### Asian Pacific 1989 - 2019 (APMO) 41p

geometry problems from Asian Pacific Mathematics Olympiad (APMO)
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)

1989 - 2019

Let A1, A2, A3 be three points in the plane, and for convenience, let A4 = A1, A5 = A2. For n = 1, 2, and 3, suppose that Bn is the midpoint of AnAn+1, and suppose that Cn is the midpoint of AnBn. Suppose that AnCn+1 and BnAn+2  meet at Dn, and that AnBn+1 and CnAn+2  meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3.

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 C1, C2 be two circles with centres O1, O2 and radii r1, r2 respectively, so that each circle Ci is internally tangent to C at Ai and so that C1, C2 are externally tangent to each other at A. Prove that the three lines OA, O1A2, and O2A1 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 CA2 = 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 P1, P2, P3, P4 be four points on a circle, and let I1 be the incentre of the triangle P2P3P4 ,I2 be the incentre of the triangle P1P3P4 , I3 be the incentre of the triangle P1P2P4 , I4 be the incentre of the triangle P1P2P3. Prove that I1, I2, I3, I4 are the vertices of a rectangle.

Triangle A1A2A3 has a right angle at A3. A sequence of points is now defined by the following  iterative process, where n is a positive integer. From An (n ≥ 3), a perpendicular line is  drawn to meet An-2 An-1 at An+1.
(a) Prove that if this process is continued indefinitely, then one and only one point P is interior to every triangle An-2 An-1 An  , n ≥ 3.
(b) Let A1 and A3 be fixed points. By considering all possible locations of A2 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 Γ1 and Γ2 be two circles intersecting at P and Q. The common tangent, closer to P, of Γ1 and Γ2 touches Γ1 at A and Γ2 at B. The tangent of Γ1 at P meets Γ2 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, X1 , … ,  Xn in the plane with AB ≠ CD that satisfy the following condition: for each i = 1, 2, … ,  n triangles ABXi  and CDXi  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 l1 and l2, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect l1 at E and F respectively. Also, sides CB and CD intersect l2 at G and H respectively. Let the perimeters of ∆AEF and ∆CGH be m1 and m2 respectively. Prove that no matter how the square was placed, m1 + m2 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 O1 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 O1 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 O2 be the circle tangent to the line BC at Q and tangent to the line segment AC. Prove that the circle O2 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 Γ1, 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 A1,B1,A2,B2,A3,B3 as follows: For each i = 1, 2, 3, Ai, Bi are distinct points on the circle ¡i such that the lines PAi and PBi are both tangents to Γi. Call the point P exceptional if, from the construction, three lines A1B1, A2B2, A3B3 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 B1 and B2, respectively, and the internal and external angle bisectors of <ACB meet the line AB at C1 and C2, respectively. Suppose that the circles with diameters B1B2 and C1C2 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 lP be the tangent line to ω at P, and let lQ be the tangent line to  Ω at Q. Prove that the circumcircle of the triangle formed by the lines lP, lQ, 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 $A$.