geometry problems from British Math Olympiads (BrMO)

with aops links

1992-93 BrMO Round 1 P2

A square piece of toast $ABCD$ of side length $1$ and centre $O$ is cut in half to form two equal pieces $ABC$ and $CDA$. If the triangle $ABC$ has to be cut into two parts of equal area, one would usually cut along the line of symmetry $BO$. However, there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle $ABC$ into two parts of equal area.

1993-94 BrMO Round 2 P3

$ AP, AQ, AR, AS$ are chords of a given circle with the property that $ \angle PAQ = \angle QAR = \angle RAS$. Prove that $ AR(AP+AR) = AQ(AQ+AS)$.

1994-95 BrMO Round 2 P2

Let $ABC$ be a triangle, and $D,E,F$ be the midpoints of $BC, CA, AB$, respectively.

Prove that if $\angle DAC=\angle ABE$ ,if and only if $\angle AFC=\angle ADB$.

1997-98 BrMO Round 2 P2

A triangle $ ABC$ has $ \angle BAC > \angle BCA$. A line $ AP$ is drawn so that $ \angle PAC =\angle BCA$ where $ P$ is inside the triangle. A point $ Q$ outside the triangle is constructed so that $ PQ$ is parallel to $ AB$, and $ BQ$ is parallel to $ AC$. $ R$ is the point on $ BC$ (separated from $ Q$ by the line $ AP$) such that $ \angle PRQ = \angle BCA$. Prove that the circumcircle of $ ABC$ touches the circumcircle of $ PQR$.

Let $ABCDEF$ be a hexagon (which may not be regular), which circumscribes a circle $S$. (That is, $S$ is tangent to each of the six sides of the hexagon.) The circle $S$ touches $AB, CD, EF$ at their midpoints $P,Q,R$ respectively. Let $X,Y,Z$ be the points of contact of $S$ with $BC, DE, FA$ respectively. Prove that $PY, QZ, RX$ are concurrent.

1999-2000 BrMO Round 1 P1

Two intersecting circles $C_1$ and $C_2$ have a common tangent which touches $C_1$ at $P$ and $C_2$ at $Q$. The two circles intersect at $M$ and $N$, where $N$ is nearer to $PQ$ than $M$ is. The line $PN$ meets the circle $C_2$ again at $R$. Prove that $MQ$ bisects angle $PMR$.

1999-2000 BrMO Round 1 P3

Two intersecting circles $C_1$ and $C_2$ have a common tangent intersecting $C_1$ in $P$ and $C_2$ in $Q$. The $2$ circles intersect in $M$ and $N$ where $N$ is nearer to $PQ$ than $M$.

Prove that the triangles $MNP$ and $MNQ$ have equal areas.

2000-01 BrMO Round 1 P2

Circle $S$ lies inside circle $T$ and touches it at $A$. From a point $P$ (distinct from $A$) on $T$, chords $PQ$ and $PR$ of $T$ are drawn touching $S$ at $X$ and $Y$ respectively. Show that $\angle QAR=2\angle XAY$.

2000-01 BrMO Round 1 P5

A triangle has sides of length $a,b,c$ and its circumcircle has radius $R$. Prove that the triangle is right angled if and only if $a^{2}+b^{2}+c^{2}=8R^{2}$

A triangle $ABC$ has $\angle ACB > \angle ABC$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. The point $E$ on $AB$ is such that $\angle EDB=90^{o}$. The point $F$ on $AC$ is such that $\angle BED=\angle DEF$. Show that $\angle BAD=\angle FDC$

2001-02 BrMO Round 1 P2

The quadrilateral $ABCD$ is inscribed in a circle. The diagonals $AC,BD$ meet at $Q$. The sides $DA$, extended beyond $A$, and $CB$, extended beyond $B$, meet at $P$. Given that $CD=CP=DQ$, prove that $\angle CAD=60$.

2001-02 BrMO Round 2 P1

The altitude from one of the vertices of an acute-angled triangle $ABC$ meets the opposite side at $D$. From $D$ perpendiculars $DE$ and $DF$ are drawn to the other two sides. Prove that the length of $EF$ is the same whichever vertex is chosen.

2002-03 BrMO Round 1 P2

The triangle $ABC$, where $AB<AC$, has a circumcircle $S$. The perpendicular from $A$ to $BC$ intersects $S$ again at $P$. The point $X$ lies on the line segment $AC$, and $BX$ intersects $S$ again at $Q$. Show that $BX=CX$ if and only if $PQ$ is a diameter of $S$.

2002-03 BrMO Round 2 P2

Let $ABC$ be a triangle and let $D$ be a point on $AB$ such that $4AD = AB$. The half-line $\ell$ is drawn on the same side of $AB$ as $C$, starting from $D$ and making an angle of $\theta$ with $DA$ where $\theta =\angle ACB$. If the circumcircle of $ABC$ meets the half-line $\ell$ at $P$, show that $PB = 2PD$.

