geometry problems from Junior Balkan Mathematical Olympiad and Shortlists

with aops links in the names

Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.

Let
${ABC}$ be an acute-angled triangle
inscribed in a circle ${k}$. It is given that the tangent from ${A}$ to the circle meets the line ${BC}$ at point ${P}$. Let ${M}$ be the midpoint of the line segment ${AP}$ and ${R}$
be the second intersection point of the circle ${k}$ with the line ${BM}$. The line ${PR}$ meets again the circle ${k}$ at point ${S}$ different from ${R}$. Prove that the lines
${AP}$ and ${CS}$ are parallel.

The
triangle ${ABC}$ is isosceles with ${AB = AC}$, and ${\angle BAC <
60^\circ}$. The points ${D}$ and ${E}$ are chosen on the side ${AC}$ such that,
${EB = ED}$, and ${\angle ABD \equiv \angle CBE}$. Denote by ${O}$ the
intersection point between the internal bisectors of the angles ${\angle BDC}$
and ${\angle ACB}$. Compute ${\angle COD}$.

Let ${ABCDE}$ be convex pentagon such that ${AB+CD=BC+DE}$ and ${k}$ half circle with center on side ${AE}$ that touches sides ${AB, BC, CD}$ and ${DE}$ of pentagon, respectively, at points ${P, Q, R}$ and ${S}$ (different from vertices of pentagon). Prove that $PS\parallel AE$.

Acute-angled
triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s
circumicircle. Diametes ${BD}$ and ${CE}$ are drawn. Circle
${c_1(A,AE)}$ interescts ${AC}$ at ${K}$ Circle ${{c}_{2}(A,AD)}$
intersects ${BA}$ at ${L}$ (${A}$ lies between ${B}$ και ${L}$).Prove
that lines ${EK}$ and ${DL}$ intersect at circle .

Theoklitos Paragyiou

Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ , ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .

Let $ABC$ be an acute triangle, $A', B'$ and $C' $ be the reflections of the vertices $A, B$ and

$C$ with respect to $BC, CA$, and $AB$, respectively, and let the circumcircles of triangles $ABB'$ and $ACC'$ meet again at $A_1$. Points $B_1$ and $C_1$ are de ned similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ have a common point.

source:

pregatirematematicaolimpiadejuniori.wordpress.com

with aops links in the names

2007-2016 collected inside aops here

JBMO 1997 - 2018

JBMO Shortlist 2000 - 2018

Let
${ABC }$ be a triangle and let ${I
}$ be the incenter. Let ${N, M }$ be
the midpoints of the sides ${AB }$ and
${CA }$ respectively. The lines ${BI
}$ and ${CI }$
meet ${MN }$ at ${K }$ and ${L }$
respectively. Prove that ${AI + BI + CI > BC + KL }$.

Determine
the triangle with sides ${a, b, c }$ and
circumradius ${R }$ for which ${R(b + c)
= a\sqrt{bc} }$.

Let
${ABCDE }$ be a convex pentagon such
that ${AB = AE = CD = 1, \angle ABC = \angle DEA = 90^\circ }$ and
${BC + DE = 1 }$. Compute the area of the pentagon.

Let
${ABC }$ be a triangle with ${AB = AC
}$. Also, let ${D \in [BC] }$ be a point
such that ${BC > BD >DC > 0 }$, and let ${C_1, C_2 }$ be the circumcircles of the triangles ${ABD
}$ and ${ADC }$ respectively. Let ${BB' }$ and ${CC' }$ be diameters in the two circles,
and let ${M }$ be the midpoint of
${B'C'}$. Prove that the area of the triangle ${MBC }$ is constant (i.e. it does not depend on the
choice of the point ${D }$).

A
half-circle of diameter ${EF }$ is
placed on the side ${BC }$ of a triangle
${ABC }$ and it is tangent to the sides
${AB }$ and ${AC }$ in the points ${Q
}$ and ${P }$ respectively. Prove that the intersection
point ${K }$ between the lines ${EP
}$ and ${F Q }$ lies on the altitude from ${A }$ of the triangle ${ABC }$.

A
triangle ${ABC}$ is given. Find all the pairs of points ${X, Y}$ so that ${X}$
is on the sides of the triangle, ${Y}$ is inside the triangle, and four
non-intersecting segments from the set ${ \{ XY,AX, AY,BX,BY,CX,CY \} }$ divide
the triangle ${ABC}$ into four triangles with equal areas.

A
triangle ${ABC}$ is given. Find all the segments ${XY}$ that lie inside the
triangle such that ${XY}$ and five of the segments ${XA,XB,XC,YA,YB,YC}$ divide
the triangle ${ABC}$ into ${5}$ regions with equal areas. Furthermore, prove
that all the segments ${XY}$ have a common point.

