geometry problems from Central American and Caribbean Mathematical Olympiads (OMCC)

with aops links in the names

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 = 150

^{o}, 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
= 2∠COD. 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. S

_{BDE}≤ S_{DEA}≤ S_{EAB}≤ S_{ABD}. 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
= 60

^{o}. 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 = 90

^{o}.
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 Γ

_{1}and Γ_{2}intersect at points A and B. Consider a circle Γ contained in Γ_{1}and Γ_{2}, 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 Γ_{2}and G be the intersection of line DC with Γ_{1}. Let H and I be the intersection points of line ED with Γ_{1}and Γ_{2}respectively. 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 Γ

_{1}be two circles internally tangent at A, with centers O and O_{1}and radii r and r_{1}, respectively (r > r_{1}). B is a point diametrically opposed to A in Γ, and C is a point on Γ such that BC is tangent to Γ_{1}at P. Let A′ the midpoint of BC. Given that O_{1}A′ is parallel to AP, find the ratio r / r_{1}.
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 F

_{1}be the symmetric of F with respect to E and E_{1}be the symmetric of E with respect to F. If 3EF = FD + DE, prove that ∠BZF_{1}= ∠CY E_{1}.
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
= 120

^{o}, 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 $\overline{CN}=2\overline{AL}$.

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $D$ be the foot of the altitude from $A$ to the side $BC$, $M$ and $N$ the midpoints of $AB$ and $AC$, and $Q$ the point on $\Gamma$ diametrically opposite to $A$. Let $E$ be the midpoint of $DQ$. Show that the lines perpendicular to $EM$ and $EN$ passing through $M$ and $N$ respectively, meet on $AD$.

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$. The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$, respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$, respectively. Show that the triangle $APQ$ is isosceles.

source: www.kalva.demon.co.uk

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