geometry problems from Iberoamerican Mathematical Olympiads

with aops links in the names

In 1986 it did not take place.

(Spain)

IberoAmerican 2018.6

Let $ABC$ be an acute triangle with $AC > AB > BC$. The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$, such that $AB = BP$ and $AC = CQ$, and let $K$ be the intersection of lines $EP$ and $DQ$. Let $M$ be the midpoint of $BC$. Show that $\angle DKA = \angle EKM$.

source: www.kalva.demon.co.uk

with aops links in the names

*Olimpíada Iberoamericana de Matemática (OIM)*

[only 10 problems were left unsolved]

1985 - 2018

A point P inside an equilateral triangle ABC satisfies PA=5, PB=7, and PC=8. Find the side length of the triangle
ABC.

Let O be the center and R the radius of the circumcircle of a
triangle ABC. The lines AO, BO, CO meet the
opposite sides of the triangle at D,E,F, respectively. Prove that
1 / AD +1 / BE+1 / CF = 2R.

In 1986 it did not take place.

In a triangle ABC, P is the centroid and M
and N the midpoints of
the sides AC and AB respectively. Prove that if a circle can be inscribed in the
quadrilateral ANPM, then △ABC is isosceles.

Let ABCD be a convex quadrilateral and let
P and Q be the points on the sides AD and BC respectively
such that AP / PD = BQ / QC = AB / CD. Prove the line PQ forms
equal angles with the lines AB and
CD.

The sides of a
triangle form an arithmetic progression, and so do the altitudes. Show that the
triangle is equilateral.

Prove that,
among all possible triangles whose vertices are 3, 5 and 7 apart from a given
point P, the ones with the
largest perimeter have P as
incenter.

Let ABC be a triangle with sides a, b, c. Each side
of the triangle is divided into n equal
parts. Denote by S the sum of
the squares of the distances from the vertices to the division points on the
opposite side. Prove that S / (a

^{2}+b^{2}+c^{2}) is a rational number.
The incircle of
the triangle ABC is tangent to
sides AB and AC at M and N, respectively. The bisectors of the angles at A and B intersect MN at
points P and Q, respectively. Let O be the incenter of △ABC. Prove that MP・OA =
BC・OQ.

In a triangle ABC, I is the incenter and D,E,F the tangency points of the incircle with the sides BC,CA,AB,
respectively. The line AD intersects
the incircle again at P. If M is the midpoint of EF,
prove that the points P, I,M and D lie on a
circle.

Let C

_{1}be a circle, AB its diameter,*t*the tangent at B, and M ≠A a variable point on C_{1}. A circle C_{2}is tangent to C_{1}at M and to the line*t*.
(a) Find the
tangency point of C

_{2}and t and find the locus of the center of C_{2}as M varies.
(b)
Prove that there is a circle orthogonal to all the circles C

_{2}.
Two
perpendicular lines divide a square into four parts, three of which have the
area 1. Show that the area of the square is 4.

Given three
non-collinear points M, N, and P such that M and
N are the midpoints of two
sides of a certain triangle and P is
the orthocenter of the triangle. Construct the triangle.

Circle G is inscribed in an equilateral
triangle of side 2.

(a) Prove that
for every point P on G, PA

^{2}+PB^{2}+PC^{2}= 5.
(b) Prove that
for every P on G the segments PA,PB,PC are the
sides of a triangle whose area is √3 /4

We are given a
circle C, the altitude h of a trapezoid ABCD inscribed in C, and the sum m of the lengths of the bases AB and CD. Show
how to construct the trapezoid ABCD.

A triangle ABC is given. Points A

_{1},A_{2},B_{1},B_{2},C_{1},C_{2}are taken on the rays BA,CA,CB,AB,AC,BC respectively such that AA_{1}= AA_{2}= BC, BB_{1}= BB_{2}=CA, and CC_{1}=CC_{2}= AB. Prove that the area of the hexagon A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}is at least 13 times the area of the triangle ABC.
Show that for
any convex polygon of unit area, there exists a parallelogram of area 2
containing the polygon.

(Mexico)

Let Γ be the incircle of an equilateral triangle ABC. If D and E are
points on the sides AB and AC, respectively, such that DE is tangent to Γ,
prove that AD / DB+ AE / EC= 1.

(Spain)

For any distinct
points P and Q of the plane, we denote m(PQ) the perpendicular bisector of the segment PQ. Let S be a finite subset of the plane, with more than one element,
which satisfies the following conditions:

(a) If P and Q are distinct points of S,
then m(PQ) meets S.

