Balkan 1984 - 2019 (BMO) 46p

geometry problems from Balkan Mathematical Olympiads
with aops links in the names

Let ABCD be a cyclic quadrilateral and let HA, HB, HC, HD  be the orthocenters of the triangles BCD, CDA, DAB and ABC respectively. Show that the quadrilaterals ABCD and HAHBHCHD  are congruent. 

Let O be the circumcircle of a triangle ABC, D be the midpoint of AB, and E be the centroid of triangle ACD. Prove that CD is perpendiculat to OE if and only if AB = AC.
 by Ivan Tonov
1985 BMO Shortlist 1 (GRE)                                    
Let e1, e2 be two lines perpendicular to the same plane. Find the locus of the points of the space , that we can draw 3 lines, perpendicular in pairs, who intersect e1 or e2 .
by Theodoros Bolis
1985 BMO Shortlist 2 (GRE)  (also)                                
Let ABC be a triangle with <A=120­­­o.  Let AD, CE be the angle bisectors of angles A,C respectively and I be the intersection point of AD, CE. If Z is the intersection point of BI and DE, calculate angle <DAZ .

by Dimitris Kontogiannis
A line through the incenter I of a triangle ABC intersects its circumcircle at F and G, and its incircle at D and E, where D is between I and F. Prove that DF EG ≥ r­­­­2 , where r is the inradius. When does equality occur? 

Let E,F,G,H,K,L respectively be points on the edges AB,BC,CA,DA, DB,DC  of a tetrahedron ABCD. If AE BE = BF CF =CGAG = DH AH = DK BK = DL CL,  prove that the points E,F,G,H,K,L lie on a sphere.

A triangle ABC and a point T are given in the plane so that the triangles TAB, TBC, TCA have the same area and perimeter. Prove that:
(a) If T is inside △ABC, then △ABC is equilateral;
(b) If T is not inside △ABC, then △ABC is right-angled.

Circles k1 (O1 ,1) and k2 (O2 , 2) with O1O2 = 2 intersect at A and B. Find the length of the chord AC of circle k2 whose midpoint lies on k­1.

Let CH,CL,CM be the altitude, angle bisector, and median of a triangle ABC, respectively, where H,L,M are on AB. Given that the ratios of the areas of HMC and LMC to the area of ABC are equal to  $\frac{1}{4}$ και $1-\frac{\sqrt{3}}{2}$, respectively, determine the angles of ABC.

Show that every tetrahedron A­­12A­­3A­­4 can be placed between two parallel planes which are at the distance at most $\frac{1}{2}\sqrt{\frac{p}{3}}$, where $P=A_{1}A_{2}^{2}+A_{1}A_{3}^{2}+A_{1}A_{4}^{2}+{A}_{2}A_{3}^{2}+A_{2}A_{4}^{2}+A_{3}A_{4}^{2}$

A line l intersects the sides AB and AC of a triangle ABC at points B1 and C1, respectively, so that the vertex A and the centroid G of ABC lie in the same half-plane determined by l. Prove that ${{S}_{B{{B}_{1}}G{{C}_{1}}}}+{{S}_{C{{C}_{1}}G{{B}_{1}}}}\ge \frac{4}{9}{{S}_{ABC}}$

by Dimitris Kontogiannis
The feet of the altitudes of a non-equilateral triangle ABC are A1,B1,C1. If A2,B2,C2 are the tangency points of the incircle of the triangle A1B1C1 with its sides, prove that the Euler lines of the triangles ABC and A2B2C2 coincide.

Let M be a point on the arc AB not containing C of the circumcircle of an acuteangled triangle ABC, and let O be the circumcenter. The perpendicular from M to OA intersects AB at K and AC at L. The perpendicular from M to OB intersects AB at N and BC at P. If KL = MN, express <MLP in terms of the angles of ABC.

A regular hexagon of area H is inscribed in a convex polygon of area P. Prove that P ≤ 3/2 H. When does equality occur?

Let D,E,F be points on the sides BC,CA,AB respectively of a triangle ABC (distinct from the vertices). If the quadrilateral AFDE is cyclic, prove that $\frac{4{{S}_{DEF}}}{{{S}_{ABC}}}\le {{\left( \frac{EF}{AD} \right)}^{2}}$

Circles C­1 and C­2 with centers O­1 and O­2, respectively, are externally tangent at point G. A circle C with center O touches C­1 at A and C­2 at B so that the centers O­1,O2 lie inside C. The common tangent to C­1 and C­2 at G intersects the circle C at K and L. If D is the midpoint of the segment KL, show that <O­1OO­2 = <ADB.

