{1}C_{1}$; let $S$ be the sum of the two smallest ones, and let $S_{1}$ be the area of quadrilateral $A_{1}B_{1}C_{1}D_{1}$. Then we always have $k S_{1} \geq S$.

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Answer key — Stage 10 · Geometry

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