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Mexico 1987- 2021 (OMM) 70p

geometry problems from Mexican Mathematical Olympiads
with aops links in the names

Olimpiada Mexicana de Matemáticas (OMM)

collected inside aops here

1987-  2021

1987 Mexican P3
Consider two  lines \ell and \ell ' and a fixed point P equidistant from these lines. What is the locus of projections M of P on AB, where A is on  \ell , B on  \ell ' , and angle \angle APB is right?

1987 Mexican P5
In a right triangle ABC, M is a point on the hypotenuse BC and P and Q the projections of M on AB and AC respectively. Prove that for no such point M do the triangles BPM, MQC and the rectangle AQMP have the same area.

1987 Mexican P8
(a) Three lines l,m,n in space pass through point S. A plane perpendicular to m intersects l,m,n at A,B,C respectively. Suppose that \angle ASB = \angle BSC = 45^o and \angle ABC = 90^o. Compute \angle ASC.
(b) Furthermore, if a plane perpendicular to l intersects  l,m,n  at P,Q,R respectively and SP = 1, find the sides of triangle PQR.

1988 Mexican P3
Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

1988 Mexican P6
Consider two fixed points B,C on a circle w. Find the locus of the incenters of all triangles ABC when point A describes w.

1988 Mexican P8
Compute the volume of a regular octahedron circumscribed about a sphere of radius 1.

1989 Mexican P1
In a triangle ABC the area is 18, the length AB is 5, and the medians from A and B are orthogonal. Find the lengths of the sides BC,AC.

1989 Mexican P5
Let C_1 and C_2 be two tangent unit circles inside a circle C of radius 2. Circle C_3 inside C is tangent to the circles C,C_1,C_2, and circle C_4 inside C is tangent to C,C_1,C_3. Prove that the centers of C,C_1,C_3 and C_4 are vertices of a rectangle.

1990 Mexican P2
ABC is a triangle with \angle B = 90^o and altitude BH. The inradii of ABC, ABH, CBH are r, r_1, r_2. Find a relation between them.

1990 Mexican P6
ABC is a triangle with \angle C = 90^o. E is a point on AC, and F is the midpoint of EC. CH is an altitude. I is the circumcenter of AHE, and G is the midpoint of BC. Show that ABC and IGF are similar.

1991 Mexican P3
Four balls of radius 1 are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?

1991 Mexican P4
The diagonals AC and BD of a convex quarilateral ABCD are orthogonal. Let M,N,R,S be the midpoints of the sides AB,BC,CD and DA respectively, and let W,X,Y,Z be the projections of the points M,N,R and S on the lines CD,DA,AB and BC, respectively. Prove that the points M,N,R,S,W,X,Y and Z lie on a circle.

1992 Mexican P1
The tetrahedron OPQR has the \angle POQ = \angle POR = \angle QOR = 90^o. X, Y, Z are the midpoints of PQ, QR and RP. Show that the four faces of the tetrahedron OXYZ have equal area.

1992 Mexican P6
ABCD is a rectangle. I is the midpoint of CD. BI meets AC at M. Show that the line DM passes through the midpoint of BC. E is a point outside the rectangle such that AE = BE and \angle AEB = 90^o. If BE = BC = x, show that EM bisects \angle AMB. Find the area of AEBM in terms of x.

1993 Mexican P1
ABC is a triangle with \angle A = 90^o. Take E such that the triangle AEC is outside ABC and AE = CE and \angle AEC = 90^o. Similarly, take D so that ADB is outside ABC and similar to AEC. O is the midpoint of BC. Let the lines OD and EC meet at D', and the lines OE and BD meet at E'. Find area DED'E' in terms of the sides of ABC.

1993 Mexican P5
OA, OB, OC are three chords of a circle. The circles with diameters OA, OB meet again at Z, the circles with diameters OB, OC meet again at X, and the circles with diameters OC, OA meet again at Y. Show that X, Y, Z are collinear.

1994 Mexican P3
ABCD is a parallelogram. Take E on the line AB so that BE = BC and B lies between A and E. Let the line through C perpendicular to BD and the line through E perpendicular to AB meet at F. Show that \angle DAF = \angle BAF.

1994 Mexican P5
ABCD is a convex quadrilateral. Take the 12 points which are the feet of the altitudes in the triangles ABC, BCD, CDA, DAB. Show that at least one of these points must lie on the sides of ABCD.

