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Stage 10 · Geometry

8 problems · Hardest shortlist tier · mathsolympiadprep.com

The answer key prints on its own page at the end.

There are 2022 equally spaced points on a circular track $\gamma$ of circumference 2022. The points are labeled $A_1, A_2, \dots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$ . She hops along $\gamma$ from $A_1$ to $A_2$ , then from $A_2$ to $A_3$ , until she reaches $A_{2022}$ , after which she hops back to $A_1$ . When hopping from $P$ to $Q$ , she always hops along the shorter of the two arcs $\text{PQ}$ of $\gamma$ ; if $\text{PQ}$ is a diameter of $\gamma$ , she moves along either semicircle.
Determine the maximal possible sum of the lengths of the 2022 arcs which Bunbun traveled, over all possible labellings of the 2022 points.

USA TST

Geometry Solution and answer checking →
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)

China National Team Selection Test

Geometry Solution and answer checking →
Let $\mathcal{P}$ be a regular 2006-gon. A diagonal of $\mathcal{P}$ is called good segment if its endpoints divide the boundary of $\mathcal{P}$ into two parts, each composed of an odd number of sides of $\mathcal{P}$ . The sides of $\mathcal{P}$ are also called good segment.
Suppose $\mathcal{P}$ has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of $\mathcal{P}$ . Find the maximum number of isosceles triangles having two good segments that could appear in such a configuration.

International Mathematical Olympiad

Geometry Solution and answer checking →
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 $i \not \equiv j \pm 1(\bmod n)$ .

International Mathematical Olympiad Spain

Geometry Solution and answer checking →
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.

International Mathematical Olympiad

Geometry Solution and answer checking →
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.

IMO Shortlist

Geometry Solution and answer checking →
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 $k S_{1} \geq S$ .

IMO Shortlist 2007

Geometry Solution and answer checking →
Let point $M$ be a point on the circumcircle of triangle $ABC$ . From point $M$ draw the (two) lines tangent to the incircle of triangle $ABC$ , meeting $BC$ at points $X_1, X_2$ respectively. Prove that the second intersection point (i.e., the intersection point different from $M$ ) of the circumcircle of triangle $MX_1X_2$ with the circumcircle of $ABC$ is exactly the tangency point of the circumcircle of $ABC$ with the mixtilinear incircle in angle $A$ .
(Note: the mixtilinear incircle in angle $A$ refers to the circle that is tangent to both sides $AB, AC$ , and is internally tangent to the circumcircle of $ABC$ .)
Let $M$ be an arbitrary point on the circumcircle of triangle $ABC$ and let the tangents from this point to the incircle of the triangle meet the sideline $BC$ at $X_1$ , and $X_2$ . Prove that the second intersection of the circumcircle of triangle $MX_1X_2$ with the circumcircle of $ABC$ (different from $M$ ) coincides with the tangency point of the circumcircle with mixtilinear incircle in angle $A$ . (As usual, the A-mixtilinear incircle names the circle tangent to $AB, AC$ and to the circumcircle of $ABC$ internally.)

IMO Training Camp Stage 3 Mock Competition (1) 2014

Geometry Solution and answer checking →

Answer key — Stage 10 · Geometry

Worked solutions for every problem are on the site, one page per problem.

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