Centroamerican 1999 - 2018 (OMCC) 37p

geometry problems from Central American and Caribbean Mathematical Olympiads (OMCC)
with aops links in the names

 Olimpiada Matemática de Centroamérica y el Caribe

1999 - 2018
ABCD is a trapezoid with AB parallel to CD. M is the midpoint of AD, MCB = 150o, BC = x and MC = y. Find area ABCD in terms of x and y.

ABCDE is a convex pentagon. Show that the centroids of the 4 triangles ABE, BCE, CDE, DAE from a parallelogram with whose area is 2/9 area ABCD.

ABC is acute-angled. The circle diameter AC meets AB again at F, and the circle diameter AB meets AC again at E. BE meets the circle diameter AC at P, and CF meets the circle diameter AB at Q. Show that AP = AQ.

C and D are points on the circle diameter AB such that AQB = 2COD. The tangents at C and D meet at P. The circle has radius 1. Find the distance of P from its center.

ABC is acute-angled. AD and BE are altitudes. SBDE ≤ SDEA ≤ SEAB ≤ SABD. Show that the triangle is isosceles.

ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and FAE = 60o. Find FEA.

AB is a diameter of a circle. C and D are points on the tangent at B on opposite sides of B. AC, AD meet the circle again at E, F respectively. CF, DE meet the circle again at G, H respectively. Show that AG = AH.

Two circles meet at P and Q. A line through P meets the circles again at A and A'. A parallel line through Q meets the circles again at B and B'. Show that PBB' and QAA' have equal perimeters.

ABC is a triangle, and E and F are points on the segments BC and CA respectively, such that           CE / CB + CF/ CA = 1 and CEF =CAB. Suppose that M is the midpoint of EF and G is the point of intersection between CM and AB. Prove that triangle FEG is similar to triangle ABC.

Let ABCD be a trapezium such that AB || CD and AB +CD = AD. Let P be the point on AD such that AP = AB and PD = CD.
a) Prove that BPC = 90o.
b) Q is the midpoint of BC and R is the point of intersection between the line AD and the circle passing through the points  B,A and Q. Show that the points B, P,R and C are concyclic.

Let ABC be a triangle. P, Q and R are the points of contact of the incircle with sides AB, BC and CA, respectively. Let L, M and N be the feet of the altitudes of the triangle PQR from R, P and Q, respectively.
a) Show that the lines AN, BL and CM meet at a point.
b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle PQR.
Aarn Ramrez, El Salvador

Let ABC be a triangle, H the orthocenter and M the midpoint of AC. Let ℓ be the parallel through M to the bisector of AHC. Prove that ℓ divides the triangle in two parts of equal perimeters.

Pedro Marrone, Panam

Let Γ and Γ΄ be two congruent circles centered at O and O΄, respectively, and let A be one of their two points of intersection. B is a point on Γ, C is the second point of intersection of AB and Γ΄, and D is a point on Γ΄  such that OBDO΄  is a parallelogram. Show that the length of CD does not depend on the position of B.

Let ABCD be a convex quadrilateral. I = AC Ç BD, and E, H, F and G are points on AB, BC, CD and DA respectively, such that EF Ç GH = I. If M = EGÇ AC, N = HF Ç  AC, show that 
AM / IM  · IN / CN =  IA / IC .
In a triangle ABC, the angle bisector of A and the cevians BD and CE concur at a point P inside the triangle. Show that the quadrilateral ADPE has an incircle if and only if AB = AC.

Consider a circle S, and a point P outside it. The tangent lines from P meet S at A and B, respectively. Let M be the midpoint of AB. The perpendicular bisector of AM meets S in a point C lying inside the triangle ABP. AC intersects PM at G, and PM meets S in a point D lying outside the triangle ABP. If BD is parallel to AC, show that G is the centroid of the triangle ABP.
Arnoldo Aguilar (El Salvador)
Let ABCD be a convex quadrilateral inscribed in a circumference centered at O such that AC is a diameter. Parallelograms DAOE and BCOF are constructed. Show that if E and F lie on the circumference then ABCD is a rectangle.
Let ABC be an acute triangle. Take points P and Q inside AB and AC, respectively, such that BPQC is cyclic. The circumcircle of ABQ intersects BC again in S and the circumcircle of APC intersects BC again in R, PR and QS intersect again in L. Prove that the intersection of AL and BC does not depend on the selection of P and Q.

Two circles Γ1 and Γ2 intersect at points  A and  B. Consider a circle  Γ contained in Γ1 and Γ2, which is tangent to both of them at  D and  E respectively. Let C be one of the intersection points of line  AB with  Γ,  F be the intersection of line  EC with Γ2and  G be the intersection of line  DC with Γ1. Let  H and  I be the intersection points of line  ED with Γ1 and Γ2respectively. Prove that  F,  G,  H and  I are on the same circle.

Given an acute and scalene triangle  ABC, let  H be its orthocenter,  O its circumcenter,  E and  F the feet of the altitudes drawn from  B and  C, respectively. Line  AO intersects the circumcircle of the triangle again at point  G and segments  FE and  BC at points  X and  Y respectively. Let  Z be the point of intersection of line  AH and the tangent line to the circumcircle at  G. Prove that  HX is parallel to  YZ.

