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Stage 8 · Geometry

10 problems · IMO Shortlist mid-range; USAMO P2/P5 · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.

   Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.

   Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.

   USA Team Selection Test

   Geometry Solution and answer checking →

2. Given positive integer $n \ge 5$ and a convex polygon $P$, namely $A_1A_2...A_n$. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $A_iA_jA_kA_l (1\le i<j<k<l\le n)$ not on diagonals of $P$, such that the $\tbinom{n}{4}$ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

   China Team Selection Test

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3. Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that
   $$\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.$$
   Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum
   $$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$$

   IMO Shortlist

   Geometry Solution and answer checking →

4. There is given a convex quadrilateral $ABCD$ and a point $P$ in the interior of the triangle $BCD$ in such a way that the quadrilateral $ABPD$ has an inscribed circle, and the three inscribed circles of the quadrilateral $ABPD$, the triangle $BCP$ and the triangle $CDP$, respectively, are pairwise tangent to each other. Denote by $Q$ and $R$ the points tangency on the line segments $BP$ and $DP$, respectively. Let the lines $BP$ and $AR$ meet at $S$, let the lines $DP$ and $AQ$ meet at $T$, and let the lines $BT$ and $DS$ meet at $U$. Show that the line $CU$ bisects the angle $BCD$.
   (5 pont)

   KöMaL problem A 2015

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5. Triangle $ABC$ has incenter $I$ and excircles $\Omega_A$, $\Omega_B$, and $\Omega_C$. Let $\ell_A$ be the line through the feet of the tangents from $I$ to $\Omega_A$, and define lines $\ell_B$ and $\ell_C$ similarly. Prove that the orthocenter of the triangle formed by lines $\ell_A$, $\ell_B$, and $\ell_C$ coincides with the Nagel point of triangle $ABC$.

   KöMaL problem A 2022

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6. In triangle $ABC$, we have $AB > AC$. The incircle $\omega$ touches $BC$ at $E$, and $AE$ intersects $\omega$ at $D$. Choose a point $F$ on $AE$ ($F$ is different from $E$), such that $CE = CF$. Let $G$ be the intersection point of $CF$ and $BD$. Prove that $CF = FG$.

   China National Team Selection Test

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7. Let $ABC$ be an acute-angled triangle with circumcenter $O$. Furthermore, let $k$ be a circle with the following properties:

   (1) The center $K$ of $k$ lies in the interior of the side $BC$.

   (2) $k$ touches $AB$ at $B'$ and $AC$ at $C'$.

   (3) $O$ lies on the shorter of the two arcs $B'C'$ of $k$.

   Prove: The circumcircle of $ABC$ and $k$ intersect each other in two distinct points.

   Auswahlklausur

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8. Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$, where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line, perpendicular to $BD$ at $E$, intersects $CD$ at $M$, prove that $KM \perp DL$.

   Balkan Mathematical Olympiad Shortlist

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9. Let $P$ be a point inside the triangle $ABC$ satisfying $\angle PAC = \angle PCB$ and $D$ be the midpoint of the line segment $PC$. Let $AP$ intersect $BC$ at $E$ and $ED$ intersect $BP$ at $Q$ such that $P$ is on the line segment $BQ$. Prove that $\angle BAP + \angle BCQ = 180^\circ$.

   Turkey — Team Selection Test

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10. Let $ABC$ be an isosceles triangle, $AB = AC$, and let $M$ and $N$ be points on the sides $BC$ and $CA$, respectively, such that the angles $BAM$ and $CNM$ are equal. The lines $AB$ and $MN$ meet at $P$. Show that the internal angle bisectors of the angles $BAM$ and $BPM$ meet at a point on the line $BC$.

    Bogdan Enescu

    

    NMO Selection Tests for BMO and IMO

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Answer key — Stage 8 · Geometry

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

1. $K \text{ is the intersection of } AB' \text{ and } BA', \text{ and } c \text{ is a constant}$ open
2. $\text{Proven}$ open
3. $4$ open
4. Prove it — see the worked solution open
5. Prove it — see the worked solution open
6. Prove it — see the worked solution open
7. Prove it — see the worked solution open
8. Prove it — see the worked solution open
9. Prove it — see the worked solution open
10. Prove it — see the worked solution open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

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