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Stage 9 · Geometry

10 problems · IMO P2/P5; hard shortlist · mathsolympiadprep.com

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

Let $\omega$ be the circumcircle of a triangle $A B C$ , and let $\Omega_{A}$ be its excircle which is tangent to the segment $B C$ . Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_{A}$ . Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_{A}$ at $X$ and $Y$ , respectively. The tangent line at $P$ to the circumcircle of the triangle $A P X$ intersects the tangent line at $Q$ to the circumcircle of the triangle $A Q Y$ at a point $R$ . Prove that $A R \perp B C$ .

IMO Shortlist 2021

Geometry Solution and answer checking →
Let $A = (0,0,0)$ be the origin in the three dimensional coordinate space. The weight of a point is the sum of the absolute values of its coordinates. A point is a primitive lattice point if all its coordinates are integers with their greatest common divisor equal to 1. A square $ABCD$ is called a unbalanced primitive integer square if it has integer side length and the points $B$ and $D$ are primitive lattice points with different weights.
Show that there are infinitely many unbalanced primitive integer squares $AB_iC_iD_i$ such that the plane containing the squares are not parallel to each other.

USA IMO

Geometry Solution and answer checking →
Let $n$ be a positive integer. Consider
$S = \{(x, y, z) : x, y, z \in \{0, 1, \dots, n\}, x + y + z > 0\}.$
as a set of $(n+1)^3 - 1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0, 0, 0)$ .
(This problem was suggested by the Netherlands.)

International Mathematical Olympiad

Geometry Solution and answer checking →
Let $n>1$ be an integer. Suppose we are given $2n$ points in a plane such that no three of them are collinear. The points are to be labelled $A_{1}, A_{2}, \ldots, A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_{1}A_{2}A_{3}, \angle A_{2}A_{3}A_{4}, \ldots, \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_{1}, \angle A_{2n}A_{1}A_{2}$ . We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$ ). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

IMO Shortlist 2019

Geometry Solution and answer checking →
Let $ABC$ be a triangle with circumcircle $(O)$ , incircle $(I)$ , and excircle $(J)$ with respect to vertex $A$ . Consider $D, E, F$ to be the tangent points of $(J)$ with $BC, CA, AB$ , respectively.
a. Let $L$ be the midpoint of $BC$ . The circle with diameter $LJ$ intersects $DE, DF$ again at $K, H$ , respectively. Prove that the circles $(BDK)$ and $(CDH)$ meet again at a point on circle $(J)$ .
b. Assume that $EF$ intersects $BC$ at $G$ . Let $M, N$ be the intersections of $GJ$ with $AB, AC$ , respectively. Consider $P, Q$ on $JB, JC$ respectively such that $\angle PAB = \angle QAC = 90^\circ$ . Denote $T$ as the intersection of $PM, QN$ and $S$ as the midpoint of the major arc $BC$ of $(O)$ . Prove that $SI, AT$ intersect at a point on the circle $(O)$ .

VN IMO Booklet

Geometry Solution and answer checking →
A rabbit is at some point $(x, y)$ in the Euclidean plane. There are some (possibly infinitely many) landmines, which are circles with any radius that do not intersect except possibly at one point (tangent). Every move, the rabbit can hop a distance of exactly $1$ , but cannot land in the interior of a landmine (but may possibly land on the edge). Suppose that neither the rabbit nor the origin is in a landmine. Find the minimum $\varepsilon$ so that the rabbit can always reach a point with a distance of at most $\varepsilon$ from the origin, regardless of the landmine configuration.

Balkan Mathematical Olympiad Shortlisted Problems

Geometry Solution and answer checking →
Given a circle $(O)$ with a fixed chord $BC$ ( $BC$ is not a diameter of the circle). Let $A$ move on the bigger arc $BC$ such that $ABC$ is an acute triangle and $AB < AC$ . Let $I$ and $H$ respectively be the midpoint of $BC$ and the orthocenter of the triangle $ABC$ . The ray $IH$ intersects the circle $(O)$ again at $K$ , the line $AH$ intersects the line $BC$ at $D$ and the line $KD$ intersects the circle $(O)$ again at $M$ . From $M$ , we draw a perpendicular line to $BC$ , intersecting $AI$ at $N$ .
a) Prove that point $N$ belongs to a fixed circle when $A$ moves along the bigger arc $BC$ .
b) The circle goes through and touching $AK$ at $A$ intersects $AB$ , $AC$ at $P$ , $Q$ , respectively. Let $J$ be the midpoint of $PQ$ . Prove that the line $AJ$ always goes through a fixed point.

IMO Team Selection Tests 2015

Geometry Solution and answer checking →
In an acute triangle $\triangle ABC$ with $|AB| \neq |AC|$ , the perpendicular bisectors of sides $AC$ and $AB$ intersect segment $BC$ at points $D$ and $E$ , respectively. The tangents to $\odot(ABC)$ at the points $B$ and $C$ intersect $\odot(ABD)$ and $\odot(ACE)$ at points $Y$ and $Z$ , respectively. Suppose that lines $YD$ and $ZE$ intersect at point $X$ . Define points $U$ and $V$ to be the intersections of $\odot(DEX)$ and the lines $AD$ and $AE$ , respectively. Prove that the lines $UV, BC$ and $XA$ are concurrent.

Baltic Way Shortlist 2023

Geometry Solution and answer checking →
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.

TST

Geometry Solution and answer checking →
Let $ABCD$ be a parallelogram and $AC$ intersects $BD$ at $I$ . Let $G$ be the point inside triangle $IAB$ that satisfies
$\angle IAG = \angle IBG \neq 45^\circ - \frac{\angle AIB}{4}.$
Let $E, F$ be projections of $C$ on $AG$ and $D$ on $BG$ . The median respect to vertex $E$ of triangle $BEF$ and the median respect to vertex $F$ of triangle $AEF$ intersects at $H$ .
a) Prove that $AF, BE$ and $IH$ are concurrent, denote the concurrent point by $L$ .
b) Let $K$ be the intersection of $CE$ and $DF$ . Let $J$ be the circumcenter of triangle $LAB$ and $M, N$ be the circumcenters of $EIJ, FIJ$ , respectively. Prove that $EM, FN$ and the line passing through the circumcenters of $GAB, KCD$ are concurrent.

IMO Team Selection Test

Geometry Solution and answer checking →

Answer key — Stage 9 · Geometry

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