2003-04 BrMO Round 1 P2

$ABCD$ is a rectangle , $P$ is the midpoint of $AB$ and $Q$ is the $PD$ such that $CQ$ is perpendicular to $PD$ . Prove that triangle $BQC$ is isosceles.

2003-04 BrMO Round 2 P1

Let $ABC$ be an equilateral triangle and $D$ an internal point of the side $BC$. A circle, tangent to $BC$ at $D$, cuts $AB$ internally at $M$ and $N$, and $AC$ internally at $P$ and $Q$.

Show that $BD + AM + AN = CD + AP + AQ$.

2004-05 BrMO Round 1 P2

Let $ABC$ be an acute-angled triangle, and let $D,E$ be the feet of the perpendiculars from $A,B$ to $BC,CA$ respectively.Let $P$ be the point where the line $AD$ meets the semicircle constructed outwardly on $BC$, and $Q$ be the point where line $BD$ meets the semicircle constructed outwardly on $AC$. Prove that $CP = CQ$.

2004-05 BrMO Round 2 P2

In triangle $ABC, \angle BAC = 120^o.$ Let the angle bisectors of angles $A,B$ and $C$ meet the opposite sides in $D,E$ and $F$ respectively. Prove that the circle on diameter $EF$ passes through $D$.

2005-06 BrMO Round 1 P3

In the cyclic quadrilateral $ABCD$, the diagonal $AC$ bisects the angle $DAB$. The side $AD$ is extended beyond $D$ to a point $E$. Show that $CE = CA$ if and only if $DE = AB$.

2005-06 BrMO Round 1 P5

Let $ABC$ be a triangle with $AC > AB$. The point $X$ lies on the side $BA$ extended through $A$, and the point $Y$ lies on the side $CA$ in such a way that $BX = CA$ and $CY = BA$. The line $XY$ meets the perpendicular bisector of side $BC$ at $P$. Show that $\angle BPC + \angle BAC = 180$.

2006-07 BrMO Round 1 P2

In the convex quadrilateral $ABCD$, points $M,N$ lie on the side $AB$ such that $AM = MN = NB$, and points $P,Q$ lie on the side $CD$ such that $CP = PQ = QD$. Prove that

Area of $AMCP =$ Area of $MNPQ = \frac{1}{3}$ Area of $ABCD$

2006-07 BrMO Round 1 P4

Two touching circles $S$ and $T$ share a common tangents which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP=PQ$.

Let $ABC$ be an acute-angled triangle with $AB > AC$ and $\angle BAC =60$ . Denote the circumcentre by $O$ and the orthocentre by $H$ and let $OH$ meet $AB$ at $P$ and $AC$ at $Q$. Prove that $PO = HQ$

2007-08 BrMO Round 1 P3

Let $ABC$ be a triangle, with an obtuse angle at $A$. Let $Q$ be a point (other than $A, B$ or $C$ ) on the circumcircle of the triangle, on the same side of chord $BC$ as $A$, and let $P$ be the other end of the diameter through $Q$. Let $V$ and $W$ be the feet of the perpendiculars from $Q$ onto $CA$ and $AB$ respectively. Prove that the triangles $PBC$ and $AWV$ are similar.

2007-08 BrMO Round 1 P5

Let $ P$ be an internal point of triangle $ ABC$. The line through $P$ parallel to $ AB$ meets $ BC$ at $ L$, the line through $ P$ parallel to $BC$ meets $ CA$ at $ M$, and the line through $ P$ parallel to $ CA$ meets $ AB$ at $ N$. Prove that $ \frac{BL}{LC}\times\frac{CM}{MA}\times\frac{AN}{NB}\le\frac{1}{8}$ and locate the position of $ P$ in triangle $ ABC$ when equality holds.

2007-08 BrMO Round 2 P2

Let triangle ABC have incentre $ I$ and circumcentre $ O$. Suppose that $ AIO =90^\circ$ and $ CIO = 45^\circ$ . Find the ratio $ AB: BC: CA$.

2008-09 BrMO Round 1 P3

Let $ ABPC$ be a parallelogram such that $ ABC$ is an acute-angled triangle. The circumcircle of triangle $ ABC$ meets the line $ CP$ again at $ Q$. Prove that $ PQ = AC$ if, and only if, $ BAC = 60^{\circ}$ .

2008-09 BrMO Round 2 P2

Let $ABC$ be an acute-angled triangle with $ \angle B = \angle C$. Let the circumcentre be $O$ and the orthocentre be $H$. Prove that the centre of the circle $BOH$ lies on the line $AB$. The circumcentre of a triangle is the centre of its circumcircle. The orthocentre of a triangle is the point where its three altitudes meet.