1995 Mexican P3
A, B, C, D are consecutive vertices of a regular 7-gon. AL and AM are tangents to the circle center C radius CB. N is the point of intersection of AC and BD. Show that L, M, N are collinear.

1995 Mexican P5
ABCDE is a convex pentagon such that the triangles ABC, BCD, CDE, DEA and EAB have equal areas. Show that (1/4) area (ABCDE) < area (ABC) < (1/3) area (ABCDE).

1996 Mexican P1
Let P and Q be the points on the diagonal BD of a quadrilateral ABCD such that BP = PQ = QD. Let AP and BC meet at E, and let AQ meet DC at F.
(a) Prove that if ABCD is a parallelogram, then E and F are the midpoints of the corresponding sides.
(b) Prove the converse of (a).

1996 Mexican P6
In a triangle ABC with AB < BC < AC, points A' ,B' ,C' are such that AA' \perp BC and AA' = BC, BB' \perp  CA and BB'=CA, and CC' \perp AB and CC'= AB, as shown on the picture. Suppose that \angle AC'B is a right angle. Prove that the points A',B' ,C' are collinear.

1997 Mexican P2
In a triangle ABC, P and P' are points on side BC, Q on side CA, and R on side AB, such that \frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B} . Let G be the centroid of triangle ABC and K be the intersection point of AP' and RQ. Prove that points P,G,K are collinear.

1997 Mexican P5
Let P,Q,R be points on the sides BC,CA,AB respectively of a triangle ABC. Suppose that BQ and CR meet at A', AP and CR meet at B', and AP and BQ meet at C', such that AB' = B'C', BC' =C'A', and CA'= A'B'. Compute the ratio of the area of \triangle PQR to the area of \triangle ABC.

1998 Mexican P2
Rays l and m forming an angle of a are drawn from the same point. Let P be a fixed point on l. For each circle C tangent to l at P and intersecting m at Q and R, let T be the intersection point of the bisector of angle QPR with C. Describe the locus of T and justify your answer.

1998 Mexican P5
The tangents at points B and C on a given circle meet at point A. Let Q be a point on segment AC and let BQ meet the circle again at P. The line through Q parallel to AB intersects BC at J. Prove that PJ is parallel to AC if and only if BC^2 = AC\cdot QC.

1999 Mexican P3
A point P is given inside a triangle ABC. Let D,E,F be the midpoints of AP,BP,CP, and let L,M,N be the intersection points of BF and CE, AF and CD, AE and BD, respectively.
(a) Prove that the area of hexagon DNELFM is equal to one third of the area of triangle ABC.
(b) Prove that DL,EM, and FN are concurrent.

1999 Mexican P5
In a quadrilateral ABCD with AB  // CD, the external bisectors of the angles at B and C meet at P, while the external bisectors of the angles at A and D meet at Q. Prove that the length of PQ equals the semiperimeter of ABCD.

2000 Mexican P1
Circles A,B,C,D are given on the plane such that circles A and B are externally tangent at P, B and C at Q, C and D at R, and D and A at S. Circles A and C do not meet, and so do not B and D.
(a) Prove that the points P,Q,R,S lie on a circle.
(b) Suppose that A and C have radius 2, B and D have radius 3, and the distance between the centers of A and C is 6. Compute the area of the quadrilateral PQRS.

Let ABC be a triangle with \angle B > 90^o such that there is a point H on side AC with AH = BH and BH perpendicular to BC. Let D and E be the midpoints of AB and BC respectively. A line through H parallel to AB cuts DE at F. Prove that \angle BCF = \angle ACD.

ABCD is a cyclic quadrilateral. M is the midpoint of CD. The diagonals meet at P. The circle through P which touches CD at M meets AC again at R and BD again at Q. The point S on BD is such that BS = DQ. The line through S parallel to AB meets AC at T. Show that AT = RC.

ABC is a triangle with AB < AC and \angle A = 2 \angle C. D is the point on AC such that CD = AB. Let L be the line through B parallel to AC. Let L meet the external bisector of \angle A at M and the line through C parallel to AB at N. Show that MD = ND.

ABCD is a parallelogram. K is the circumcircle of ABD. The lines BC and CD meet K again at E and F. Show that the circumcenter of CEF lies on K.

Let ABCD be a quadrilateral with \measuredangle DAB=\measuredangle ABC=90^{\circ}. Denote by M the midpoint of the side AB, and assume that \measuredangle CMD=90^{\circ}. Let K be the foot of the perpendicular from the point M to the line CD. The line AK meets BD at P, and the line BK meets AC at Q. Show that \angle{AKB}=90^{\circ} and \frac{KP}{PA}+\frac{KQ}{QB}=1.