Let
${ABC}$ be a triangle. Find all the triangles ${XYZ}$ with vertices inside
triangle ${ABC}$ such that ${XY,YZ,ZX}$ and six non-intersecting segments from
the following ${AX, AY,AZ,BX,BY,BZ,CX,CY,CZ}$ divide the triangle ${ABC}$ into
seven regions with equal areas.

Let
${ABC}$ be a triangle and let ${a, b, c}$ be the lengths of the sides
${BC,CA,AB}$ respectively. Consider a triangle ${DEF}$ with the side lengths
${EF = \sqrt{au}, FD = \sqrt{bu}, DE = \sqrt{cu}}$. Prove that ${\angle A
>\angle B > \angle C}$ implies ${\angle A > \angle D > \angle E
> \angle F > \angle C}$.

All
the angles of the hexagon ${ABCDEF}$ are equal. Prove that
${AB-DE=EF-BC=CD-FA}$

Consider
a quadrilateral with ${\angle DAB = 60^\circ, \angle ABC = 90^\circ}$ and
${\angle BCD = 120^\circ }$. The diagonals ${AC}$ and ${BD}$ intersect at ${M}$. If ${MB = 1}$
and ${MD = 2}$, find the area of the quadrilateral ${ABCD}$.

The
point ${P}$ is inside of an equilateral triangle with side length ${10}$ so
that the distance from ${P}$ to two of the sides are ${1}$ and ${3}$. Find the
distance from ${P}$ to the third side.

Let
${ABC }$ be a triangle with ${\angle C =
90^\circ }$ and ${CA \ne CB }$. Let ${CH }$ be an altitude and ${CL }$ be an interior angle bisector. Show that for
${X \ne C }$ on the line ${CL }$, we
have ${\angle XAC \ne \angle XBC }$. Also show that for ${Y \ne C
}$ on the line ${CH }$ we have ${\angle Y AC \ne \angle YBC }$.

Let ${ABC}$ be an equilateral triangle and
${D, E }$ points on the sides ${ [AB]}$
and ${ [AC] }$ respectively. If ${DF, EF }$
(with ${F \in AE, G \in AD }$) are the interior angle bisectors of the
angles of the triangle ${ADE }$, prove that the sum of the areas of the
triangles ${DEF }$ and ${DEG }$ is at most equal with the area of the
triangle ${ABC}$. When does the equality hold?

Prove
that no three points with integer coordinates can be the vertices of an
equilateral triangle.

Consider
a convex quadrilateral ${ABCD}$ with ${AB = CD}$ and ${\angle BAC = 30^\circ}$.
If ${\angle ADC = 150^\circ }$, prove that ${\angle BCA =\angle ACD}$.

A
triangle ${ABC}$ is inscribed in the circle ${C(O,R) }$. Let $\alpha< 1$ be
the ratio of the radii of the circles tangent to ${C}$, and both of the rays
${(AB}$ and ${(AC}$. The numbers ${ \beta < 1}$ and ${\gamma< 1}$ are defined analogously.
Prove that ${\alpha+\beta+ \gamma = 1}$.

Dan Brânzei

Consider
a triangle ${ABC}$ with ${AB = AC}$, and ${D}$ the foot of the altitude from
the vertex ${A}$. The point ${E}$ lies on the side ${AB}$ such that ${\angle
ACE = \angle ECB = 18^\circ }$. If ${AD = 3}$, find the length of the segment
${CE}$.

Sorin Peligrad, Pitești

Consider
the triangle ${ABC}$ with ${\angle A = 90^\circ}$ and ${\angle B \ne \angle C}$. A circle ${C(O,R) }$ passes
through ${B}$ and ${C}$ and intersects the sides ${AB}$ and ${AC}$ at ${D}$ and
${E}$, respectively. Let ${S}$ be the foot of the perpendicular from ${A}$ to
${BC}$ and let ${K}$ be the intersection point of ${AS}$ with the segment
${DE}$. If ${M}$ is the midpoint of ${BC}$, prove that ${AKOM}$ is a
parallelogram.

The
triangle ${ABC }$ has ${CA = CB }$. ${P
}$ is a point on the circumcircle
between ${A }$ and ${B }$ (and on the opposite side of the line ${AB
}$ to ${C}$). ${D }$ is the foot of the perpendicular from ${C
}$ to ${PB}$. Show that ${PA + PB = 2
\cdot PD }$.

Two
circles with centers ${O_1}$ and
${O_2}$ meet at two points ${A}$ and ${B}$
such that the centers of the circles are on opposite sides of the line
${AB}$. The lines ${BO_1}$ and ${BO_2}$ meet their respective circles again at
${B_1}$ and ${B_2}$. Let ${M}$ be the midpoint of ${B_1B_2 }$. Let ${M_1,
M_2}$ be points on the circles of
centers ${O_1}$ and ${O_2}$ respectively, such that ${\angle AO_1M_1
=\angle AO_2M_2}$, and ${B_1}$ lies on
the minor arc ${AM_1}$ while ${B }$ lies
on the minor arc ${AM_2}$. Show that ${\angle MM_1B = \angle MM_2B}$.

