### geometry problems from Balkan Mathematical Olympiads

with aops links in the names

Balkan MO Geometry 1984 - 2017 EN in pdf with aops links

Balkan MO Geometry 1984 - 2017 GR in pdf with aops links

Balkan MO all 1984-2017 EN in pdf aops

Balkan MO all 1984-2009 EN in pdf imomath

Balkan MO all 2008-19 EN in pdf with solutions

Balkan MO Geometry 1984 - 2017 GR in pdf with aops links

Balkan MO all 1984-2017 EN in pdf aops

Balkan MO all 1984-2009 EN in pdf imomath

Balkan MO all 2008-19 EN in pdf with solutions

Let ABCD be a cyclic quadrilateral and let H

_{A}, H_{B}, H_{C}, H_{D}be the orthocenters of the triangles BCD, CDA, DAB and ABC respectively. Show that the quadrilaterals ABCD and H_{A}H_{B}H_{C}H_{D}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 e

_{1}, e_{2}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 e_{1}or e_{2 }.
by Theodoros Bolis

Let ABC be a triangle with
<A=120_{}

^{o}. 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 ≥ r

_{}_{}^{2 }, 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 =CG・AG = 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.

In a triangle ABC, the excircle ω

Circles k

_{1}(O_{1},1) and k_{2}(O_{2}, √2) with O_{1}O_{2}= 2 intersect at A and B. Find the length of the chord AC of circle k_{2}whose midpoint lies on k_{1}.
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 A

_{1}A_{2}A_{3}A_{4}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 B_{1}and C_{1}, 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 A

_{1},B_{1},C_{1}. If A_{2},B_{2},C_{2}are the tangency points of the incircle of the triangle A_{1}B_{1}C_{1}with its sides, prove that the Euler lines of the triangles ABC and A_{2}B_{2}C_{2}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 C

_{1}and C_{2}with centers O_{1}and O_{2}, respectively, are externally tangent at point G. A circle C with center O touches C_{1}at A and C_{2}at B so that the centers O_{1},O2 lie inside C. The common tangent to C_{1}and C_{2}at G intersects the circle C at K and L. If D is the midpoint of the segment KL, show that <O_{1}OO_{2}= <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 AP

^{2}.
Circles c

_{1 }(O_{1}, r_{1}) and c_{2}(O2, r_{2}), r_{2}> r_{1}, intersect at A and B so that <O_{1 }AO_{2}= 90◦. The line O_{1 }O_{2}meets c_{1}at C and D, and c2 at E and F (in the order C−E−D−F). The line BE meets c_{1}at K and AC at M, and the line BD meets c_{2}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 OA

^{2}+OB^{2}+OC^{2}+OD^{2}= 2S_{ABCD , }where S_{ABCD }denotes the area of ABCD. Prove that ABCD is a square and O its center.
Circles C

_{1}and C_{2}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 C_{2}at D. Lines AB and AC meet C_{1}and C_{2}again at K and L, respectively, and the line BC meets C_{1}again at M and C_{2}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 = 60

^{o}.
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
A

_{1},B_{1},C_{1}, respectively. Show that the lines A_{1}E, B_{1}F, C_{1}D 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 C

_{1}on AB is such that CC_{1}is an altitude of the triangle ABC. Let H_{1}be the reection of H in AB. The orthogonal projections of C_{1}onto the lines AH_{1}, AC and BC are P, Q and R, respectively. Let M_{1}be the point such that the circumcentre of triangle PQR is the midpoint of the segment MM_{1}. Prove that M_{1}lies on the segment BH_{1}.
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 ω

_{a}opposite A touches AB at P and AC at Q, and the excircle ω

_{b}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.

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 t

_{B}and t_{C}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 t_{C}in point D. The straight line passing through the point C and parallel to AB intersects t_{B}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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