Pan African 2000-22 (PAMO) + SHL 2018 39p

geometry problems from Pan African Mathematics Olympiads (PAMO) and recent shortlists
with aops links in the names

PAMO geometry problems 2000-17 EN in pdf with aops links

It was not held in 2011, 2014 and 2020

collected inside aops here

2000 - 2022

Shortlists inside aops: 2017, 2018

Pan African 2000.5

Let γ be circle and let P be a point outside . Let PA and PB be the tangents from P to  γ (where A,B ä γ). A line passing through P intersects γ  at points Q and R. Let S be a point on γ  such that BS // QR. Prove that SA bisects QR.

Pan African 2001. 3

Let P_o be a point outside an equilateral triangle ABC such that AP_oC is an isosceles triangle with a right angle at P_o. A grasshopper starts at P_o and turns around the triangle as follows. From P_o, the grasshopper jumps to the point P₁ symmetric to P_o with respect to A, then it jumps to the point P₂ symmetric to P₁ with respect to B, then to the point P₃ symmetric to P₂ with respect to C, etc. For each n ä N, compare the distances P_oP₁ and P₀P_n.

Pan African 2001. 6

Let S₁ be a semicircle with center O and diameter AB. A circle C₁ with center P is tangent to S₁ and to AB at O. A semicircle S₂ with center Q on AB is tangent to S₁ and to C₁. A circle C₂ with center R is internally tangent to S₁ and externally tangent to S₂ and C₁. Prove that OPRQ is a rectangle.

Pan African 2002.3

△AOB is a right triangle with ÐAOB = 90^o. C and D are moving on AO and BO respectively such that AC = BD. Show that there is a fixed point P through which the perpendicular bisector of CD always passes.

Pan African 2002.6

Let △ABC be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

Pan African 2003.2

The circumference of a circle is arbitrarily divided into four parts. The midpoints of the arcs are connected by line segments. Show that two of these segments are perpendicular.

Pan African 2004.6

Let ABCD be a cyclic quadrilateral such that AB is a diameter of it’s circumcircle. Suppose that AB and CD intersect at I, AD and BC at J, AC and BD at K, and let N be a point on AB. Show that IK is perpendicular to JN if and only if N is the midpoint of AB.

Pan African 2005.3

Let ABC be a triangle and let P be a point on one of the sides of ABC. Show how to construct a line passing through P that divides triangle ABC into two parts of equal area.

Pan African 2006.1

Let AB and CD be two perpendicular diameters of a circle with centre O. Consider a point M on the diameter AB, different from A and B. The line CM cuts the circle again at N. The tangent at N to the circle and the perpendicular at M to AM intersect at P. Show that OP = CM.

Pan African 2006.6

Let ABC be a right angled triangle at A. Denote D the foot of the altitude through A and O₁,O₂ the incentres of triangles ADB and ADC. The circle with centre A and radius AD cuts AB in K and AC in L. Show that O₁, O₂, K and L are on a line.

Pan African 2007.2

Let A, B and C be three fixed points, not on the same line. Consider all  triangles AB′C′ where B′ moves on a given straight line (not containing A), and C′ is determined such that ÐB′ = ÐB and ÐC′ = ÐC. Find the locus of C′.

Pan African 2007.6

An equilateral triangle of side length 2 is divided into four pieces by two perpendicularlines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?

Pan African 2008.2

Let C₁ be a circle with centre O, and let AB be a chord of the circle that is not a diameter. M is the midpoint of AB. Consider a point T on the circle C₂ with diameter OM. The tangent to C₂ at the point T intersects C₁ at two points. Let P be one of these points. Show that PA² + PB² = 4PT².

Pan African 2009.2

Point P lies inside a triangle ABC. Let D, E and F be reflections of the point P in the lines BC, CA and AB, respectively. Prove that if the triangle DEF is equilateral, then the lines AD, BE and CF intersect in a common point.

Pan African 2009.6

Points C, E, D and F lie on a circle with center O. Two chords CD and EF intersect at a point N. The tangents at C and D intersect at A, and the tangents at E and F intersect at B. Prove that ON $\perp$ AB.

Pan African 2010.3

In an acute-angled triangle ABC, CF is an altitude, with F on AB, and BM is a median, with M on CA. Given that BM = CF and ÐMBC = ÐFCA, prove that triangle ABC is equilateral.

Pan African 2010.4

Seven distinct points are marked on a circle of circumference c. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal to c / 24  .

In 2011 did not take place.

Pan African 2012.1

AB is a chord (not a diameter) of a circle with centre O. Let T be a point on segment OB. The line through T perpendicular to OB meets AB at C and the circle at D and E. Denote by S the orthogonal projection of T onto AB . Prove that AS · BC = TE · TD.

Pan African 2012.6

(i) Find the angles of △ABC if the length of the altitude through B is equal to the length of the median through C and the length of the altitude through C is equal to the length of the median through B.