2009-10 BrMO Round 1 P2

Points $A,B,C,D$ and $E$ lie, in that order, on a circle and the lines $AB$ and $ED$ are parallel. Prove that $\angle{ABC} = 90^{\circ}$ if, and only if, $AC^2 = BD^2 + CE^2$.

2009-10 BrMO Round 1 P4

In triangle $ ABC$ the centroid is $ G$ and $ D$ is the midpoint of $ CA$. The line through $ G$ parallel to $ BC$ meets $ AB$ at $ E$. Prove that $ \angle AEC = \angle DGC$ if, and only if, $ \angle ACB = 90^{\circ}$

2010-11 BrMO Round 1 P3

Let $ABC$ be a triangle with $\angle CAB$ a right-angle. The point $L$ lies on the side $BC$ between $B$ and $C$. The circle $ABL$ meets the line $AC$ again at $M$ and the circle $CAL$ meets the line $AB$ again at $N$. Prove that $L, M$ and $N$ lie on a straight line.

2010-11 BrMO Round 1 P5

Let $ABC$ be a triangle and $X$ be a point inside the triangle. The lines $AX,BX$ and $CX$ meet the circle $ABC$ again at $ P,Q$ and $R$ respectively. Choose a point $U$ on $XP$ which is between $X$ and $P$. Suppose that the lines through $U$ which are parallel to $AB$ and $CA$ meet $XQ$ and $XR$ at points $V$ and $W$ respectively. Prove that the points $R,W, V$ and $Q$ lie on a circle.

2011-12 BrMO Round 1 P3

Consider a circle $S$. The point $P$ lies outside $S$ and a line is drawn through $P$, cutting $S$ at distinct points $X$ and $Y$ . Circles $S_1$ and $S_2$ are drawn through $P$ which are tangent to $S$ at $X$ and $Y$ respectively.Prove that the difference of the radii of $S_1$ and $S_2$ is independent of the positions of $P$,$ X$ and $Y$

2011-12 BrMO Round 1 P6

Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A$, $B$ and $C$ are $D , E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds.

2011-12 BrMO Round 2 P1

The diagonals $AC$ and $BD$ of a cyclic quadrilateral meet at $E$. The midpoints of the sides $AB, BC, CD$ and $DA$ are $P, Q, R$ and $S$ respectively. Prove that the circles $EPS$ and $EQR$ have the same radius.

2012-13 BrMO Round 1 P2

Two circles $S$ and $T$ touch at $X$. They have a common tangent which meets $S$ at $A$ and $T$ at $B$. The points $A$ and $B$ are different. Let $AP$ be a diameter of $S$. Prove that $B, X$ and $P$ lie on a straight line.

A triangle has sides of length at most $2,3$ & $4$ respectively. Determine with proof the max. possible area of the triangle.

2012-13 BrMO Round 1 P6

Let $ABC$ be a triangle. Let $S$ be the circle through $B$ tangent to $CA$ at $A$ and let $T$ be the circle through $C$ tangent to $AB$ at $A$. The circles $S$ and $T$ intersect at $A$ and $D$. Let $E$ be the point where the line $AD$ meets the circle $ABC$. Prove that $D$ is the midpoint of $AE$.

2012-13 BrMO Round 2 P2

The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.

2013-14 BrMO Round 1 P2

In the acute-angled triangle $ABC$, the foot of the perpendicular from $B$ to $CA$ is $E$. Let $l$ be the tangent to the circle $ABC$ at $B$. The foot of the perpendicular from $C$ to $l$ is $F$, Prove that $EF$ is parallel to $AB$.

2013-14 BrMO Round 1 P5

Let $ABC$ be an equilateral triangle, and let $P$ be a point inside this triangle. Let $D, E$ and $F$ be the feet of the perpendiculars from $P$ to the sides $BC, CA$ and $AB$ respectively. Prove that

(a) $AF + BD + CE = AE + BF + CD$ and

(b) $[APF] + [BPD] + [CPE] = [APE] + [BPF] + [CPD]$.

The area of triangle $XYZ$ is denoted $[XYZ]$.

2013-14 BrMO Round 2 P4

Let $ABC$ be a triangle with $P$ a point in it s interior. Let $AP$ meet the circumcircle of $ABC$ at $A'$. The points $B'$ and $C'$ defined similiarly. Let $O_A$ be the circumcenter of BCP. The circumcenters $O_B$, $O_C$ are defined similiarly. Let $O_A'$ be the circumcenter of $B'C'P$, $O_B'$ and $O_C$ define similiarly. Prove that lines $O_AO_A'$, $O_B'O_B$ and $O_C'O_C$ concur.