(b) If P

_{1}Q_{1}, P_{2}Q_{2}and P_{3}Q_{3}are three distinct segments with endpoints in S, then no point of S belongs simultaneously to the three lines m(P_{1}Q_{1}), m(P_{2}Q_{2}), m(P_{3}Q_{3}).
(Mexico)

Let ABCD be a cyclic quadrilateral.
Suppose that there exists a circle with center in AB that is tangent to the other sides of the quadrilateral.

(a) Prove that AB = AD+BC.

(b) Determine, in terms of x = AB and y =CD, the maximum possible area of the quadrilateral.

(Brazil)

An acute-angled
triangle ABC is inscribed in a
circle k. For a point P inside circle k, the lines AP,BP,CP meet k again at X,Y,Z. Determine the point P
for which triangle XYZ is
equilateral.

(Brazil)

Let r and s be two orthogonal lines, not belonging to the same plane. Let AB be their common perpendicular,
with A ∈ r and
B ∈ s.
The points M ∈ r and
N ∈ s are
variable so that MN is tangent
to the sphere with diameter AB at
some point T. Find the locus of
T.

(Brazil)

The incircle of
a triangle ABC is tangent to BC,CA and AB at D,E and F,
respectively. Suppose that the incircle passes through the midpoint X of AD. The lines XB and
XC meet the incircle again at Y and Z, respectively. Show that EY = FZ.

(Spain)

Let M be the midpoint of the median AD of a triangle ABC.
Line BM meets side AC at N. Prove that AB is
tangent to the circumcircle of triangle NBC
if and only if the equality BM /
MN = BC

^{2 }/ BN^{2}holds.
(Spain)

A circle
centered at the incenter I of a
triangle ABC meets all three
sides of the triangle: side BC at D and P (with D nearer to B), side CA at E and Q (with E nearer
to C), and side AB at F and R (with F nearer to A). The diagonals of the quadrilaterals EQFR, FRDP, and DPEQ meet at S, T, and U, respectively. Show that the
circumcircles of the triangles FRT,
DPU and EQS have a single point in common.

In a triangle ABC, AE and BF are
altitudes and H the orthocenter.
The line symmetric to AE with
respect to the bisector of ÐA and the line
symmetric to BF with respect to
the bisector of ÐB intersect at a
point O. The lines AE and AO meet the circumcircle of △ABC again at M and N,
respectively. The lines BC and HN meet at P, BC and OM at R, and HR and OP at S. Prove that AHSO is
a parallelogram.

The incircle of
a triangle ABC is tangent to BC, CA, AB at points
D, E, F,
respectively. Line AD meets the
incircle again at Q. Prove that
EQ passes through the midpoint
of AF if and only if AC = BC

.

.

We say that
circle M bisects circle N if their common chord is a diameter
of N. Consider two
non-concentric circles C

_{1}and C_{2}.
(a) Prove that
there exist infinitely many circles B that
bisect both C

_{1}and C_{2}.
(b)
Find the locus of the centers of such circles B.

An acute-angled
triangle ABC is inscribed in a
circle with center O. Let AD, BE, CF be its
altitudes. The line EF intersects
the circle at P and Q.

(a) Prove that OA is perpendicular to PQ.

(b) Prove that
if M is the midpoint of BC, then AP

^{2}= 2AD·OM.
Two circles S

_{1}and S_{2}with the respective centers O_{1}and O_{2}intersect at M and N. Their common tangent t, closer to M, touches S_{1}at A and S_{2}at B. Point C is diametrically opposite to B, and D is the intersection of line O_{1}O_{2}with the perpendicular to AM from B. Prove that M,D and C are collinear.
The incircle of
a triangle ABC is centered at O and tangent to BC,CA,AB at points X,Y,Z, respectively. The lines BO and CO meet
the line YZ at P and Q. Prove that if X is
equidistant from P and Q, then △ABC is isosceles.

A point P inside an equilateral triangle ABC satisfies ÐAPC =
120

^{o}. The rays CP and AP meet AB and BC respectively at points M and N. Find the locus of the circumcenter of triangle MBN when P assumes all possible positions.
In a scalene
triangle ABC, BD is an inner angle bisector with D on AC. Let E and F be the respective projections of A and C on the line BD,
and let M be the projection of D on BC. Prove that ÐEMD = ÐDMF.

Let C and D be points on the semicircle with diameter AB such that B and C are on
different sides of AD. Denote
by M,N,P the
midpoints of AC,DB,CD, respectively. If OA
and OB are the
circumcenters of the triangles ACP and
BDP, show that the lines OAOB and MN are parallel.

In a square ABCD, P and Q are
points on sides BC and CD respectively, distinct from their
endpoints, such that BP =CQ. Let X and Y be
arbitrary points on the segments AP and
AQ, respectively. Show that there
always exists a triangle with the sides congruent to BX,XY, and DY.

In the plane are
given a circle with center O and
radius r and a point A outside the circle. For any point M on the circle, let N be the diametrically opposite point.
Find the locus of the circumcenter of triangle AMN when M describes
the circle.