Let
${ABC}$ be a triangle with centroid
${G}$ and ${A_1,B_1,C_1}$ midpoints of the sides ${BC,CA,AB}$. A
parallel through ${A_1}$ to ${BB_1}$
intersects ${B_1C_1}$ at ${F}$. Prove
that triangles ${ABC}$ and
${FA_1A}$ are similar if and only if
quadrilateral ${AB_1GC_1}$ is cyclic.

In
triangle ${ABC, H,I,O}$ are orthocenter,
incenter and circumcenter, respectively. ${CI}$
cuts circumcircle at ${L}$. If ${AB = IL}$ and ${AH = OH}$, find angles of triangle
${ABC}$.

Let
${ABC}$ be a triangle with area
${S}$ and points ${D,E, F}$ on the sides ${BC,CA,AB}$. Perpendiculars at
points ${D,E, F}$ to the
${BC,CA,AB}$ cut circumcircle of the
triangle ${ABC at points (D_1,D_2), (E_1,E2), (F_1, F_2) }$. Prove that:
${|D_1B \cdot D_1C - D_2B \cdot D_2C| + |E_1A \cdot E_1C – E_2A \cdot E_2C| +
|F_1B \cdot F_1A - F_2B \cdot F_2A| > 4S }$

Let
${ABC}$ be an isosceles triangle with ${AB = AC}$ and ${\angle A = 20^\circ}$.
On the side ${AC}$ consider point ${D}$ such that ${AD = BC}$. Find ${\angle
BDC}$.

Let
${ABCD}$ be a convex quadrilateral with ${AB = AD}$ and ${BC = CD}$. On the
sides ${AB,BC,CD,DA}$ we consider points ${K,L,L_1,K_1}$ such that
quadrilateral ${KLL_1K_1}$ is rectangle. Then consider rectangles ${MNPQ}$
inscribed in the triangle ${BLK}$, where ${M \in KB,N \in BL, P,Q \in LK}$ and
${M_1N_1P_1Q_1}$ inscribed in triangle
${DK_1L_1}$ where ${P_1 }$ and ${Q_1}$ are situated on the ${L_1K_1,
M}$ on the ${DK_1}$ and ${N_1}$
on the ${DL_1}$. Let ${S, S_1, S_2, S_3}$ be the areas of the
${ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1}$ respectively. Find the maximum possible
value of the expression: ${\frac{S_1+S_2+S_3}{S}}$

Let
${A_1,A_2, ...,A_{2002}}$ be arbitrary points in the plane. Prove that for
every circle of radius ${1}$ and for
every rectangle inscribed in this circle, there exist ${3}$ vertices ${M,N, P}$ of the rectangle such that ${MA_1 + MA2 + ..
+ MA_{2002}+ NA_1 + NA_2 + … + NA_{2002}+ PA_1 + PA_2 + … + PA_{2002 }\ge
6006}$.

Let
${D, E, F}$ be the midpoints of the arcs
${BC, CA, AB}$ on the circumcircle of a
triangle ${ABC}$ not containing the
points ${A, B, C}$, respectively. Let the line ${DE}$ meets ${BC}$
and ${CA}$ at ${G}$ and ${H}$, and let ${M}$ be the midpoint of the segment ${GH}$. Let
the line ${FD}$ meet ${BC}$ and
${AB}$ at ${K}$ and ${J}$, and let ${N}$ be the midpoint of the segment ${KJ}$.

a)
Find the angles of triangle ${DMN}$,

b) Prove that if ${P}$ is the point of
intersection of the lines ${AD}$ and
${EF}$, then the circumcenter of triangle ${DMN}$ lies on the circumcircle of triangle ${PMN}$.

Ch. Lozanov

Is
there is a convex quadrilateral which the diagonals divide into four triangles
with areas of distinct primes?

Is
there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?

Let
$G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if
and only if $GA \perp GC$.

Three
equal circles have a common point $M$ and intersect in pairs at points $A, B,
C$. Prove that that $M$ is the orthocenter of triangle $ABC$.

Let
$ABC$ be an isosceles triangle with $AB = AC$. A semi-circle of diameter $[EF]
$ with $E, F \in [BC]$, is tangent to the sides $AB,AC$ in $M, N$
respectively and $AE$ intersects the
semicircle at $P$. Prove that $PF$ passes through the midpoint of $[MN]$.

Parallels
to the sides of a triangle passing through an interior point divide the inside
of a triangle into $6$ parts with the marked areas as in the figure. Show that
$\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$

**2004 JBMO problem 2**

Let
${ABC}$ be an isosceles triangle with
${AC = BC}$, let ${M}$ be the midpoint
of its side ${AC}$, and let ${Z}$ be the
line through ${C}$ perpendicular to
${AB}$. The circle through the points ${B, C}$, and ${M}$ intersects the line ${Z}$ at the points ${C}$ and ${Q}$. Find the radius of the
circumcircle of the triangle ${ABC}$ in
terms of ${m = CQ}$.