2014-15 BrMO Round 1 P5

Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.

2014-15 BrMO Round 2 P3

Two circles touch one another internally at $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Prove that the locus of the incentre of triangle $APQ$ is another circle touching the given circles at $A$.

2015-16 BrMO Round 1 P2

Let $ABCD$ be a cyclic quadrilateral and let the lines $CD$ and $BA$ meet at $E$. The line through $D$ which is tangent to the circle $ADE$ meets the line $CB$ at $F$. Prove that triangle $CDF$ is isosceles.

2015-16 BrMO Round 1 P5

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to $BC, CA$ and $AB$ respectively. Let $P, Q, R$ and $S$ be the feet of the perpendiculars from $D$ to $BA, BE, CF$ and $CA$ respectively. Prove that $P, Q, R$ and $S$ are collinear.

2015-16 BrMO Round 2 P3

Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA$ and $CB$ produced meet at $Q$. The midpoint of $AB$ is $E$. Prove that if $PQ$ is perpendicular to $AC$, then $PE$ is perpendicular to $BC$.

2016-17 BrMO Round 1 P5

Let $ABC$ be a triangle with $\angle A < \angle B < 90^{\circ}$ and let $\Gamma$ be the circle through $A, B$ and $C$. The tangents to $\Gamma$ at $A$ and $C$ meet at $P$. The line segments $AB$ and $PC$ produced meet at $Q$. It is given that $[ACP] = [ABC] = [BCQ]$.

Prove that $\angle BCA = 90^{\circ}$.

Here $[XYZ]$ denotes the area of triangle $XYZ$.

2016-17 BrMO Round 2 P3

Consider a cyclic quadrilateral $ABCD$. The diagonals $AC$ and $BD$ meet at $P$, and the rays $AD$ and $BC$ meet at $Q$. the internal angle bisector of $\angle BQA$ meets $AC$ at $R$ and the internal angle bisector of $\angle APD$ meets $AD$ at $S$. Prove that $RS$ is parallel to $CD$.

2017-18 BrMO Round 1 P3

The triangle $ABC$ has $AB = CA$ and $BC$ is its longest side. The point $N$ is on the side $BC$ and $BN = AB$. The line perpendicular to $AB$ which passes through $N$ meets $AB$ at $M$. Prove that the line $MN$ divides both the area and the perimeter of triangle $ABC$ into equal parts.

2017-18 BrMO Round 2 P1

Consider triangle $ABC$. The midpoint of $AC$ is $M$. The circle tangent to $BC$ at $B$ and passing through $M$ meets the line $AB$ again at $P$. Prove that $AB \times BP = 2BM^2$.

2018-19 BrMO Round 1 P1

Let $\Gamma$ be a semicircle with diameter $AB$. The point $C$ lies on the diameter $AB$ and points $E$ and $D$ lie on the arc $BA$, with $E$ between $B$ and $D$. Let the tangents to $\Gamma$ at $D$ and $E$ meet at $F$. Suppose that $\angle ACD = \angle ECB$. Prove that $\angle EFD = \angle ACD + \angle ECB$.

2018-19 BrMO Round 2 P1

Let $ABC$ be a triangle. Let $L$ be the line through $B$ perpendicular to $AB$. The perpendicular from $A$ to $BC$ meets $L$ at the point $D$. The perpendicular bisector of $BC$ meets $L$ at the point $P$. Let $E$ be the foot of the perpendicular from $D$ to $AC$. Prove that triangle $BPE$ is isosceles.

source: bmos.ukmt.org.uk/home/bmolot.pdf

with aops links

1993 - 2018

A square piece of toast $ABCD$ of side length $1$ and centre $O$ is cut in half to form two equal pieces $ABC$ and $CDA$. If the triangle $ABC$ has to be cut into two parts of equal area, one would usually cut along the line of symmetry $BO$. However, there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle $ABC$ into two parts of equal area.

Two circles touch internally at $ M$. A straight line touches the inner circle at $ P$ and cuts the outer circle at $ Q$ and $ R$. Prove that angle $ QMP =$ angle $ RMP$.

1992-93 BrMO Round 2 P3

Let $P$ be an internal point of a triangle $ABC$. Let's define $\alpha = \angle BPC - \angle BAC$ , $\beta = \angle APC - \angle ABC$ , $\gamma = \angle APB - \angle ACB$.

Prove that $\displaystyle PA \frac{\text{sin} \, \angle BAC}{\text{sin} \, \alpha} = PB \frac{\text{sin} \, \angle ABC}{\text{sin} \, \beta} =PC \frac{\text{sin} \, \angle ACB}{\text{sin} \, \gamma}$

Let $P$ be an internal point of a triangle $ABC$. Let's define $\alpha = \angle BPC - \angle BAC$ , $\beta = \angle APC - \angle ABC$ , $\gamma = \angle APB - \angle ACB$.