In a scalene
triangle ABC, points A′,B′,C′ are the
intersection points of the internal bisectors of angles A,B,C with the opposite sides, respectively.
Let BC meet the perpendicular
bisector of AA′ at
A′′, CA meet the perpendicular bisector of
BB′ at
B′′, and AB meet the perpendicular bisector of
CC′ at
C′′. Prove that A′′,B′′ and C′′ are collinear.

Let O be the circumcenter of the
acute-angled triangle ABC, and
let A

_{1}be a point on the smaller arc BC of the circumcircle of △ABC. Denote by A_{2}and A_{3}on the sides ABC and AC, respectively, such that ÐBA_{1}A_{2}= ÐOAC and ÐCA_{1}A_{3}=ÐOAB. Prove that the line A_{2}A_{3}passes through the orthocenter of △ABC.
IberoAmerican
2006.1

In a scalene triangle ABC with ÐA = 90

In a scalene triangle ABC with ÐA = 90

^{o}, the tangent line at A to its circumcircle meets line BC at M and the incircle touches AC at S and AB at R. The lines RS and BC intersect at N, while the lines AM and SR intersect at U. Prove that the
triangle
UMN is isosceles.

The sides AD and CD of a tangent quadrilateral ABCD touch the incircle φ at P and Q, respectively. If M is
the midpoint of the chord XY determined
by φ on the diagonal BD, prove that ÐAMP = ÐCMQ.

Let I be the incenter of △ABC and let Γ be a circle centered at I whose radius is greater than the inradius but does not pass
through any of the vertices. Out of two intersection points of Γ and AB,
denote by X

_{1}the one closer to B. Let X_{2}and X_{3}be the intersections of Γ with BC, with X_{2}closer to B, and let X_{4}be one of the intersection points of Γ_{ }with CA that is closer to C. Let K be the intersection of X_{1}X2 and X_{3}X_{4}. Prove that AK bisects X_{2}X_{3}.
Given a triangle
ABC, let r be the external bisector of ÐABC.
Let P and Q be the feet of perpendiculars from A and C to r. If CP∩BA =
M and AQ∩BC = N, show that MN, r, and AC pass through the same point.

Let X, Y, and Z be the
points of the sides BC, CA, and AB of △ABC.
Let A′,B′, and C′ be the
circumcenters of △AZY,
△BXZ,
and CYX, respectively. Prove
that 4S

_{A}_{′}_{B}_{′}_{C}_{′ }≥ S_{ABC }with equality if and only if AA′, BB′, and CC′ pass through the same point.
Let C

_{1}and C_{2}be two congruent circles with centers O_{1}and O_{2}, which intersect at A and B. Let P be a point of the arc AB of C_{2 }which is contained in the interior of C_{1}. AP intersects C_{1}at C, CB intersects C_{2}at D, and the bisector of ÐCAD intersects C_{1}and C_{2}at E and L, respectively. Let F be the symmetric point to D with respect to the midpoint of PE. Prove that there exists a point X satisfying ÐXFL = ÐXDC = 30^{ο}and CX = O_{1}O_{2}.
Given a triangle
ABC with incenter I, let P be the intersection point of the external bisector of ÐA and
the circumcircle of △ABC.
Let J be the second
intersection point of PI and
the circumcircle of △ABC.
Show that the circumcircles of △JIB and △JIC are tangent to IC and IB,
respectively.

The circle Γ
is inscribed to the scalene triangle ABC. is tangent to the sides
BC,CA and AB at D,E and F respectively. The line EF intersects the line BC at G.
The circle of diameter GD intersects Γ in R (R ≠ D). Let P, Q (P
≠R, Q ≠R) be the
intersections of with BR and CR, respectively. The lines BQ and CP
intersects at X. The circumcircle of CDE meets QR at M, and the circumcircle of
BDF meet PR at N. Prove that PM, QN and RX are concurrent.

(Arnoldo Aguilar, El Salvador)

Let
ABCD be a cyclic quadrilateral whose diagonals AC and BD are perpendicular. Let
O be the circumcenter of ABCD, K the intersection of the diagonals, L ≠ O the intersection of the circles circumscribed to OAC and OBD,
and G the intersection of the diagonals of the quadrilateral whose vertices are
the midpoints of the sides of ABCD. Prove that O,K,L and G are collinear

Let ABC be a
triangle and X,Y,Z be the tangency points of its inscribed circle with the
sides BC,CA,AB, respectively. Suppose that C

_{1},C_{2},C_{3}are circle with chords Y Z,ZX,XY , respectively, such that C_{1}and C_{2}intersect on the line CZ and that C_{1}and C_{3}intersect on the line BY . Suppose that C_{1}intersects the chords XY and ZX at J and M, respectively; that C_{2 }intersects the chords Y Z and XY at L and I, respectively; and that C_{3}intersects the chords Y Z and ZX at K and N, respectively. Show that I, J,K,L,M,N lie on the same circle.
Let ABC be an
acute-angled triangle, with AC ≠ BC and let O be
its circumcenter. Let P and Q be points such that BOAP and COPQ are
parallelograms. Show that Q is the orthocenter of ABC.