Two circles $C_1$ and $C_2$ intersect in points $A$
and $B$. A circle $C$ with center in $A$ intersect $C_1$ in $M$ and $P$ and
$C_2$ in $N$ and $Q$ so that $N$ and $Q$ are located on different sides wrt
$MP$, and $AB> AM$. Prove that
$\angle MBQ = \angle NBP$.

Let
$E, F$ be two distinct points inside a parallelogram $ABCD$ . Determine the
maximum possible number of triangles having the same area with three vertices
from points $A, B, C, D, E, F$.

Let $ABC$ be a triangle inscribed in circle $C$.
Circles $C_1, C_2, C_3$ are tangent internally with circle $C$ in $A_1, B_1,
C_1$ and tangent to sides $[BC], [CA], [AB]$ in points $A_2, B_2, C_2$
respectively, so that $A, A_1$ are on one side of $BC$ and so on. Lines
$A_1A_2, B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at
points $A’,B’$ and $C’$, respectively. If $ M = BB’ \cap CC’$, prove that $m
(\angle MAA’) = 90^\circ$ .

**2005 JBMO problem 2**

Let
$ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$
wrt $AC$. Prove that the quadrilateral $ANDE$
is cyclic.

Let
$ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m
(\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of
$\frac{DM}{ME}$.

Let $ABC$
be an isosceles triangle $(AB=AC)$ so that
$\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height
$AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$,
and let $K$ the orthogonal projection of $Z$ on $AE$.
Prove that $ \angle KDZ = \angle KDB
= \angle KZB$.

**2005 JBMO Shortlist 8**

Let $O$ be
the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$
respectively. Let $A\in C_1, B\in
C_2$ and $C$ point so that triangle
$ABC$ is equilateral. Find the maximum length of $ [OC] $.

Let
$C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be
the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random
line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of
triangle $DEP$ lies on the circumcircle of triangle $OKP$.

**2005 JBMO Shortlist 10**

Let
$ABCD$ be a parallelogram.

$P \in (CD), Q \in (AB), M= AP \cap DQ, N=BP
\cap CQ, K=MN \cap AD, L= MN \cap BC$.

Prove
that $BL=DK$.

**2006 JBMO Shortlist 9**

Let
${ABCD}$ be a trapezoid with ${AB // CD, AB > CD}$ and ${\angle A + \angle B
= 90^\circ}$. Prove that the distance between the midpoints of the bases is
equal to the semidifference of the bases.

Let
${ABCD}$ be an isosceles trapezoid inscribed in a circle ${C}$ with ${AB // CD,
AB = 2CD}$. Let ${Q = AD \cap BC}$ and let ${P}$ be the intersection of the
tangents to ${C}$ at ${B}$ and ${D}$. Calculate the area of the quadrilateral
${ABPQ}$ in terms of the area of the triangle ${PDQ}$.

Circles
${C_1}$ and ${C_2}$ intersect at ${A}$ and ${B}$. Let ${M \in AB}$. A line
through ${M}$ (different from ${AB)}$ cuts circles in ${C_1}$ and ${C_2}$ in
${Z,D,E,C}$ respectively such that ${D,E \in ZC}$. Perpendiculars at ${B}$ to
the lines ${EB,ZB}$ and ${AD}$ respectively cut circle ${C_2}$ in ${F,K}$ and
${N}$. Prove that ${KF = NC}$.

Let
${ABC}$ be an equilateral triangle of center ${O}$, and ${M \in BC}$. Let
${K,L}$ be projections of ${M}$ onto the sides ${AB}$ and ${AC}$ respectively.
Prove that line ${OM}$ passes through the midpoint of the segment ${KL}$.

**2007 JBMO Shortlist G1**

Let $M$ be an interior point of the triangle $\angle
ABC$ with angles $\angle BAC={{70}^{o}}$.
And $\angle ABC={{80}^{o}}$. If $\angle ACM={{10}^{o}}$. and $\angle CBM={{20}^{o}}$., prove that
$\angle AB = M C$.

Let$ABCD$be
a convex quadrilateral with $\angle DAC=\angle BDC={{36}^{o}},\angle
CBD={{18}^{o}}$ and $\angle BAC={{72}^{o}}$. If $P$ is the point of
intersection of the diagonals $AC$ and $BD$, find the measure of $\angle AP D$.

Let the inscribed
circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at
$N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such
that $\frac{DP}{DN}=\frac{BD}{CD}$ .
Show that $DM \perp PN$ .