(ii) Find all possible values of ÐABC of △ABC if the length of the altitude through A is equal to the length of the median through C and the length of the altitude through C is equal to the length of the median through B.

Pan African 2013.3

Let ABCDEF be a convex hexagon with ÐA = ÐD and ÐB = ÐE . Let K and L  be the midpoints of the sides AB and DE respectively. Prove that the sum of the areas of triangles FAK, KCB and CFL is equal to half of the area of the hexagon if and only if BC / CD = EF / FA

Pan African 2013.4

Let ABCD be a convex quadrilateral with AB parallel to CD. Let P and Q be the midpoints of AC and BD, respectively. Prove that if ÐABP = ÐCBD, then ÐBCQ = ÐACD.

In 2014 did not take place.

Pan African 2015.2

A convex hexagon ABCDEF is such that AB = BC , CD = DE , EF = FA and ÐABC = 2ÐAEC,                                                             

  ÐCDE = 2ÐCAE ,  ÐEFA = 2ÐACE . Show that AD, CF and EB are concurrent.

Pan African 2015.6

Let ABCD be a quadrilateral (with non-perpendicular diagonals).

The perpendicular from A to BC meets CD at K.

The perpendicular from A to CD meets BC at L.

The perpendicular from C to AB meets AD at M.

The perpendicular from C to AD meets AB at N.

(i) Prove that KL is parallel to MN.

(ii) Prove that KLMN is a parallelogram if ABCD is cyclic.

Pan African 2016.1

Two circles C₁ and C₂ intersect each other at two distinct points M and N. A common tangent lines touches C₁ at P and C2 at Q, the line being closer to N than to M. The line PN meets the circle C₂ again at the point R. Prove that the line MQ is a bisector of the angle ÐPMR.

Pan African 2016.5

Let ABCD be a trapezium such that the sides AB and CD are parallel and the side AB is longer than the side CD. Let M and N be on the segments AB  and BC respectively, such that each of the segments CM and AN divides the trapezium in two parts of equal area. Prove that the segment MN intersects the segment BD at its midpoint.

Pan African 2017.6
Let ABC be a triangle with H its orthocenter. The circle with diameter [AC]  cuts the circumcircle of triangle ABH at K. Prove that the point of intersection of the lines CK and BH is the midpoint of the segment [BH]

Pan African 2018.4
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

Pan African 2019.3
Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$  \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}.$$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle

Pan African 2019.4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let  $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

In 2020 did not take place.

Pan African 2021.2

Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between

$\Gamma$ and the tangents from $P$ to $\Gamma$. Let $K$ be a point on the line $AB$, distinct from

$A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the

triangle $PBK$.Also, let $P'$ be the reflection of $P$ in point $A$. Show that $\angle PBT=\angle P'KA$ Pan African 2021.6

Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$.

Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$. Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.

Pan African 2022.1

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the

orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the

second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$

meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and

$BH$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.

PAMO Shortlist 2017 - 2018

(2017 incomplete)

PAMO Shortlist 2017  G1
We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.

PAMO Shortlist 2017  G3
Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that $  \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2},

$and that $FD + FB + FA = FE + FC$.

PAMO Shortlist 2017 (problem 6)

Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$ .

(2018 complete)

PAMO Shortlist 2018  G1
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively, and let $F$ be the foot of the altitude through $A$. Show that the line $DE$, the angle bisector of $\angle ACB$ and the circumcircle of $ACF$ pass through a common point.


Alternate version: In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively. The line $DE$ and the angle bisector of $\angle ACB$ meet at $G$. Show that $\angle AGC$ is a right angle.

PAMO Shortlist 2018  G2
Let $P$ be a point on the median $AM$ of a triangle $ABC$. Suppose that the tangents to the circumcircles of $ABP$ and $ACP$ at $B$ and $C$ respectively meet at $Q$. Show that $\angle PAB = \angle CAQ$.

PAMO Shortlist 2018  G3 (problem 4)
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

PAMO Shortlist 2018  G4
Let $ABC$ be a triangle and $\Gamma$ be the circle with diameter $[AB]$. The bisectors of $\angle BAC$ and $\angle ABC$ cut the circle $\Gamma$ again at $D$ and $E$, respectively. The incicrcle of the triangle $ABC$ cuts the lines $BC$ and $AC$ in $F$ and $G$, respectively. Show that the points $D, E, F$ and $G$ lie on the same line.

PAMO Shortlist 2018  G5
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.

PAMO Shortlist 2018  G6
Let $\Gamma$ be the circumcircle of an acute triangle $ABC$. The perpendicular line to $AB$ passing through $C$ cuts $AB$ in $D$ and $\Gamma$ again in $E$. The bisector of the angle $C$ cuts $AB$ in $F$ and $\Gamma$ again in $G$. The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I$. Prove that $AI = EB$.