Prove that $\displaystyle PA \frac{\text{sin} \, \angle BAC}{\text{sin} \, \alpha} = PB \frac{\text{sin} \, \angle ABC}{\text{sin} \, \beta} =PC \frac{\text{sin} \, \angle ACB}{\text{sin} \, \gamma}$

1993-94 BrMO Round 1 P2

In triangle $ ABC$ the point $ X$ lies on $ BC$.

(i) Suppose that $ \angle BAC = 90^{\circ}$, that $ X$ is the midpoint of $ BC$, that $ \angle BAX$ is one third of $ \angle BAC$. What can you say (and prove) about triangle $ ACX$?

(ii) Suppose that $ \angle BAC = 60^{\circ}$, that $ X$ lies one third of the way from $ B$ to $ C$ and that $ AX$ bisects $ \angle BAC$. What can you say (and prove) about triangle $ ACX$?

In triangle $ ABC$ the point $ X$ lies on $ BC$.

(i) Suppose that $ \angle BAC = 90^{\circ}$, that $ X$ is the midpoint of $ BC$, that $ \angle BAX$ is one third of $ \angle BAC$. What can you say (and prove) about triangle $ ACX$?

(ii) Suppose that $ \angle BAC = 60^{\circ}$, that $ X$ lies one third of the way from $ B$ to $ C$ and that $ AX$ bisects $ \angle BAC$. What can you say (and prove) about triangle $ ACX$?

1993-94 BrMO Round 1 P4

The points $ Q, R$ lie on the circle $ \gamma$, and $ P$ is a point such that $ PQ, PR$ are tangents to $ \gamma$. $ A$ is a point on the extension of $ PQ$ and $ \gamma '$ is the circumcircle of triangle $ PAR$. The circle $ \gamma '$ cuts $ \gamma$ again at $ B$ and $ AR$ cuts $ \gamma$ at the point $ C$. Prove that $ \angle PAR = \angle ABC$.

The points $ Q, R$ lie on the circle $ \gamma$, and $ P$ is a point such that $ PQ, PR$ are tangents to $ \gamma$. $ A$ is a point on the extension of $ PQ$ and $ \gamma '$ is the circumcircle of triangle $ PAR$. The circle $ \gamma '$ cuts $ \gamma$ again at $ B$ and $ AR$ cuts $ \gamma$ at the point $ C$. Prove that $ \angle PAR = \angle ABC$.

$ AP, AQ, AR, AS$ are chords of a given circle with the property that $ \angle PAQ = \angle QAR = \angle RAS$. Prove that $ AR(AP+AR) = AQ(AQ+AS)$.

1994-95 BrMO Round 1 P2

$ ABCDEFGH$ is a cube of side $ 2$ with $ A, B, C, D$ above $ F, G, H, E$, respectively.

(i) Find the area of the quadrilateral $ AMHN$ where $ M$ is the midpoint of $ BC$ and $ N$ is the midpoint of $ EF$.

(ii) Let $ P$ be the midpoint of $ AB$, and $ Q$ the midpoint of $ HE$. Let $ AM$ meet $ CP$ at $ X$, and $ HN$ meet $ FQ$ at $ Y$. Find the length of $ XY$.

$ ABCDEFGH$ is a cube of side $ 2$ with $ A, B, C, D$ above $ F, G, H, E$, respectively.

(i) Find the area of the quadrilateral $ AMHN$ where $ M$ is the midpoint of $ BC$ and $ N$ is the midpoint of $ EF$.

(ii) Let $ P$ be the midpoint of $ AB$, and $ Q$ the midpoint of $ HE$. Let $ AM$ meet $ CP$ at $ X$, and $ HN$ meet $ FQ$ at $ Y$. Find the length of $ XY$.

1994-95 BrMO Round 1 P4

$ABC$ is a triangle, right- angled at $C$. The internal bisectors of angles $BAC$ and $ABC$ meet

$BC$ and $CA$ at $P$ and $Q$ respectively. The points $M$ and $N$ are the feet of the perpendiculars from $P$ and $Q$ to $AB$. Find angle $MCN$.

$ABC$ is a triangle, right- angled at $C$. The internal bisectors of angles $BAC$ and $ABC$ meet

$BC$ and $CA$ at $P$ and $Q$ respectively. The points $M$ and $N$ are the feet of the perpendiculars from $P$ and $Q$ to $AB$. Find angle $MCN$.

Let $ABC$ be a triangle, and $D,E,F$ be the midpoints of $BC, CA, AB$, respectively.

Prove that if $\angle DAC=\angle ABE$ ,if and only if $\angle AFC=\angle ADB$.

