geometry problems from Mathematics Regional Olympiad of Mexico Center Zone with aops links in the names
2007-2012, 2014-16, 2018-21
missing 2013
it didn't take place in 2017
Consider the triangle ABC with circumcenter O. Let D be the intersection of the angle bisector of \angle{A} with BC. Show that OA, the perpendicular bisector of AD and the perpendicular to BC passing through D are concurrent.
Consider a triangle ABC with \angle ACB = 2 \angle CAB and \angle ABC> 90 ^ \circ. Consider the perpendicular on AC that passes through A and intersects BC at D, prove that\frac {1} {BC} - \frac {2} {DC} = \frac {1} {CA}
Let ABC be a triangle with incenter I , the line AI intersects BC at L and the circumcircle of ABC at L'. Show that the triangles BLI and L'IB are similar if and only if AC = AB + BL.
In the quadrilateral ABCD, we have AB = AD and \angle B = \angle D = 90 ^ \circ . The points P and Q lie on BC and CD, respectively, so that AQ is perpendicular on DP. Prove that AP is perpendicular to BQ.
Let \Gamma be a circle with the center O and let A, A ^ \prime be two diametrically opposite points in \Gamma. Let P be the midpoint of OA ^ \prime and \ell a line that passes through P, different from the line AA ^ \prime and different from the line perpendicular on AA ^ \prime. Let B and C be the intersection points of \ell with \Gamma, let H be the foot of the altitude from A on BC, let M be the midpoint of BC, and let D be the intersection of the line A ^ \prime M with AH. Show that the angle \angle ADO = 90 ^ \circ .
Let ABC be a triangle and let D be the foot of the altitude from A. Let points E and F on a line through D such that AE is perpendicular to BE, AF is perpendicular to CF, where E and F are points other than the point D. Let M and N be the midpoints of BC and EF, respectively. Prove that AN is perpendicular to NM.
In the acute triangle ABC, \angle BAC is less than \angle ACB . Let AD be a diameter of \omega, the circle circumscribed to said triangle. Let E be the point of intersection of the ray AC and the tangent to \omega passing through B. The perpendicular to AD that passes through E intersects the circle circumscribed to the triangle BCE, again, at the point F. Show that CD is an angle bisector of \angle BCF.
Let ABC be an equilateral triangle and D the midpoint of BC. Let E and F be points on AC and AB respectively such that AF=CE. P=BE \cap CF. Show that \angle APF= \angle BPD.
Let ABC be a triangle and let L, M, N be the midpoints of the sides BC, CA and AB , respectively. The points P and Q lie on AB and BC, respectively; the points R and S are such that N is the midpoint of PR and L is the midpoint of QS. Show that if PS and QR are perpendicular, then their intersection lies on in the circumcircle of triangle LMN.
Given a circle C and a diameter AB in it, mark a point P on AB different from the ends. In one of the two arcs determined by AB choose the points M and N such that \angle APM = 60 ^ \circ = \angle BPN. The segments MP and NP are drawn to obtain three curvilinear triangles; APM , MPN and NPB (the sides of the curvilinear triangle APM are the segments AP and PM and the arc AM). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of C.
In the parallelogram ABCD, \angle BAD =60 ^ \circ. Let E be the intersection point of the diagonals. The circle circumscribed to the triangle ACD intersects the line AB at the point K (different from A), the line BD at the point P (different from D), and to the line BC in L (different from C). The line EP intersects the circumscribed circle of the triangle CEL at the points E and M. Show that the triangles KLM and CAP are congruent.
On an acute triangle ABC we draw the internal bisector of \angle ABC, BE, and the altitude AD, (D on BC), show that <CDE it's bigger than 45 degrees.
2013 missing
Let AB be a triangle and \Gamma the excircle, relative to the vertex A, with center D. The circle \Gamma is tangent to the lines AB and AC at E and F, respectively. Let P and Q be the intersections of EF with BD and CD, respectively. If O is the point of intersection of BQ and CP, show that the distance from O to the line BC is equal to the radius of the inscribed circle in the triangle ABC.
Let ABCD be a square and let M be the midpoint of BC. Let C ^ \prime be the reflection of C wrt to DM. The parallel to AB passing through C ^ \prime intersects AD at R and BC at S. Show that \frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}
In the triangle ABC, we have that \angle BAC is acute. Let \Gamma be the circle that passes through A and is tangent to the side BC at C. Let M be the midpoint of BC and let D be the other point of intersection of \Gamma with AM. If BD cuts back to \Gamma at E, show that AC is the bisector of \angle BAE.