An acute angle XAY and a point P inside it are given. Construct (by a ruler and a compass) a line that passes through P and intersects the rays AX and AY at B and C such that the area of the triangle ABC equals AP2.

Circles c1 (O1, r1) and c2 (O2, r2), r2> r1, intersect at A and B so that <O1 AO2 = 90◦. The line O1 O2 meets c1 at C and D, and c2 at E and F (in the order CEDF). The line BE meets c1 at K and AC at M, and the line BD meets c2 at L and AF at N. Prove that $\frac{{{r}_{2}}}{{{r}_{1}}}=\frac{KE}{KM}\cdot \frac{LN}{LD}$

Let O be the circumcenter and G be the centroid of a triangle ABC. If R and r are the circumradius and inradius of the triangle, respectively, prove that  

In a convex pentagon ABCDE, M,N,P,Q,R are the midpoints of the sides AB, BC, CD, DE, EA, respectively. If the segments AP, BQ, CR, DM pass through a single point, prove that EN contains that point as well. 

Suppose that O is a point inside a convex quadrilateral ABCD such that OA2+OB2+OC2+OD2 = 2SABCD , where SABCD  denotes the area of ABCD. Prove that ABCD is a square and O its center.

Circles C1 and C2 touch each other externally at D, and touch a circle G internally at B and C, respectively. Let A be an intersection point of G and the common tangent to C1 and C2 at D. Lines AB and AC meet C1 and C2 again at K and L, respectively, and the line BC meets C1 again at M and C2 again at N. Show that the lines AD, KM, LN are concurrent.
Let L denote the set of points inside or on the border of a triangle ABC, without a fixed point T inside the triangle. Show that L can be partitioned into disjoint closed segments.

Let D be the midpoint of the shorter arc BC of the circumcircle of an acuteangled triangle ABC. The points symmetric to D with respect to BC and the circumcenter are denoted by E and F, respectively. Let K be the midpoint of EA.
(a) Prove that the circle passing through the midpoints of the sides of ABC also passes through K.
(b) The line through K and the midpoint of BC is perpendicular to AF.

Let M,N,P be the orthogonal projections of the centroid G of an acute-angled triangle ABC onto AB,BC,CA, respectively. Prove that $\frac{4}{27}<\frac{{{S}_{MNP}}}{{{S}_{ABC}}}\le \frac{1}{4}$

Let ABC be a scalene triangle and E be a point on the median AD. Point F is the orthogonal projection of E onto BC. Let M be a point on the segment EF, and N,P be the orthogonal projections of M onto AC and AB respectively. Prove that the bisectors of the angles PMN and PEN are parallel.

Prove that a convex pentagon that satisfies the following two conditions must be regular:
(i) All its interior angles are equal.
(ii) The lengths of all its sides are rational numbers.

Two circles with different radii intersect at A and B. Their common tangents MN and ST touch the first circle at M and S and the second circle at N and T. Show  that the orthocenters of triangles AMN, AST, BMN, and BST are the vertices of a rectangle.

Let ABC be a triangle with AB AC. The tangent at A to the circumcircle of the triangle ABC meets the line BC at D. Let E and F be the points on the perpendicular bisectors of the segments AB and AC respectively, such that BE and CF are both perpendicular to BC. Prove that the points D,E, and F are collinear.
by Valentin Vornicu
Let O be an interior point of an acute-angled triangle ABC. The circles centered at the midpoints of the sides of the triangle ABC and passing through point O, meet in points K,L,M different from O. Prove that O is the incenter of the triangle KLM if and only if O is the circumcenter of the triangle ABC.

The incircle of an acute-angled triangle ABC touches AB at D and AC at E. Let the bisectors of the angles <ACB and <ABC intersect the line DE at X and Y respectively, and let Z be the midpoint of BC. Prove that the triangle XYZ is equilateral if and only if <A = 60o.

A line m intersects the sides AB, AC and the extension of BC beyond C of the triangle ABC at points D,F,E, respectively. The lines through points A,B,C which are parallel to m meet the circumcircle of triangle ABC again at points A1,B1,C1, respectively. Show that the lines A1E, B1F, C1D are concurrent.
by Dimitris Kontogiannis
2007 BMO Problem 1 (ALB) 
In a convex quadrilateral ABCD with AB = BC =CD, the diagonals AC and BD are of different length and intersect at point E. Prove that AE = DE if and only if <BAD + <ADC = 120 o.

