## Σελίδες

### Pan African 2000-19 (PAMO) + SHL 2018 36p

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

It was not held in 2011 and 2014

2000 - 2019

Shortlists inside aops: 2017, 2018

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.

Let Po be a point outside an equilateral triangle ABC such that APoC is an isosceles triangle with a right angle at Po. A grasshopper starts at Po and turns around the triangle as follows. From Po, the grasshopper jumps to the point P1 symmetric to Po 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 PoP1 and P0Pn.

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.

AOB is a right triangle with ÐAOB = 90o. 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.

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.

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.

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.

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.

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.

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.

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

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?

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.

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.

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.

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.

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.

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.

(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.

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

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.

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.

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.

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.

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 $EF$.

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 $AI = EB$.