Let $S$ be a
point inside $\angle pOq$, and let $k$be a circle which contains $S$ and
touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight
line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$
be the point of intersection of the ray $PS$ and circumscribed circle of
$\vartriangle SQR$ and $T \ne S$. Prove
that $OT // SQ$ and $OT$ is a tangent of the circumscribed circle of
$\vartriangle SQR$.

Two perpendicular chords of a circle, $AM, BN$ , which intersect at point
$K$, define on the circle four arcs with pairwise different length, with $AB$
being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C,
D$ different from $N, M$ . If $L$ is the point of intersection of $DN, M C$ and
$T$ the point of intersection of $DC, KL,$ prove that $\angle KTC = \angle
KNL$.

For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$),
a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$)
in a way such that $AM -BC = BN- AC = CP – AB$. Prove that the angles of
triangle $MNP$ do not depend on the choice of $M, N, P$ .

The vertices $A$ and $B$ of an equilateral $\vartriangle ABC$ lie on a
circle $k$ of radius $1$, and the vertex $C$ is inside $k$. The point $D \ne B$
lies on $k$, $AD = AB$ and the line $DC$ intersects $k$ for the second time in
point $E$. Find the length of the segment $CE$.

Let $ABC$ be a triangle, ($BC < AB$). The line

*$l$*passing trough the vertices $C$ and orthogonal to the angle bisector $BE$ of $\angle B$, meets $BE$ and the median $BD$ of the side $AC$ at points $F$ and $G$, respectively. Prove that segment $DF$ bisects the segment $EG$.
Is it possible to cover a given square with a few congruent right-angled
triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not
overlap and may not exceed the margins of the square.)

Let $ABC$ be a triangle with $\angle A<{{90}^{o}} $. Outside of a
triangle we consider isosceles triangles $ABE$ and $ACZ$ with bases $AB$ and
$AC$, respectively. If the midpoint $D$ of the side $BC$ is such that $DE \perp
DZ$ and $EZ = 2 \cdot ED$, prove that $\angle AEB = 2 \cdot \angle AZC$ .

Let $ABC$ be an isosceles triangle with $AC = BC$. The point $D$ lies on the side $AB$ such that
the semicircle with diameter $BD$ and center $O$ is tangent to the side $AC$ in
the point $P$ and intersects the side $BC$ at the point $Q$. The radius $OP$
intersects the chord $DQ$ at the point $E$ such that $5 \cdot PE = 3 \cdot DE$.
Find the ratio $\frac{AB}{BC}$ .

The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$
and $y$.

Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left(
\frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left(
\frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.

Let $O$ be a point inside the parallelogram $ABCD$ such that $\angle AOB + \angle
COD = \angle BOC + \angle AOD$. Prove that there exists a circle $k$ tangent to
the circumscribed circles of the triangles $\vartriangle AOB, \vartriangle BOC,
\vartriangle COD$ and $\vartriangle DOA$.

Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not
intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto
$\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$
the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of
$\gamma$ and $\Gamma$, and by $Y$ the
(other than $A$) intersection point of $\gamma$
and $\delta$. Prove that the line $XY$
passes through a fixed point.

Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the
midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively
and let $P$ be the intersection point of the lines $DE$ and $BC$. The
perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at
point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

Parallelogram ${ABCD}$ is given with ${AC>BD}$,
and ${O}$ point of intersection of ${AC}$ and ${BD}$. Circle with center at ${O}$and
radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$
be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle
ZCA={{90}^{{}^\circ }}$.

In right trapezoid ${ABCD \left(AB\parallel
CD\right)}$ the angle at vertex B measures ${{75}^{{}^\circ }}$. Point ${H}$is
the foot of the perpendicular from point ${A}$ to the line ${BC}$. If ${BH=DC}$
and${AD+AH=8}$, find the area of ${ABCD}$.

Parallelogram ${ABCD}$with obtuse angle $\angle
ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$,
we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear.
Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects
the straight line ${BD}$at point ${P}$. Prove that ${PC}$is the bisector of the
angle $\angle BP{D}'$.

Let ${ABCDE}$ be convex pentagon such that ${AB+CD=BC+DE}$ and ${k}$ half circle with center on side ${AE}$ that touches sides ${AB, BC, CD}$ and ${DE}$ of pentagon, respectively, at points ${P, Q, R}$ and ${S}$ (different from vertices of pentagon). Prove that $PS\parallel AE$.

Let ${A, B, C}$ and ${O}$ be four points in plane,
such that $\angle ABC>{{90}^{{}^\circ }}$ and ${OA=OB=OC}$.Define the point ${D\in
AB}$ and the line ${l}$ such that ${D\in l, AC\perp DC}$ and ${l\perp AO}$. Line
${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and
${CFD}$are tangent at ${F}$.

