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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 $\overline{AH_1}$ , $\overline{BH_2}$ , and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$ . The incircle of triangle $ABC$ is tangent to $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ at $T_1, T_2$ , and $T_3$ , respectively. For $k = 1, 2, 3$ , let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$ ) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$ . Prove that the circumcircles of triangles $T_1P_1T_2$ , $T_2P_2T_3$ , $T_3P_3T_1$ pass through a common point.

Geometry Solution and answer checking →
Let $ABC$ be an acute scalene triangle. The incircle of $ABC$ touches $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ respectively. Let $X$ , $Y$ , $Z$ be feet of the altitudes from $A$ , $B$ , $C$ to the sides $BC$ , $CA$ , $AB$ respectively. Let $A'$ , $B'$ , $C'$ be the reflections of $X$ , $Y$ , $Z$ in $EF$ , $FD$ , $DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.

Geometry Solution and answer checking →
A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.
Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles triangles with two odd sides.
(Serbia)

Geometry Solution and answer checking →
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$ . Let $I_{a}$ , $I_{b}$ , $I_{c}$ , and $I_{d}$ be the incenters of the triangles $DAB$ , $ABC$ , $BCD$ , and $CDA$ , respectively. Suppose that the common external tangents of the circles $A I_{b} I_{d}$ and $C I_{b} I_{d}$ meet at $X$ , and the common external tangents of the circles $B I_{a} I_{c}$ and $D I_{a} I_{c}$ meet at $Y$ . Prove that $\angle X I Y = 90^{\circ}$ .

Geometry Solution and answer checking →
Given an acute, non-isosceles triangle $ABC$ , and a point $P$ inside the triangle such that $\angle APB = \angle APC = \alpha$ with $\alpha > 80^\circ - \angle BAC$ . The circle $(APB)$ intersects the line $AC$ at $E$ , the circle $(APC)$ intersects the line $AB$ at $F$ . Let $Q$ be the inside point of the triangle $AEF$ such that $\angle AQE = \angle AQF = \alpha$ . Let $D$ be the symmetric point of $Q$ through $EF$ . The bisector of $\angle EDF$ intersects $AP$ at $T.
a) Prove that $\angle DET = \angle ABC$ , $\angle DFT = \angle ACB$ .
b) The line $AP$ intersects the line $DE$ , $DF$ respectively at $M$ , $N$ . Let $I$ and $J$ be centers of incircles of $PEM$ and $PFN$ respectively. Let $K$ be the center of $(DIJ)$ . The line $DT$ intersects $(K)$ at $H$ . Prove that $HK$ goes through the incircle center of the triangle $DMN$ .

Geometry Solution and answer checking →
Let $M$ be a subset of a plane sufficing following properties:
1) There is no single line $k$ , such that $M \subset k$ .
2) For any parallelogram $ABCD$ if $A, B, C \in M$ , then $D \in M$ .
3) If $A, B \in M$ , then $|AB| > 1$ .
Prove, that there are two families of parallel lines, such that $M$ is a set consisting of all intersection points of lines from the first family with lines from the second family.

Geometry Solution and answer checking →
The incircle $\omega$ of acute-angled scalene triangle $A B C$ has centre $I$ and meets sides $B C$ , $C A$ , and $A B$ at $D, E$ , and $F$ , respectively. The line through $D$ perpendicular to $E F$ meets $\omega$ again at $R$ . Line $A R$ meets $\omega$ again at $P$ . The circumcircles of triangles $P C E$ and $P B F$ meet again at $Q \neq P$ . Prove that lines $D I$ and $P Q$ meet on the external bisector of angle $B A C$ .

Geometry Solution and answer checking →
Let $ABC$ be a triangle with $AB < AC < BC$ , and let $D$ be a point in the interior of segment $BC$ . Let $E$ be a point on the circumcircle of triangle $ABC$ such that $A$ and $E$ lie on opposite sides of line $BC$ and $\angle BAD = \angle EAC$ . Let $I, I_{B}, I_{C}, J_{B}$ , and $J_{C}$ be the incentres of triangles $ABC, ABD, ADC, ABE$ , and $AEC$ , respectively.
Prove that $I_{B}, I_{C}, J_{B}$ , and $J_{C}$ are concyclic if and only if $AI, I_{B}J_{C}$ , and $J_{B}I_{C}$ concur.

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.

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 $(O)$ .

Geometry Solution and answer checking →

Answer key — Stage 9 · Geometry

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