In the triangle ABC, we have that M and N are points on AB and AC, respectively, such that BC is parallel to MN. A point D is chosen inside the triangle AMN. Let E and F be the points of intersection of MN with BD and CD, respectively. Show that the line joining the centers of the circles circumscribed to the triangles DEN and DFM is perpendicular to AD.
Let ABC be a triangle with orthocenter H and \ell a line that passes through H, and is parallel to BC. Let m and n be the reflections of \ell on the sides of AB and AC, respectively, m and n are intersect at P. If HP and BC intersect at Q, prove that the parallel to AH through Q and AP intersect at the circumcenter of the triangle ABC.
Let A be one of the two points where the circles whose centers are the points M and N intersect. The tangents in A to such circles intersect them again in B and C, respectively. Let P be a point such that the quadrilateral AMPN is a parallelogram. Show that P is the circumcenter of triangle ABC.
2017 didn't take place
Let \vartriangle ABC be a triangle and let \Gamma its circumscribed circle. Let M be the midpoint of the side BC and let D be the point of intersection of the line AM with \Gamma. By D a straight line is drawn parallel to BC, which intersects \Gamma at a point E. Let N be the midpoint of the segment AE and let P be the point of intersection of CN with AM. Show that AP = PC.
Let \vartriangle ABC be a triangle with orthocenter H and altitudes AD, BE and CF. Let D', E' and F' be the intersections of the heights AD, BE and CF, respectively, with the circumcircle of \vartriangle ABC , so that they are different points from the vertices of triangle \vartriangle ABC. Let L, M and N be the midpoints of BC, AC and AB, respectively. Let P, Q and R be the intersections of the circumcircle with LH, MH and NH, respectively, such that P and A are on opposite sides of BC, Q and A are on opposite sides of AC and R and C are on opposite sides of AB. Show that there exists a triangle whose sides have the lengths of the segments D' P, E'Q, and F'R.
Let ABC be an acute triangle and D a point on the side BC such that \angle BAD = \angle DAC. The circumcircles of the triangles ABD and ACD intersect the segments AC and AB at E and F, respectively. The internal bisectors of \angle BDF and \angle CDE intersect the sides AB and AC at P and Q, respectively. Points X and Y are chosen on the side BC such that PX is parallel to AC and QY is parallel to AB. Finally, let Z be the point of intersection of BE and CF. Prove that ZX = ZY.
Let ABC be a triangle with \angle BAC> 90 ^ \circ and D a point on BC. Let E and Fbe the reflections of the point D about AB and AC, respectively. Suppose that BE and CF intersect at P. Show that AP passes through the circumcenter of triangle ABC.
In an acute triangle ABC, an arbitrary point P is chosen on the altitude AH. The points E and F are the midpoints of AC and AB, respectively. The perpendiculars from E on CP and from F on BP intersect at the point K. Show that KB = KC.
Let \Gamma_1 be a circle with center O and A a point on it. Consider the circle \Gamma_2 with center at A and radius AO. Let P and Q be the intersection points of \Gamma_1and \Gamma_2. Consider the circle \Gamma_3 with center at P and radius PQ. Let C be the second intersection point of \Gamma_3 and \Gamma_1. The line OP cuts \Gamma_3 at R and S, with R outside \Gamma_1. RC cuts \Gamma_1 into B. CS cuts \Gamma_1 into D. Show that ABCD is a square.
Let W,X,Y and Z be points on a circumference \omega with center O, in that order, such that WY is perpendicular to XZ; T is their intersection. ABCD is the convex quadrilateral such that W,X,Y and Z are the tangency points of \omega with segments AB,BC,CD and DA respectively. The perpendicular lines to OA and OB through A and B, respectively, intersect at P; the perpendicular lines to OB and OC through B and C, respectively, intersect at Q, and the perpendicular lines to OC and OD through C and D, respectively, intersect at R. O_1 is the circumcenter of triangle PQR. Prove that T,O and O_1 are collinear.
by CDMX
Let ABCD be a parallelogram. Half-circles \omega_1,\omega_2,\omega_3 and \omega_4 with diameters AB,BC,CD and DA, respectively, are erected on the exterior of ABCD. Line l_1 is parallel to BC and cuts \omega_1 at X, segment AB at P, segment CD at R and \omega_3 at Z. Line l_2 is parallel to AB and cuts \omega_2 at Y, segment BC at Q, segment DA at S and \omega_4 at W. If XP\cdot RZ=YQ\cdot SW, prove that PQRS is cyclic.
by José Alejandro Reyes González
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