Let ABC be a triangle and L, M, N be the midpoints of BC, CA and AB, respectively. The tangent to the circumcircle of ABC at A intersects LM and LN at P and Q, respectively. Show that CP is parallel to BQ.

Let Γ and Γ1 be two circles internally tangent at A, with centers O and O1 and radii r and r1, respectively (r > r1). B is a point diametrically opposed to A in Γ, and C is a point on Γ such that BC is tangent to Γ1 at P. Let A′ the midpoint of BC. Given that O1A′ is parallel to AP, find the ratio r / r1.

In a scalene triangle ABC, D is the foot of the altitude through A, E is the intersection of AC with the bisector of ABC and F is a point on AB. Let O the circumcenter of ABC and  X = AD Ç BE, Y = BE Ç CF, Z = CF Ç AD. If XYZ is an equilateral triangle, prove that one of the triangles OXY , OY Z, OZX must be equilateral.

Let ABC be an acute triangle and D, E, F be the feet of the altitudes through A, B, C respectively. Call Y and Z the feet of the perpendicular lines from B and C to FD and DE, respectively. Let F1 be the symmetric of F with respect to E and E1 be the symmetric of E with respect to F. If  3EF = FD + DE, prove that BZF1 = CY E1.

Let γ be the circumcircle of the acute triangle ABC. Let P be the midpoint of the minor arc BC. The parallel to AB through P cuts BC,AC and γ  at points R, S and T, respectively. Let K ≡ AP Ç BT and L ≡ BS Ç AR. Show that KL passes through the midpoint of AB if and only if CS = PR.
Let ABC be a triangle with AB < BC, and let E and F be points in AC and AB such that BF = BC = CE, both on the same halfplane as A with respect to BC. Let G be the intersection of BE and CF. Let H be a point in the parallel through G to AC such that HG = AF (with H and C in opposite halfplanes with respect to BG). Show that EHG = BAC / 2 .

Let ABCD be a convex quadrilateral and let M be the midpoint of side AB. The circle passing through D and tangent to AB at A intersects the segment DM at E. The circle passing through C and tangent to AB at B intersects the segment CM at F. Suppose that the lines AF and BE intersect at a point which belongs to the perpendicular bisector of side AB. Prove that A, E, and
C are collinear if and only if B, F, and D are collinear.

Let ABC be an acute triangle and let Γ be its circumcircle. The bisector of \A intersects BC at D, Γ at K (different from A), and the line through B tangent to Γ at X. Show that K is the midpoint of AX if and only if AD  / DC = √2.

Let ABCD be a trapezoid with bases AB and CD, inscribed in a circle of center O. Let P be the intersection of the lines BC and AD. A circle through O and P intersects the segments BC and AD at interior points F and G, respectively. Show that BF = DG.

Points A, B, C and D are chosen on a line in that order, with AB and CD greater than BC. Equilateral triangles APB, BCQ and CDR are constructed so that P, Q and R are on the same side with respect to AD. If PQR = 120o, show that 1 / AB +1 / CD = 1 / BC
Let ABCD be a cyclic quadrilateral with AB < CD, and let P be the point  of intersection of the lines AD and BC. The circumcircle of the triangle PCD intersects the line AB at the points Q and R. Let S and T be the points where the tangents from P to the circumcircle of ABCD touch that circle.
(a) Prove that PQ = PR.
(b) Prove that QRST is a cyclic quadrilateral.

Let ABC be a triangle such that AC = 2AB. Let D be the point of intersection of the angle bisector of the angle CAB with BC. Let F be the point of intersection of the line parallel to AB passing through C with the perpendicular line to AD passing through A. Prove that FD passes through the midpoint of AC.

Let ABC be an acute-angled triangle, Γ its circumcircle and M the midpoint of BC. Let N be a point in the arc BC of Γ not containing A such that NAC = BAM. Let R be the midpoint of AM, S the midpoint of AN and T the foot of the altitude through A. Prove that R, S and T are collinear.
Let ABC be triangle with incenter I and circumcircle Γ. Let M = BI ∩ Γ and N = CI ∩ Γ, the line parallel to MN through I cuts AB, AC in P and Q. Prove that the circumradius of  BNP and CMQ are equal.

Given a triangle ABC, let D be the foot of the altitude from A, and l be the line through the midpoints of AC and BC. Let E be the reflection of point D with respect to l. Show that the circumcentre of the triangle ABC lies on the line AE.

Let ABC be a triangle with a right angle at B. Let B΄ be the reflection of B with respect to the line AC, and M be the midpoint of AC. The segment BM is extended beyond M to a point D such that BD = AC. Show that B΄ C is the bisector of MB΄D.

Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$. Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$. Line $BT'$ intersects $\omega$ again at $R$. The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$. Let $N$ be the intersection of lines $TR$ and $AC$. Prove that $\overline{CN}=2\overline{AL}$.

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