geometry problems from Pan African Mathematics Olympiads (PAMO) and recent shortlists

with aops links in the names

In 2011 did not take place.

with aops links in the names

It was not held in 2011 and 2014

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 P

_{o}be a point outside an equilateral triangle ABC such that AP_{o}C 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 P_{1}symmetric to P_{o}with respect to A, then it jumps to the point P_{2}symmetric to P_{1}with respect to B, then to the point P_{3}symmetric to P_{2}with respect to C, etc. For each n ä N, compare the distances P_{o}P_{1}and P_{0}P_{n}.
Let S

_{1}be a semicircle with center O and diameter AB. A circle C_{1}with center P is tangent to S_{1}and to AB at O. A semicircle S_{2}with center Q on AB is tangent to S_{1}and to C_{1}. A circle C_{2}with center R is internally tangent to S_{1}and externally tangent to S_{2}and C_{1}. Prove that OPRQ is a rectangle.
△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.
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 O

_{1},O_{2}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 O_{1}, O_{2}, K and L are on a line.
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′.

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 C

_{1}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 C_{2}with diameter OM. The tangent to C_{2}at the point T intersects C_{1}at two points. Let P be one of these points. Show that PA^{2}+ PB^{2}= 4PT^{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.

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 C

_{1}and C_{2}intersect each other at two distinct points M and N. A common tangent lines touches C_{1}at P and C2 at Q, the line being closer to N than to M. The line PN meets the circle C_{2}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

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]

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 $EF$.

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 $BF + DE = AE$.

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

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.

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 $\angle PAB = \angle CAQ$.

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 $\angle BAP = \angle CBP$.

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.

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 $AK = KL$.

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 $AI = EB$.

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 $AI = EB$.

## Δεν υπάρχουν σχόλια:

## Δημοσίευση σχολίου