A, B, C are collinear with B betweeen A and C. K_{1} is the circle with diameter AB, and K_{2} is the circle with diameter BC. Another circle touches AC at B and meets K_{1} again at P and K_{2} again at Q. The line PQ meets K_{1} again at R and K_{2} again at S. Show that the lines AR and CS meet on the perpendicular to AC at B.

The quadrilateral ABCD has AB parallel to CD. P is on the side AB and Q on the side CD such that \frac{AP}{PB}= \frac{DQ}{CQ}. M is the intersection of AQ and DP, and N is the intersection of PC and QB. Find MN in terms of AB and CD.

Let Z and Y be the tangency points of the incircle of the triangle ABC with the sides AB and CA, respectively. The parallel line to Y Z through the midpoint M of BC, meets CA in N. Let L be the point in CA such that NL = AB (and L on the same side of N than A). The line ML meets AB in K. Prove that KA = NC.

Let \omega_1 and \omega_2 be two circles such that the center O of \omega_2 lies in \omega_1. Let C and D be the two intersection points of the circles. Let A be a point on \omega_1 and let B be a point on \omega_2 such that AC is tangent to \omega_2 in C and BC is tangent to \omega_1 in C. The line segment AB meets \omega_2 again in E and also meets \omega_1 again in F. The line CE meets \omega_1 again in G and the line CF meets the line GD in H. Prove that the intersection point of GO and EH is the center of the circumcircle of the triangle DEF.

Let O be the center of the circumcircle of an acute triangle ABC, let P be any point inside the segment BC. Suppose the circumcircle of triangle BPO intersects the segment AB at point R and the circumcircle of triangle COP intersects CA at point Q.
(i) Consider the triangle PQR, show that it is similar to triangle ABC and that O is its orthocenter.
(ii) Show that the circumcircles of triangles BPO, COP, PQR have the same radius.

Let ABC be a triangle and AD be the angle bisector of <BAC, with D on BC. Let E be a point on segment BC such that BD = EC. Through E draw l a parallel line to AD and let P be a point in l inside the triangle. Let G be the point where BP intersects AC and F be the point where CP intersects AB. Show BF = CG.

Let ABC be a right triangle with a right angle at A, such that AB < AC. Let M be the midpoint of BC and D the intersection of AC with the perpendicular on BC passing through M. Let E be the intersection of the parallel to AC that passes through M, with the perpendicular on BD passing through B. Show that the triangles  AEM and MCA are similar if and only if \angle ABC = 60^o.

Let ABC be an acute triangle , with altitudes AD,BE and CF. Circle of diameter AD intersects the sides AB,AC in M,N respevtively. Let P,Q be the intersection points of AD with EF and MN respectively. Show that Q is the midpoint of PD.

2007 Mexican P2
Given an equilateral \triangle ABC, find the locus of points P such that \angle APB=\angle BPC.

2007 Mexican P6
Let ABC be a triangle with AB>AC>BC. Let D be a point on AB such that CD=BC, and let M be the midpoint of AC. Show that BD=AC \longleftrightarrow \angle BAC=2\angle ABM.

2008 Mexican P2
Consider a circle \Gamma, a point A on its exterior, and the points of tangency B and C from A to \Gamma. Let P be a point on the segment AB, distinct from A and B, and let Q be the point on AC such that PQ is tangent to \Gamma. Points R and S are on lines AB and AC, respectively, such that PQ\parallel RS and RS is tangent to \Gamma as well. Prove that [APQ]\cdot[ARS] does not depend on the placement of point P.


The internal angle bisectors of A, B, and C in \triangle ABC concur at I and intersect the circumcircle of \triangle ABC at L, M, and N, respectively. The circle with diameter IL intersects BC at D and E; the circle with diameter IM intersects CA at F and G; the circle with diameter IN intersects AB at H and J. Show that D, E, F, G, H, and J are concyclic.

2009 Mexican P1
In \triangle ABC, let D be the foot of the altitude from A to BC. A circle centered at D with radius AD intersects lines AB and AC at P and Q, respectively. Show that \triangle AQP\sim\triangle ABC.

2009 Mexican P5
Consider a triangle ABC and a point M on side BC. Let P be the intersection of the perpendiculars from M to AB and from B to BC, and let Q be the intersection of the perpendiculars from M to AC and from C to BC. Show that PQ is perpendicular to AM if and only if M is the midpoint of BC.

