geometry problems from Asian Pacific Mathematics Olympiad (APMO)

with aops links in the names

with aops links in the names

APMO all 1989-2019 EN in pdf

APMO all 1989-2004 EN in pdf with solutions by John Scholes (kalva)

APMO all 1989-2004 EN in pdf with solutions by John Scholes (kalva)

1989 - 2019

Let A

_{1}, A_{2}, A_{3}be three points in the plane, and for convenience, let A_{4}= A_{1}, A_{5}= A_{2}. For n = 1, 2, and 3, suppose that B_{n}is the midpoint of A_{n}A_{n}_{+1}, and suppose that C_{n}is the midpoint of A_{n}B_{n}. Suppose that A_{n}C_{n}_{+1}and B_{n}A_{n}_{+2 }meet at D_{n}, and that A_{n}B_{n}_{+1 }and C_{n}A_{n}_{+2 }meet at E_{n}. Calculate the ratio of the area of triangle D_{1}D_{2}D_{3}to the area of triangle E_{1}E_{2}E_{3}.
Given triangle ABC, let D, E, F be the
midpoints of BC, AC, AB respectively and let G
be the centroid of the triangle. For each value of <BAC, how many non-similar triangles
are there in which AEGF is a cyclic
quadrilateral?

Consider all the triangles ABC which
have a fixed base AB and whose
altitude from C is a constant h. For which of these triangles is
the product of its altitudes a maximum?

Show that for every integer n ≥
6, there exists a convex hexagon which can be dissected into exactly n congruent triangles

Let G be the centroid of
triangle ABC and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X,
Y , and G are collinear and XY and
BC are parallel. Suppose that XC and GB intersect at Q and
Y B and GC intersect at P.
Show that triangle MPQ is
similar to triangle ABC.

Given are two tangent circles and a point P on their common tangent perpendicular to the lines joining
their centres. Construct with ruler and compass all the circles that are
tangent to these two circles and pass through the point P.

In a circle C with
centre O and radius r, let C

_{1}, C_{2}be two circles with centres O_{1}, O_{2}and radii r_{1}, r_{2}respectively, so that each circle C_{i}is internally tangent to C at A_{i}and so that C_{1}, C_{2}are externally tangent to each other at A. Prove that the three lines OA, O_{1}A_{2}, and O_{2}A_{1}are concurrent.
Let ABCD be a
quadrilateral such that all sides have equal length and angle ABC is 60 deg. Let

*l*be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of*l*with AB and BC respectively. Let M be the point of intersection of CE and AF. Prove that CA^{2}= CM · CE.
Given a nondegenerate triangle ABC,
with circumcentre O,
orthocentre H, and circumradius R, prove that |OH|
< 3R.

Let PQRS be a cyclic
quadrilateral such that the segments PQ
and RS are not parallel.
Consider the set of circles through P and
Q, and the set of circles
through R and S. Determine the set A of points of tangency of circles in
these two sets.

Let C be a circle with
radius R and centre O, and S a fixed point in the interior of C. Let AΑ΄ and BB΄ be perpendicular chords through S. Consider the rectangles SAMB, SBN΄A΄, SA΄M΄B΄, and SB΄NA. Find the set of all points M,
N΄, M΄, and N when A moves around the whole
circle.

Let ABCD be a
quadrilateral AB = BC = CD = DA. Let MN and PQ be two segments perpendicular to the diagonal BD and such that the distance between
them is d > BD / 2, with M ä
AD, N ä DC, P ä
AB, and Q ä BC. Show that the
perimeter of hexagon AMNCQP does
not depend on the position of MN and
PQ so long as the distance
between them remains constant.

Let P

_{1}, P_{2}, P_{3}, P_{4}be four points on a circle, and let I_{1}be the incentre of the triangle P_{2}P_{3}P_{4},I_{2}be the incentre of the triangle P_{1}P_{3}P_{4}, I_{3}be the incentre of the triangle P_{1}P_{2}P_{4}, I_{4}be the incentre of the triangle P_{1}P_{2}P_{3}. Prove that I_{1}, I_{2}, I_{3}, I_{4}are the vertices of a rectangle.
Triangle A

_{1}A_{2}A_{3}has a right angle at A_{3}. A sequence of points is now defined by the following iterative process, where n is a positive integer. From A_{n}(n ≥ 3), a perpendicular line is drawn to meet A_{n-}_{2 }A_{n-}_{1}at A_{n}_{+1}.
(a) Prove that if this process is continued indefinitely, then one and
only one point P is interior to
every triangle A

_{n-}_{2 }A_{n-}_{1}A_{n }, n ≥ 3.
(b) Let A

_{1}and A_{3}be fixed points. By considering all possible locations of A_{2}on the plane, find the locus of P.
Let ABC be a triangle
and D the foot of the altitude
from A. Let E and F be on a line through D
such that AE is
perpendicular to BE, AF is perpendicular to CF, and E and F are
different from D. Let M and N be the midpoints of the line segments BC and EF,
respectively. Prove that AN is
perpendicular to NM.