1995-96 BrMO Round 1 P3

Let $ABC$ be an acute-angled triangle, and let $O$ be its circumcentre. The circle through $C,O$ and $B$ is called $S$. The lines $AC$ and $AB$ meet the circle $S$ again at $P$ and $Q$ respectively. Prove that the lines $AO$ and $PQ$ are perpendicular.

Let $ABC$ be an acute-angled triangle, and let $O$ be its circumcentre. The circle through $C,O$ and $B$ is called $S$. The lines $AC$ and $AB$ meet the circle $S$ again at $P$ and $Q$ respectively. Prove that the lines $AO$ and $PQ$ are perpendicular.

1995-96 BrMO Round 2 P3

Two circles $k_1$ and $k_2$ touch each other externally at $K.$ They also touch a circle $\omega$ internally at $A_1$ and $A_2,$ respectively. Let $P$ be one point of intersection of $\omega$ with the common tangent to $k_1$ and $k_2$ at $K.$ The line $PA_1$ meets $k_1$ again at $B_1$ and the line $PA_2$ meets $k_2$ again at $B_2.$ Prove that $B_1B_2$ is a common tangent to $k_1$ and $k_2.$

Two circles $k_1$ and $k_2$ touch each other externally at $K.$ They also touch a circle $\omega$ internally at $A_1$ and $A_2,$ respectively. Let $P$ be one point of intersection of $\omega$ with the common tangent to $k_1$ and $k_2$ at $K.$ The line $PA_1$ meets $k_1$ again at $B_1$ and the line $PA_2$ meets $k_2$ again at $B_2.$ Prove that $B_1B_2$ is a common tangent to $k_1$ and $k_2.$

1996-97 BrMO Round 1 P4

Let $ ABCD$ be a convex quadrilateral. The midpoints of $ AB, BC, CD$ and $ DA$ are $ P, Q, R$ and $ S$, respectively. Given that the quadrilateral $ PQRS$ has area $ 1$, prove that the area of the quadrilateral $ ABCD$ is $ 2$

Let $ ABCD$ be a convex quadrilateral. The midpoints of $ AB, BC, CD$ and $ DA$ are $ P, Q, R$ and $ S$, respectively. Given that the quadrilateral $ PQRS$ has area $ 1$, prove that the area of the quadrilateral $ ABCD$ is $ 2$

1996-97 BrMO Round 2 P2

In the acute-angled triangle $ ABC$, $ CF$ is an altitude, with $ F$ on $ AB$, and $ BM$ is a median with $ M$ on $ CA$. Given that $ BM= CF$ and $ \angle MBC = \angle FCA$, prove that the triangle $ ABC$ is equilateral.

In the acute-angled triangle $ ABC$, $ CF$ is an altitude, with $ F$ on $ AB$, and $ BM$ is a median with $ M$ on $ CA$. Given that $ BM= CF$ and $ \angle MBC = \angle FCA$, prove that the triangle $ ABC$ is equilateral.

1997-98 BrMO Round 1 P3

$ ABP$ is an isosceles triangle with $ AB=AP$ and $ \angle PAB$ acute. $ PC$ is the line through $ P$ perpendicular to $ BP$, and $ C$ is a point on this line on the same side of $ BP$ as $ A$. (You may assume that $ C$ is not on the line$ AB$.) $ D$ completes the parallelogram $ ABCD$. $ PC$ meets $ DA$ at $ M$. Prove that $ M$ is the midpoint of $ DA$.

1997-98 BrMO Round 1 P5

$ ABP$ is an isosceles triangle with $ AB=AP$ and $ \angle PAB$ acute. $ PC$ is the line through $ P$ perpendicular to $ BP$, and $ C$ is a point on this line on the same side of $ BP$ as $ A$. (You may assume that $ C$ is not on the line$ AB$.) $ D$ completes the parallelogram $ ABCD$. $ PC$ meets $ DA$ at $ M$. Prove that $ M$ is the midpoint of $ DA$.

1997-98 BrMO Round 1 P5

In triangle $ ABC$, $ D$ is the midpoint of $ AB$ and $ E$ is the point of trisection of $ BC$ nearer to $ C$. Given that $ \angle ADC = \angle BAE$ find $ \angle BAC$

A triangle $ ABC$ has $ \angle BAC > \angle BCA$. A line $ AP$ is drawn so that $ \angle PAC =\angle BCA$ where $ P$ is inside the triangle. A point $ Q$ outside the triangle is constructed so that $ PQ$ is parallel to $ AB$, and $ BQ$ is parallel to $ AC$. $ R$ is the point on $ BC$ (separated from $ Q$ by the line $ AP$) such that $ \angle PRQ = \angle BCA$. Prove that the circumcircle of $ ABC$ touches the circumcircle of $ PQR$.