Let ABCD be a rectangle.
Construct equilateral triangles BCX and DCY , in such a way that both of these
triangles share some of their interior points with some interior points of the
rectangle. Line AX intersects line CD on P, and line AY intersects line BC on
Q. Prove that triangle APQ is equilateral.

Let ABC be a
triangle, P and Q the intersections of the parallel line to BC that passes
through A with the external angle bisectors of angles B and C, respectively.
The perpendicular to BP at P and the perpendicular to CQ at Q meet at R. Let I
be the incenter of ABC. Show that AI = AR.

Let X and Y be
the diameter’s extremes of a circunference Γ
and N be the midpoint of one of the arcs XY of Γ.
Let A and B be two points on the segment XY . The lines NA and NB cuts Γ
again in C and D, respectively. The tangents to Γ
at C and at D meets in P. Let M the the intersection point between XY and NP.
Prove that M is the midpoint of the segment AB.

Let Γ
be a circunference and O its center. AE is a diameter of Γ
and B the midpoint of one of the arcs AE of Γ.
The point D ≠ E in on the segment OE. The point C is
such that the quadrilateral ABCD is a parallelogram, with AB parallel to CD and
BC parallel to AD. The lines EB and CD meets at point F. The line OF cuts the
minor arc EB of Γ
at I. Prove that the line EI is the angle bissector of ÐBEC.

Let ABC be an
acute triangle and H its orthocenter. Let D be the intersection of the altitude
from A to BC. Let M and N be the midpoints of BH and CH, respectively. Let the
lines DM and DN intersect AB and AC at points X and Y respectively. If P is the
intersection of XY with BH and Q the intersection of XY with CH, show that H,
P,D,Q lie on a circumference.

A line r
contains the points A, B, C, D in that order. Let P be a point not in r such
that ÐAPB
= ÐCPD.
Prove that the angle bisector of ÐAPD intersects the line r at a point G such that:

1 / GA + 1 / GC
= 1 /GB + 1 /GD

Let ABC be an
acute triangle and let D be the foot of the perpendicular from A to side BC.
Let P be a point on segment AD. Lines BP and CP intersect sides AC and AB at E
and F, respectively. Let J and K be the feet of the perpendiculars from E and F
to AD, respectively. Show that FK/ KD = EJ / JD.

Let ABC be an
acute triangle and Γ its circumcircle. The lines
tangent to Γ through B and C meet at P. Let M be a point on the
arc AC that does not contain B such that M ≠A and M ≠C,
and K be the point where the lines BC and AM meet. Let R be the point
symmetrical to P with respect to the line AM and Q the point of intersection of
lines RA and PM. Let J be the midpoint of BC and L be the intersection point of
the line PJ and the line through A parallel to PR. Prove that L, J, A,Q, and K
all lie on a circle.

The
circumferences C

_{1}and C_{2}cut each other at different points A and K. The common tangent to C_{1}and C_{2}nearer to K touches C_{1}at B and C_{2}at C. Let P be the foot of the perpendicular from B to AC, and let Q be the foot of the perpendicular from C to AB. If E and F are the symmetric points of K with respect to the lines PQ and BC, respectively, prove that A,E and F are collinear.
Let ABC be an
acute angled triangle and Γ its
circumcircle. Led D be a point on segment BC, different from B and C, and let M
be the midpoint of AD. The line perpendicular to AB that passes through D
intersects AB in E and Γ in F, with point D between E and
F. Lines FC and EM intersect at point X. If ÐDAE
= ÐAFE, show that
line AX is tangent to Γ.

Let
ABC be an acute triangle with AC > AB and O its circumcenter. Let D be a
point on segment BC such that O lies inside triangle ADC and ÐDAO
+ ÐADB
= ÐADC.
Let P and Q be the circumcenters of triangles ABD and ACD respectively, and let
M be the intersection of lines BP and CQ. Show that lines AM, PQ and BC are
concurrent.

IberoAmerican 2018.2

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.

Let $ABC$ be an acute triangle with $AC > AB > BC$. The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$, such that $AB = BP$ and $AC = CQ$, and let $K$ be the intersection of lines $EP$ and $DQ$. Let $M$ be the midpoint of $BC$. Show that $\angle DKA = \angle EKM$.

source: www.kalva.demon.co.uk

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