2010 Mexican P3
Let \mathcal{C}_1 and \mathcal{C}_2 be externally tangent at a point A. A line tangent to \mathcal{C}_1 at B intersects \mathcal{C}_2 at C and D; then the segment AB is extended to intersect \mathcal{C}_2 at a point E. Let F be the midpoint of arc CD that does not contain E, and let H be the intersection of BF with \mathcal{C}_2. Show that CD, AF, and EH are concurrent.

Let ABC be an acute triangle with AB\neq AC, M be the median of BC, and H be the orthocenter of \triangle ABC. The circumcircle of B, H, and C intersects the median AM at N. Show that \angle ANH=90^\circ

2011 Mexican P2
Let ABC be an acute triangle and \Gamma its circumcircle. Let l be the line tangent to \Gamma at A. Let D and E be the intersections of the circumference with center B and radius AB with lines l and AC, respectively. Prove the orthocenter of ABC lies on line DE.

Let \mathcal{C}_1 and \mathcal{C}_2 be two circumferences intersecting at points A and B. Let C be a point on line AB such that B lies between A and C. Let P and Q be points on \mathcal{C}_1 and \mathcal{C}_2 respectively such that CP and CQ are tangent to \mathcal{C}_1 and \mathcal{C}_2 respectively, P is not inside \mathcal{C}_2 and Q is not inside \mathcal{C}_1. Line PQ cuts \mathcal{C}_1 at R and \mathcal{C}_2 at S, both points different from P, Q and B. Suppose CR cuts \mathcal{C}_1 again at X and CS cuts \mathcal{C}_2 again at Y. Let Z be a point on line XY. Prove SZ is parallel to QX if and only if PZ is parallel to RX.

2012 Mexican P1
Let \mathcal{C}_1 be a circumference with center O, P a point on it and \ell the line tangent to \mathcal{C}_1 at P. Consider a point Q on \ell different from P, and let \mathcal{C}_2 be the circumference passing through O, P and Q. Segment OQ cuts \mathcal{C}_1 at S and line PS cuts \mathcal{C}_2 at a point R diffferent from P. If r_1 and r_2 are the radii of \mathcal{C}_1 and \mathcal{C}_2 respectively, Prove
\frac{PS}{SR} = \frac{r_1}{r_2}.

Consider an acute triangle ABC with circumcircle \mathcal{C}. Let H be the orthocenter of ABC and M the midpoint of BC. Lines AH, BH and CH cut \mathcal{C} again at points D, E, and F respectively; line MH cuts \mathcal{C} at J such that H lies between J and M. Let K and L be the incenters of triangles DEJ and DFJ respectively. Prove KL is parallel to BC.

2013 Mexican P2
Let ABCD be a parallelogram with the angle at A obtuse. Let P be a point on segment BD. The circle with center P passing through A cuts line AD at A and Y and cuts line AB at A and X. Line AP intersects BC at Q and CD at R. Prove \angle XPY = \angle XQY + \angle XRY.

Let A_1A_2 ... A_8 be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each i = 1, ... , 8, define B_i as the intersection between segments A_iA_{i+4} and A_{i-1}A_{i+1}, where A_{j+8} = A_j and B_{j+8} = B_j for all j. Show some number i, amongst 1, 2, 3, and 4 satisfies \frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}

2014 Mexican P3
Let \Gamma_1 be a circle and P a point outside of \Gamma_1. The tangents from P to \Gamma_1 touch the circle at A and B. Let M be the midpoint of PA and \Gamma_2 the circle through P, A and B. Line BM cuts \Gamma_2 at C, line CA cuts \Gamma_1 at D, segment DB cuts \Gamma_2 at E and line PE cuts \Gamma_1 at F, with E in segment PF. Prove lines AF, BP, and CE are concurrent. 

Let ABCD be a rectangle with diagonals AC and BD. Let E be the intersection of the bisector of \angle CAD with segment CD, F on CD such that E is midpoint of DF, and G on BC such that BG = AC (with C between B and G). Prove that the circumference through D, F and G is tangent to BG

2015 Mexican P1
Let ABC be an acuted-angle triangle and let H be it's orthocenter. Let PQ be a segment through H such that P lies on AB and Q lies on AC and such that \angle PHB= \angle CHQ. Finally, in the circumcircle of \triangle ABC, consider M such that M is the mid point of the arc BC that doesn't contain A. Prove that MP=MQ

by Eduardo Velasco/Marco Figueroa
Let I be the incenter of an acute-angled triangle ABC. Line AI cuts the circumcircle of BIC again at E. Let D be the foot of the altitude from A to BC, and let J be the reflection of I across BC. Show D, J and E are collinear.