1998-99 BrMO Round 1 P2

A circle has diameter $AB$ and $X$ is a fixed point of $AB$ lying in between $A$ and $B$. A point $P$, distinct from $A$ and $B$, lies on the circumference of the circle. Prove that, for all possible positions of $P$, $\frac{\tan \angle APX}{\tan \angle PAX}$ remains constant.

A circle has diameter $AB$ and $X$ is a fixed point of $AB$ lying in between $A$ and $B$. A point $P$, distinct from $A$ and $B$, lies on the circumference of the circle. Prove that, for all possible positions of $P$, $\frac{\tan \angle APX}{\tan \angle PAX}$ remains constant.

1999-2000 BrMO Round 1 P1

Two intersecting circles $C_1$ and $C_2$ have a common tangent which touches $C_1$ at $P$ and $C_2$ at $Q$. The two circles intersect at $M$ and $N$, where $N$ is nearer to $PQ$ than $M$ is. The line $PN$ meets the circle $C_2$ again at $R$. Prove that $MQ$ bisects angle $PMR$.

Triangle $ABC$ has a right angle at $A$. Among all points $P$ on the perimeter of the triangle, find the position of P such that $AP + BP + CP$ is minimized.

1999-2000 BrMO Round 2 P1Two intersecting circles $C_1$ and $C_2$ have a common tangent intersecting $C_1$ in $P$ and $C_2$ in $Q$. The $2$ circles intersect in $M$ and $N$ where $N$ is nearer to $PQ$ than $M$.

Prove that the triangles $MNP$ and $MNQ$ have equal areas.

2000-01 BrMO Round 1 P2

Circle $S$ lies inside circle $T$ and touches it at $A$. From a point $P$ (distinct from $A$) on $T$, chords $PQ$ and $PR$ of $T$ are drawn touching $S$ at $X$ and $Y$ respectively. Show that $\angle QAR=2\angle XAY$.

2000-01 BrMO Round 1 P5

A triangle has sides of length $a,b,c$ and its circumcircle has radius $R$. Prove that the triangle is right angled if and only if $a^{2}+b^{2}+c^{2}=8R^{2}$

A triangle $ABC$ has $\angle ACB > \angle ABC$. The internal bisector of $\angle BAC$ meets $BC$ at $D$. The point $E$ on $AB$ is such that $\angle EDB=90^{o}$. The point $F$ on $AC$ is such that $\angle BED=\angle DEF$. Show that $\angle BAD=\angle FDC$

2001-02 BrMO Round 1 P2

The quadrilateral $ABCD$ is inscribed in a circle. The diagonals $AC,BD$ meet at $Q$. The sides $DA$, extended beyond $A$, and $CB$, extended beyond $B$, meet at $P$. Given that $CD=CP=DQ$, prove that $\angle CAD=60$.

2001-02 BrMO Round 2 P1

The altitude from one of the vertices of an acute-angled triangle $ABC$ meets the opposite side at $D$. From $D$ perpendiculars $DE$ and $DF$ are drawn to the other two sides. Prove that the length of $EF$ is the same whichever vertex is chosen.

2002-03 BrMO Round 1 P2

The triangle $ABC$, where $AB<AC$, has a circumcircle $S$. The perpendicular from $A$ to $BC$ intersects $S$ again at $P$. The point $X$ lies on the line segment $AC$, and $BX$ intersects $S$ again at $Q$. Show that $BX=CX$ if and only if $PQ$ is a diameter of $S$.

2002-03 BrMO Round 2 P2

Let $ABC$ be a triangle and let $D$ be a point on $AB$ such that $4AD = AB$. The half-line $\ell$ is drawn on the same side of $AB$ as $C$, starting from $D$ and making an angle of $\theta$ with $DA$ where $\theta =\angle ACB$. If the circumcircle of $ABC$ meets the half-line $\ell$ at $P$, show that $PB = 2PD$.

2003-04 BrMO Round 1 P2

$ABCD$ is a rectangle , $P$ is the midpoint of $AB$ and $Q$ is the $PD$ such that $CQ$ is perpendicular to $PD$ . Prove that triangle $BQC$ is isosceles.

2003-04 BrMO Round 2 P1

Let $ABC$ be an equilateral triangle and $D$ an internal point of the side $BC$. A circle, tangent to $BC$ at $D$, cuts $AB$ internally at $M$ and $N$, and $AC$ internally at $P$ and $Q$.

Show that $BD + AM + AN = CD + AP + AQ$.

2004-05 BrMO Round 1 P2

Let $ABC$ be an acute-angled triangle, and let $D,E$ be the feet of the perpendiculars from $A,B$ to $BC,CA$ respectively.Let $P$ be the point where the line $AD$ meets the semicircle constructed outwardly on $BC$, and $Q$ be the point where line $BD$ meets the semicircle constructed outwardly on $AC$. Prove that $CP = CQ$.