Consider a triangle ${ABC}$ with${\angle ACB=90^{\circ}.}$ Let ${F}$ be the foot
of the altitude from ${C}$. Circle ${\omega}$ touches the line segment ${FB}$at
point ${P,}$ the altitude ${CF}$at point
${Q}$ and the circumcircle of ${ABC}$at point ${R.}$ Prove that points ${A,Q,R}$
are collinear and ${AP=AC}$.

Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$
Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the
measure of ${\angle ABC}$.

Let ${ABC}$be an acute-angled triangle. A circle ${\omega_1(O_1,R_1)}$
passes through points ${B}$ and ${C}$ and meets the sides ${AB}$and ${AC}$at
points ${D}$ and ${E,}$ respectively. Let ${\omega_2(O_2,R_2)}$be the
circumcircle of the triangle ${ADE.}$. Prove that ${O_1O_2}$is equal to the
circumradius of the triangle ${ABC}$.

Let ${AL}$and ${BK}$be angle bisectors in the
non-isosceles triangle ${ABC}$ ($L\in BC,$ $K\in AC$). The perpendicular
bisector of ${BK}$ intersects the line ${AL}$ at point ${M}$. Point ${N}$ lies on
the line ${BK}$such that $LN\parallel MK$. Prove that ${LN=NA}$.

Let $ABC$ be an isosceles triangle with $AB=AC$. On the
extension of the side ${CA}$ we consider the point ${D}$

*such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$**at the points ${E}$and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$**are concyclic.*
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle
ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at
the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle
${\angle{CGH}}$.

Let $ABC$ be a triangle in which (${BL}$is the angle
bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle
ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It
is known that the midpoints of the
segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle
$\vartriangle ABC$.

Point ${D}$ lies on the side ${BC}$ of $\vartriangle
ABC$. The circumcenters of $\vartriangle ADC$ and $\vartriangle BAD$ are ${O_1}$ and
${O_2}$, respectively and ${O_1O_2\parallel AB}$. The orthocenter of $\vartriangle
ADC$is ${H}$ and ${AH=O_1O_2}.$ Find the angles of $\vartriangle ABC$ if $2m\left( \angle C \right)=3m\left( \angle B
\right).$

Inside the square ${ABCD}$, the equilateral triangle $\vartriangle
ABE$is constructed. Let ${M}$ be an interior point of the triangle $\vartriangle
ABE$ such that ${MB=\sqrt{2}, MC=\sqrt{6}, MD=\sqrt{5}}$ and ${ME=\sqrt{3}}$.
Find the area of the square ${ABCD}$.

Let ${ABCD}$ be a convex quadrilateral, $E$ and $F$ points on the
sides $AB$ and ${CD}$, respectively,
such that ${AB:AE=CD:DF=n}$. Denote by ${S}$ the area of the quadrilateral${AEFD}$.
Prove that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+nDA\cdot BC}{2n^2}}$

Let $ABC$ be an equlateral triangle and ${P}$ a point on the circumcircle of the
triangle $ABC$, and distinct from ${A, B}$ and ${C}$. If the lines through ${P}$
and parallel to $BC,CA,AB$ intersect the lines $CA,AB,BC$ at $M,N,Q$
respectively, prove that ${M, N}$ and ${Q}$ are collinear.

Let $ABC$ be an isosceles triangle with $AB=AC$.
Let also ${c\left(K, KC\right)}$ be a circle tangent to the line ${AC}$ at point${C}$ which it intersects the segment ${BC}$ again at an interior
point ${H}$. Prove that ${HK\perp AB}$.

Let $AB$ and $CD$ be chords in a circle of
center ${O}$ with $A,B,C,D$
distinct, and let the lines $AB$ and $CD$
meet at a right angle at point ${E}$. Let also $M$ and $N$ be the midpoints of $AC$
and $BD$, respectively. If ${MN\perp OE}$, prove that ${AD\parallel BC}$.

Let $ABC$
be an acute-angled triangle with circumcircle $\Gamma $, and let ${O, H}$ be the triangle’s
circumcenter and orthocenter respectively. Let also ${A'}$ be the point where
the angle bisector of angle ${\angle BAC}$ meets $\Gamma $. If ${A'H=AH}$, find the measure of the angle${\angle
BAC}$.

**2012 JBMO Shortlist G5 problem 2**

Let the circles ${{k}_{1}}$ and ${{k}_{2}}$ intersect
at two distinct points ${A}$ and ${B}$ , and let $t$t be a common tangent of ${{k}_{1}}$ and
${{k}_{2}}$, that touches ${{k}_{1}}$ and ${{k}_{2}}$ at ${M}$ and ${N}$,
respectively. If $t\bot AM$ and ${MN=2AM}$, evaluate ${\angle{NMB}}$.