Let Γ

_{1}and Γ_{2}be two circles intersecting at P and Q. The common tangent, closer to P, of Γ_{1}and Γ_{2}touches Γ_{1}at A and Γ_{2}at B. The tangent of Γ_{1}at P meets Γ_{2}at C, which is different from P, and the extension of AP meets BC at R. Prove that the circumcircle of triangle PQR is tangent to BP and BR.
Let ABC be a triangle.
Let M and N be the points in which the median
and the angle bisector, respectively, at A meet
the side BC. Let Q and P be the points in which the perpendicular at N to NA meets MA and BA, respectively, and O
the point in which the perpendicular at P to BA meets AN produced.
Prove that QO is perpendicular
to BC.

Find the greatest integer n,
such that there are n+4 points A, B, C, D, X

_{1 }, … , X_{n}in the plane with AB ≠ CD that satisfy the following condition: for each i = 1, 2, … , n triangles ABX_{i}and CDX_{i }are equal.
Let ABC be an
equilateral triangle. Let P be
a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be
the orthocentre of triangle ABP and
S be the orthocenter of
triangle ACQ. Let T be the point common to the segments
BP and CQ. Find all possible values of 6 CBP and 6 BCQ such
that triangle TRS is
equilateral.

Suppose ABCD is a square
piece of cardboard with side length a.
On a plane are two parallel lines

*l*_{1}and*l*_{2}, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect*l*_{1 }at E and F respectively. Also, sides CB and CD intersect*l*_{2 }at G and H respectively. Let the perimeters of ∆AEF and ∆CGH be m_{1}and m_{2}respectively. Prove that no matter how the square was placed, m_{1}+ m_{2}remains constant.
Let O be the
circumcentre and H the
orthocentre of an acute triangle ABC.
Prove that the area of one of the triangles
AOH, BOH and COH is equal
to the sum of the areas of the other two.

In a triangle ABC,
points M and N are on sides AB and AC, respectively, such that MB = BC = CN. Let R and r denote
the circumradius and the inradius of the triangle ABC, respectively. Express the ratio MN=BC in terms of R and
r.

Let A,B be two distinct
points on a given circle O and
let P be the midpoint of the
line segment AB. Let O

_{1}be the circle tangent to the line AB at P and tangent to the circle O. Let ` be the tangent line, different from the line AB, to O_{1}passing through A. Let C be the intersection point, different from A, of ` and O. Let Q be the midpoint of the line segment BC and O_{2}be the circle tangent to the line BC at Q and tangent to the line segment AC. Prove that the circle O_{2}is tangent to the circle O.
Let ABC be an acute
angled triangle with <BAC =
60þ and AB > AC. Let I be
the incenter, and H the
orthocenter of the triangle ABC.
Prove that 2<AHI = 3<ABC:

Let ABC be a triangle
with <A < 60þ. Let X and
Y be the points on the sides AB and AC, respectively, such that CA+AX = CB +BX and BA+AY = BC + CY . Let P be the point in the plane such that
the lines PX and PY are perpendicular to AB and AC, respectively. Prove that <BPC < 120þ.

Let Γ be the circumcircle of a triangle ABC. A circle passing through points A and C meets
the sides BC and BA at D and E,
respectively. The lines AD and CE meet Γ again at G and H, respectively. The tangent lines of
Γ at A and C meet the line DE at L and M,
respectively. Prove that the lines LH and
MG meet at Γ.

Let three circles Γ

_{1}, Γ2, Γ3, which are non-overlapping and mutually external, be given in the plane. For each point P in the plane, outside the three circles, construct six points A_{1},B_{1},A_{2},B_{2},A_{3},B_{3}as follows: For each i = 1, 2, 3, A_{i}, B_{i}are distinct points on the circle ¡i such that the lines PA_{i}and PB_{i}are both tangents to Γ_{i}. Call the point P exceptional if, from the construction, three lines A_{1}B_{1}, A_{2}B_{2}, A_{3}B_{3}are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.
Let ABC be a triangle
with < BAC ≠ 90þ. Let O be the
circumcenter of the triangle ABC and
let Γ be the circumcircle of the triangle BOC. Suppose that Γ intersects the
line segment AB at P different from B, and the line segment AC at Q different from C.
Let ON be a diameter of the
circle Γ. Prove that the
quadrilateral APNQ is a
parallelogram.

