Iberoamerican 1985 - 2018 (OIM) 61p

geometry problems from Iberoamerican Mathematical Olympiads
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 / (a2+b2+c2)  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 MPOA = BCOQ.

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 C1 be a circle, AB its diameter, t the tangent at B, and M A a variable point on C1. A circle C2 is tangent to C1 at M and to the line t.
(a) Find the tangency point of C2 and t and find the locus of the center of C2 as M varies.
(b) Prove that there is a circle orthogonal to all the circles C2.

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, PA2+PB2+PC2 = 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 A1,A2,B1,B2,C1,C2 are taken on the rays BA,CA,CB,AB,AC,BC respectively such that AA1 = AA2 = BC, BB1 = BB2 =CA, and CC1 =CC2 = AB. Prove that the area of the hexagon A1A2B1B2C1C2 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 P1Q1, P2Q2 and P3Q3 are three distinct segments with endpoints in S, then no point of S belongs simultaneously to the three lines m(P1Q1), m(P2Q2), m(P3Q3).
(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 = BC2 / BN2 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 C1 and C2.
(a) Prove that there exist infinitely many circles B that bisect both C1 and C2.
(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 AP2 = 2AD·OM.

Two circles S1 and S2 with the respective centers O1 and O2 intersect at M and N. Their common tangent t, closer to M, touches S1 at A and S2 at B. Point C is diametrically opposite to B, and D is the intersection of line O1O2 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 = 120o. 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,Care the intersection points of the internal bisectors of angles A,B,C with the opposite sides, respectively. Let BC meet the perpendicular bisector of AAat A′′, CA meet the perpendicular bisector of BBat B′′, and AB meet the perpendicular bisector of CCat C′′. Prove that A′′,B′′ and C′′ are collinear.

Let O be the circumcenter of the acute-angled triangle ABC, and let A1 be a point on the smaller arc BC of the circumcircle of ABC. Denote by A2 and A3 on the sides ABC and AC, respectively, such that ÐBA1A2 = ÐOAC and ÐCA1A3 =ÐOAB. Prove that the line A2A3 passes through the orthocenter of ABC.

IberoAmerican 2006.1
In a scalene triangle ABC with ÐA = 90o, 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 X1 the one closer to B. Let X2 and X3 be the intersections of Γ with BC, with X2 closer to B, and let X4 be one of the intersection points of Γ with CA that is closer to C. Let K be the intersection of X1X2 and X3X4. Prove that AK bisects X2X3.

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 CPBA = M and AQBC = 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 Cbe the circumcenters of AZY, BXZ, and CYX, respectively. Prove that 4SABCSABC with equality if and only if AA, BB, and CCpass through the same point.

Let C1 and C2 be two congruent circles with centers O1 and O2, which intersect at A and B. Let P be a point of the arc AB of C2 which is contained in the interior of C1. AP intersects C1 at C, CB intersects C2 at D, and the bisector of ÐCAD intersects C1 and C2 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 = 30ο and CX = O1O2.

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 C1,C2,C3 are circle with chords Y Z,ZX,XY , respectively, such that C1 and C2 intersect on the line CZ and that C1 and C3 intersect on the line BY . Suppose that C1 intersects the chords XY and ZX at J and M, respectively; that C2 intersects the chords Y Z and XY at L and I, respectively; and that C3 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 C1 and C2 cut each other at different points A and K. The common tangent to C1 and C2 nearer to K touches C1 at B and C2 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 $BE = CF$.

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