Let C_1 and C_2 be two circumferences externally tangents at S such that the radius of C_2 is the triple of the radius of C_1. Let a line be tangent to C_1 at P \neq S and to C_2 at Q \neq S. Let T be a point on C_2 such that QT is diameter of C_2. Let the angle bisector of \angle SQT meet ST at R. Prove that QR=RT

Let ABCD a quadrilateral inscribed in a circumference, l_1 the parallel to BC through A, and l_2 the parallel to AD through B. The line DC intersects l_1 and l_2 at E and F, respectively. The perpendicular to l_1 through A intersects BC at P, and the perpendicular to l_2 through B cuts AD at Q. Let \Gamma_1 and \Gamma_2 be the circumferences that pass through the vertex of triangles ADE and BFC, respectively. Prove that \Gamma_1 and \Gamma_2 are tangent to each other if and only if DP is perpendicular to CQ.

Let ABC be an acute triangle with orthocenter H. The circle through B, H, and C intersects lines AB and AC at D and E respectively, and segment DE intersects HB and HC at P and Q respectively. Two points X and Y, both different from A, are located on lines AP and AQ respectively such that X, H, A, B are concyclic and Y, H, A, C are concyclic. Show that lines XY and BC are parallel.

On a circle \Gamma, points A, B, N, C, D, M are chosen in a clockwise order in such a way that N and M are the midpoints of clockwise arcs BC and AD respectively. Let P be the intersection of AC and BD, and let Q be a point on line MB such that PQ is perpendicular to MN. Point R is chosen on segment MC such that QB = RC, prove that the midpoint of QR lies on AC.

Let A and B be two points on a line \ell, M the midpoint of AB, and X a point on segment AB other than M. Let \Omega be a semicircle with diameter AB. Consider a point P on \Omega and let \Gamma be the circle through P and X that is tangent to AB. Let Q be the second intersection point of \Omega and \Gamma. The internal angle bisector of \angle PXQ intersects \Gamma at a point R. Let Y be a point on \ell such that RY is perpendicular to \ell. Show that MX > XY

Let ABC be an acute-angled triangle with circumference \Omega. Let the angle bisectors of \angle B and \angle C intersect \Omega again at M and N. Let I be the intersection point of these angle bisectors. Let M' and N' be the respective reflections of M and N in AC and AB. Prove that the center of the circle passing through I, M', N' lies on the altitude of triangle ABC from A.

Let H be the orthocenter of acute-angled triangle ABC and M be the midpoint of AH. Line BH cuts AC at D. Consider point E such that BC is the perpendicular bisector of DE. Segments CM and AE intersect at F. Show that BF is perpendicular to CM

Let ABC be a triangle such that \angle BAC = 45^{\circ}. Let H,O be the orthocenter and circumcenter of ABC, respectively. Let \omega be the circumcircle of ABC and P the point on \omega such that the circumcircle of PBH is tangent to BC. Let X and Y be the circumcenters of PHB and PHC respectively. Let O_1,O_2 be the circumcenters of PXO and PYO respectively. Prove that O_1 and O_2 lie on AB and AC, respectively.

Let ABC be a triangle with incenter I. The line BI meets AC at D. Let P be a point on CI such that DI=DP (P\ne I), E the second intersection point of segment BC with the circumcircle of ABD and Q the second intersection point of line EP with the circumcircle of AEC. Prove that \angle PDQ=90^\circ.
Let ABC be a triangle with \angle ACB > 90^{\circ}, and let D be a point on BC such that AD is perpendicular to BC. Consider the circumference \Gamma with with diameter BC. A line \ell passes through D and is tangent to \Gamma at P, cuts AC at M (such that M is in between A and C), and cuts the side AB at N. Prove that M is the midpoint of DP if and only if N is the midpoint of AB.

Let ABC be an acutangle scalene triangle with \angle BAC = 60^{\circ} and orthocenter H. Let \omega_b be the circumference passing through H and tangent to AB at B, and \omega_c the circumference passing through H and tangent to AC at C.
 Prove that \omega_b and \omega_c only have H as common point.
Prove that the line passing through H and the circumcenter O of triangle ABC is a common tangent to \omega_b and \omega_c.


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