Let ${O_1}$ be a point in the exterior of the circle
${c\left(O, R\right)}$ and let ${O_1N, O_1D}$ be the tangent segments from ${O_1}$
to the circle. On the segment ${O_1N}$ consider the point ${B}$ such that ${BN=R}$.
Let the line from ${B}$ parallel to ${ON}$, intersect the segment ${O_1D}$ at ${C}$.
If ${A}$ is a point on the segment ${O_1D}$, other than ${C}$ so that ${BC=BA=a}$,
and if ${c'\left(K, r\right)}$ is the
incircle of the triangle ${{O}_{1}}AB$ find
the area of $ABC$ in terms of $a,R,r$.

Let ${MNPQ}$ be a square of side length 1, and ${A, B, C, D}$

*points on the sides $MN,NP,PQ$ and $QM$**respectively such that ${AC\cdot BD=\dfrac{5}{4}}$.**Can the set ${\left\{AB, BC, CD, DA\right\}}$**be partitioned into two subsets ${{S}_{1}}$and ${{S}_{2}}$of two elements each such that both the sum of the elements of ${{S}_{1}}$and the sum of the elements of ${{S}_{2}}$ are positive integers?*
by Flavian Georgescu

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to
$AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the
second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the
intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove
that points ${M,O,P,N}$ are cocyclic.

Circles
${\omega_1}$ , ${\omega_2}$ are
externally tangent at point M and tangent internally with circle
${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and
${B}$be the points that their common
tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with
circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$

Theoklitos Paragyiou

Let ${ABC}$ be an acute triangle with ${AB<AC}$ and ${O}$
be the center of its circumcircle $\omega $. Let ${D}$ be a point on the line
segment ${BC}$ such
that $\angle BAD=\angle CAO$. Let ${E}$ be the second point of intersection of ${\omega}$ and the line $AD$.
If ${M, N}$ and ${P}$ are the midpoints of the line segments ${BE, OD}$ and$\left[
AC \right]$ respectively, show that the points ${M, N}$ and ${P}$ are
collinear.

Stefan
Lozanovski

Let ${I}$ be the incenter and ${AB}$ the shortest side of a
triangle${ABC.}$ The circle with center ${I}$ and passing through ${C}$ intersects the ray ${AB}$ at the point ${P}$ and the ray
${BA}$ at the point$Q$. Let ${D}$ be the point
where the excircle of the triangle ${ABC}$ belonging to angle ${A}$ touches the side${BC}$, and let ${E}$ be the
symmetric of the point ${C}$ with respect to $D$. Show that the lines ${PE}$ and ${CQ}$ are perpendicular.

A circle passing through the midpoint ${M}$of
side ${BC}$ and the vertex ${A}$ of a triangle ${ABC}$, intersects sides ${AB}$
and ${AC}$ for the second time at points ${P}$ and ${Q,}$ respectively. Prove
that if $\angle BAC={{60}^{{}^\circ }}$ then ${AP+AQ+PQ<AB+AC+\frac{1}{2}BC.}$

Let ${P}$and ${Q}$be the
midpoints of the sides ${BC}$and ${CD}$, respectively in a rectangle $ABCD$.
Let ${K}$
and ${M}$be the
intersections of the line ${PD}$with the lines ${QB}$and $QA$respectively, and
let ${N}$ be
the intersection of the lines ${PA}$and ${QB.}$. Let $X,Y$and ${X,Y,Z}$be the midpoints of the segments , $AN,KN$and $AM$
respectively. Let ${l_1}$be the line passing through ${X}$and perpendicular to $MK,\,\,{{l}_{2}}$
be the line passing through ${Y}$and perpendicular to ${AM}$ and ${l_3}$ the line passing through ${Z}$and perpendicular to ${KN.}$.
Prove that the lines ${{l}_{1}},{{l}_{2}}$and ${l_1,l_2,l_3}$ are concurrent.

Theoklitos Paragyiou

Let
${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C
\right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects
${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

Evangelos Psychas

Consider
an acute triangle ${ABC}$ with area ${S}$. Let ${CD\perp AB \; (D\in
AB), DM\perp AC \; (M\in AC)}$ and ${DN \perp BC \; (N\in BC)}$.
Denote by ${H_1}$ and ${H_2}$ the orthocenters of the triangles ${MNC}$
and ${MND}$ respectively. Find the area of the quadrilateral
${AH_1BH_2}$ in terms of ${S}$.

Let
${ABC}$ be a triangle such that ${AB\ne AC}$. Let ${M}$ be the midpoint
of ${BC,H}$ be the orthocenter of triangle $ABC$,${O_1}$ be the
midpoint of ${AH}$, ${O_2}$ the circumcentre of triangle $BCH$. Prove
that ${O_1AMO_2}$ is a parallelogram.

Let
$ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal
bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$
the midpoint of the arc ${AC}$ which contains point ${B}$. The
circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the
segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of
${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove
that the points ${J, B, M, O}$ belong to the same circle.