An acute-angled scalene triangle ABC with AB > BC is given. Let O be its circumcenter, H its orthocenter, and F the foot of the altitude from C. Let P be the point (other than A) on the line AB such that AF = PF, and M be a point on AC. We denote the intersection of PH and BC by X, the intersection of OM and FX by Y, and the intersection of OF and AC by Z. Prove that the points F, M, Y, and Z are concyclic.
by Theoklitos Paragyiou
Let MN be a line parallel to the side BC of triangle ABC, with M on the side AB and N on the side AC. The lines BN and CM meet at point P. The circumcircles of triangles BMP and CNP meet at two distinct points P and Q. Prove that <BAQ = <CAP.
by Liubomir Chiriac
Let ABC be an acute triangle with orthocentre H, and let M be the midpoint of AC. The point C1 on AB is such that CC1 is an altitude of the triangle ABC. Let H1 be the reection of H in AB. The orthogonal projections of C1 onto the lines AH1, AC and BC are P, Q and R, respectively. Let M1 be the point such that the circumcentre of triangle PQR is the midpoint of the segment MM1. Prove that M1 lies on the segment BH1.

Let ABCD be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at E. The midpoints of AB and CD are F and G respectively, and l is the line through G parallel to AB. The feet of the perpendiculars from E onto the lines l and CD are H and K, respectively. Prove that the lines EF and HK are perpendicular.

Let ABCDEF be a convex hexagon of area 1, whose opposite sides are parallel. The lines AB, CD and EF meet in pairs to determine the vertices of a triangle. Similarly, the lines BC, DE and FA meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least 3 / 2.

Let A, B and C be points lying on a circle Γ with centre  O. Assume that <ABC > 90. Let D be the point of intersection of the line AB with the line perpendicular to AC at C. Let l be the line through D which is perpendicular to AO. Let E be the point of intersection of l with the line AC, and let F be the point of intersection of Γ with l that lies between D and E. Prove that the circumcircles of triangles BFE and CFD are tangent at F.

2013 BMO Problem 1 (BUL) 
In a triangle ABC, the excircle ωa opposite A touches AB at P and AC at Q, and the excircle ωb opposite B touches BA at M and BC at N. Let K be the projection of C onto MN, and let L be the projection of C onto PQ. Show that the quadrilateral MKLP is cyclic.

Let ABCD be a trapezium inscribed in a circle Γ with diameter AB. Let E be the intersection point of the diagonals AC and BD . The circle with center B and radius BE meets Γ at the points K and L (where K is on the same side of AB as C). The line perpendicular to BD at E intersects CD at M. Prove that KM is perpendicular to DL.
by Silouanos Brazitikos  
Let ABC be a scalene triangle with incentre I and circumcircle (ω).The lines AI,BI,CI intersect (ω) for the second time at the points D,E, F, respectively. The lines through I parallel to the sides BC,AC,AB intersect the lines EF,DF,DE at the points K, L,M, respectively. Prove that  the points K, L,M are collinear.
by Theoklitos Paragyiou
Let ABCD be a cyclic quadrilateral with AB < CD. The diagonals intersect at the point F and lines AD and BC intersect at the point E. Let K and L be the orthogonal projections of F onto lines AD and BC respectively, and let M, S and T be the midpoints of EF, CF and DF respectively. Prove that the second intersection point of the circumcircles of triangles MKT and MLS lies on the segment CD.

by Silouanos Brazitikos
Consider an acute-angled triangle ABC with AB<AC and let ω be its circumscribed circle. Let tB and tC be the tangents to the circle ω at points B and C, respectively, and let L be their intersection. The straight line passing through the point B and parallel to AC intersects tC in point D. The straight line passing through the point C and parallel to AB intersects tB in point E. The circumcircle of the triangle BDC intersects AC in T, where T is located between A and C. The circumcircle of the triangle BEC intersects the line AB (or its extension) in S, where B is located between S and A. Prove that ST, AL, and BC are concurrent.

by Evangelos Psychas and Silouanos Brazitikos

A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$.

by Emil Stoyanov, Bulgaria

Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
i) $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
ii) $ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$..

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