Let ABC be an acute
triangle satisfying the condition AB
> BC and AC > BC.
Denote by O and H the circumcenter and the
orthocenter, respectively, of the triangle ABC. Suppose that the circumcircle of the triangle AHC intersects the line AB at M different from A, and
that the circumcircle of the triangle AHB
intersects the line AC at
N different from A. Prove that the circumcenter of the
triangle MNH lies on the line OH:

Let ABC be an acute triangle with <BAC = 30þ. The internal and external angle bisectors of <ABC meet the line AC
at B

_{1}and B_{2}, respectively, and the internal and external angle bisectors of <ACB meet the line AB at C_{1}and C_{2}, respectively. Suppose that the circles with diameters B_{1}B_{2}and C_{1}C_{2}meet inside the triangle ABC at point P. Prove that <BPC = 90þ.
Let P be a point in the interior of a triangle ABC, and let D,E, F be
the point of intersection of the line AP and the side BC of the triangle, of
the line BP and the side CA, and of the line CP and the side AB, respectively.
Prove that the area of the triangle ABC must be 6 if the area of each of the
triangles PFA, PDB and PEC is 1.

Let ABC be an acute triangle. Denote by D the foot of the perpendicular
line drawn from the point A to the side BC, by M the midpoint of BC, and by H
the orthocenter of ABC. Let E be the point of intersection of the circumcircle Γ of the triangle ABC and the half line MH, and F be the point of intersection
(other than E) of the line ED and the circle Γ. Prove that BF / CF = AB / AC must hold. Here we denote by XY the
length of the line segment XY .

Let ABC be an acute triangle with altitudes AD,BE and CF, and let O be
the center of its circumcircle. Show that the segments OA,OF,OB,OD,OC,OE dissect
the triangle ABC into three pairs of triangles that have equal areas.

Let ABCD be a quadrilateral inscribed in a circle ω, and let P be a point on the
extension of AC such that PB and PD are tangent to ω. The tangent at C intersects PD at Q and the line AD at R. Let E be the
second point of intersection between AQ and ω. Prove that B,E,R are collinear.

Circles ω and Ω meet at points A and B. Let M be the midpoint of the arc AB of circle ω (M lies inside Ω). A chord MP of circle ω intersects Ω at Q (Q lies inside ω). Let

*l*_{P}be the tangent line to ω at P, and let*l*_{Q}be the tangent line to Ω at Q. Prove that the circumcircle of the triangle formed by the lines*l*_{P},*l*_{Q}, and AB is tangent to Ω.
by Ilya Bogdanov, Russia and Medeubek Kungozhin,
Kazakhstan

Let ABC be a triangle, and let D be a point on side BC. A line through D
intersects side AB at X and ray AC at Y . The circumcircle of triangle BXD
intersects the circumcircle ω of triangle ABC
again at point Z ≠ B. The lines ZD and
ZY intersect ω again at V and W, respectively. Prove that AB = VW.

by Warut Suksompong,
Thailand

We say that a triangle ABC is great if the following holds: for any point
D on the side BC, if P and Q are the feet of the perpendiculars from D to the lines
AB and AC, respectively, then the reflection of D in the line PQ lies on the circumcircle
of the triangle ABC. Prove that triangle ABC is great if and only if <A =
90◦ and AB = AC.

Let AB and AC be two distinct rays not lying on the same line, and let ω
be a circle with center O that is tangent to ray AC at E and ray AB at F. Let R
be a point on segment EF. The line through O parallel to EF intersects line AB
at P. Let N be the intersection of lines PR and AC, and let M be the
intersection of line AB and the line through R parallel to AC. Prove that line
MN is tangent to ω.

by Warut Suksompong,
Thailand

Let ABC be a triangle with AB < AC. Let D be the intersection point of
the internal bisector of angle BAC and the circumcircle of ABC. Let Z be the intersection
point of the perpendicular bisector of AC with the external bisector of angle <BAC.
Prove that the midpoint of the segment AB lies on the circumcircle of triangle
ADZ.

by Equipo Nicaragua,
Nicaragua

APMO 2018 / 1

Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.

APMO 2019 / 3

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

APMO 2019 / 3

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

source: apmo.ommenlinea.org , www.apmo-official.org

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