Let
${ABCD}$ be a quadrilateral whose diagonals are not perpendicular and
whose sides ${AB\nparallel CD}$ and ${AB\nparallel CD}$ are not
parallel. Let ${O}$ be the intersection of its diagonals. Denote with
${H_1}$and ${H_2}$ the orthocenters of triangles $\vartriangle OAB$ and
$\vartriangle OCD$, respectively. If ${M}$ and ${N}$ are the midpoints
of the segment lines $\left[ AB \right]$ and $\left[ CD \right]$,
respectively, prove that the lines ${H_1H_2}$ and ${MN}$are parallel if
and only if $AC=BD$.

Flavian Georgescu

Around the triangle $ABC$ the
circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to
this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$,
respectively. Prove that the points $A,B,D,E$ belong to the same circle.

The point ${P}$ is outside the
circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the
circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in
BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$
intersects again the circle ${\Omega}$ at
the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.

Let ${c\equiv c\left(O,
R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two
points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$
intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the
circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.

Evangelos
Psychas

Let $\vartriangle ABC$ be an
acute triangle. Lines ${l_1, l_2}$ are perpendicular to ${AB}$ at the points ${A}$
and ${B}$, respectively. The perpendicular lines from the midpoints ${M}$ of ${AB}$
to the sides of the triangle ${AC}$ , ${BC}$ intersect ${l_1}$ , ${l_2}$ at the
points ${E}$ , ${F}$, respectively. If ${D}$ είναι το σημείο τομής των ${EF}$ και ${MC}$, να αποδείξετε ότι $\angle
ADB=\angle EMF$.

Theoklitos Paragyiou

Let $ABC$ be an acute triangle
with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle κύκλος touches the sides ${BC, CA}$ and ${AB}$ at ${D,
E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$onto ${BC}$
meets ${EF}$at ${M}$, and similarly the perpendicular line erected at ${B}$onto
${BC}$ meets ${EF}$at${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$,
and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$.

Ruben Dario and Leo Giugiuc

Let ${ABC}$ be an acute angled triangle, let ${O}$be its
circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$,
respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses
the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again
at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses
the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$.
Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the
segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$.
Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic,
and their circumcircles share a common point.

Evangelos Psychas

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$.
Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external
angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be
the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the
triangles ${ADE}$and ${BOC}$ are tangent to each other.

A trapezoid ${ABCD}$ (${AB\parallel CD}$,${AB>CD}$)
is circumscribed. The incircle of triangle ${ABC}$ touches the lines ${AB}$and ${AC}$ at ${M}$ and ${N}$, respectively. Prove that the
incenter of the trapezoid lies on the line ${MN}$.

Let ${ABC}$ be an acute angled triangle whose shortest side
is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$
and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and
${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle
${ADE}$ passes through a fixed point.

Let $ABC$ be an acute angled triangle with orthocenter ${H}$
and circumcenter ${O}$. Assume the
circumcenter ${X}$ of ${BHC}$lies on the circumcircle of ${ABC}$. Reflect O
across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[
XB \right],\left[ XC \right]$] and $\left[ BC \right]$, respectively. Prove
that the points $K,L,M$ and ${K,L,M,N}$ are cocyclic.

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$
externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle
CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet
of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the
midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the
circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.

Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and
let ${K}$ be a point on the segment ${AB}$
such that ${AK<BK}$. Two circles through ${K}$, internally tangent to
${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one
of the points of intersection of the line ${KL}$and the circle ${(c)}$, and let the lines ${AB}$and ${LO}$meet at ${M}$.
Prove that the line ${MP}$is tangent to the circle ${(c)}$.

Theoklitos Paragyiou

**2017 JBMO Shortlist G1**

Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ , ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .

Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $ M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.

Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$ . Let $S$ be the midpoint of segment $AD$. If $\angle ADE = \angle ABC - \angle ACB$ prove that $\angle BSC = 2 \angle BAC$ .

**2017 JBMO Shortlist G4 problem 3**

Let ${ABC}$ be an acute
triangle such that ${AB \neq AC}$, with circumcircle ${\Gamma}$ and
circumcenter ${O}$. Let ${M}$ be the midpoint of ${BC}$and ${D}$ be a point on ${\Gamma}$
such that ${AD \perp BC}$. Let ${T}$ be a point such that ${BDCT}$is a
parallelogram and ${Q}$ a point on the same side of ${BC}$as ${A}$ such that ${
\angle BQM = \angle BCA}$ and ${ \angle CQM = \angle CBA}$. Let the line ${AO}$ intersect
${\Gamma}$ at ${E}$, (${E \neq A}$) and let the circumcircle of ${ETQ}$ intersect
${\Gamma}$ at point ${X \neq E}$. Prove that the points ${A,M}$ and ${X}$ are collinear.

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

$C$ with respect to $BC, CA$, and $AB$, respectively, and let the circumcircles of triangles $ABB'$ and $ACC'$ meet again at $A_1$. Points $B_1$ and $C_1$ are de ned similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ have a common point.

source:

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