2004-05 BrMO Round 2 P2

In triangle $ABC, \angle BAC = 120^o.$ Let the angle bisectors of angles $A,B$ and $C$ meet the opposite sides in $D,E$ and $F$ respectively. Prove that the circle on diameter $EF$ passes through $D$.

2005-06 BrMO Round 1 P3

In the cyclic quadrilateral $ABCD$, the diagonal $AC$ bisects the angle $DAB$. The side $AD$ is extended beyond $D$ to a point $E$. Show that $CE = CA$ if and only if $DE = AB$.

2005-06 BrMO Round 1 P5

Let $G$ be a convex quadrilateral. Show that there is a point $X$ in the plane of $G$ with the property that every straight line through $X$ divides $G$ into two regions of equal area if and only if $G$ is a parallelogram.

2005-06 BrMO Round 2 P3Let $ABC$ be a triangle with $AC > AB$. The point $X$ lies on the side $BA$ extended through $A$, and the point $Y$ lies on the side $CA$ in such a way that $BX = CA$ and $CY = BA$. The line $XY$ meets the perpendicular bisector of side $BC$ at $P$. Show that $\angle BPC + \angle BAC = 180$.

2006-07 BrMO Round 1 P2

In the convex quadrilateral $ABCD$, points $M,N$ lie on the side $AB$ such that $AM = MN = NB$, and points $P,Q$ lie on the side $CD$ such that $CP = PQ = QD$. Prove that

Area of $AMCP =$ Area of $MNPQ = \frac{1}{3}$ Area of $ABCD$

2006-07 BrMO Round 1 P4

Two touching circles $S$ and $T$ share a common tangents which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP=PQ$.

Let $ABC$ be an acute-angled triangle with $AB > AC$ and $\angle BAC =60$ . Denote the circumcentre by $O$ and the orthocentre by $H$ and let $OH$ meet $AB$ at $P$ and $AC$ at $Q$. Prove that $PO = HQ$

2007-08 BrMO Round 1 P3

Let $ABC$ be a triangle, with an obtuse angle at $A$. Let $Q$ be a point (other than $A, B$ or $C$ ) on the circumcircle of the triangle, on the same side of chord $BC$ as $A$, and let $P$ be the other end of the diameter through $Q$. Let $V$ and $W$ be the feet of the perpendiculars from $Q$ onto $CA$ and $AB$ respectively. Prove that the triangles $PBC$ and $AWV$ are similar.

2007-08 BrMO Round 1 P5

Let $ P$ be an internal point of triangle $ ABC$. The line through $P$ parallel to $ AB$ meets $ BC$ at $ L$, the line through $ P$ parallel to $BC$ meets $ CA$ at $ M$, and the line through $ P$ parallel to $ CA$ meets $ AB$ at $ N$. Prove that $ \frac{BL}{LC}\times\frac{CM}{MA}\times\frac{AN}{NB}\le\frac{1}{8}$ and locate the position of $ P$ in triangle $ ABC$ when equality holds.

2007-08 BrMO Round 2 P2

Let triangle ABC have incentre $ I$ and circumcentre $ O$. Suppose that $ AIO =90^\circ$ and $ CIO = 45^\circ$ . Find the ratio $ AB: BC: CA$.

2008-09 BrMO Round 1 P3

Let $ ABPC$ be a parallelogram such that $ ABC$ is an acute-angled triangle. The circumcircle of triangle $ ABC$ meets the line $ CP$ again at $ Q$. Prove that $ PQ = AC$ if, and only if, $ BAC = 60^{\circ}$ .

2008-09 BrMO Round 2 P2

Let $ABC$ be an acute-angled triangle with $ \angle B = \angle C$. Let the circumcentre be $O$ and the orthocentre be $H$. Prove that the centre of the circle $BOH$ lies on the line $AB$. The circumcentre of a triangle is the centre of its circumcircle. The orthocentre of a triangle is the point where its three altitudes meet.

2009-10 BrMO Round 1 P2

Points $A,B,C,D$ and $E$ lie, in that order, on a circle and the lines $AB$ and $ED$ are parallel. Prove that $\angle{ABC} = 90^{\circ}$ if, and only if, $AC^2 = BD^2 + CE^2$.

2009-10 BrMO Round 1 P4

Two circles of different radius with centers at $B$ & $C$ respectively touch each other externally at $A$. A common tangent not through $A$ touches the first circle at $D$ and second at $E$. The line through $A$ which is perpendicular to $DE$ and the perpendicular bisector of $BC$ meet at $F$. Prove that $BC=2AF$.

2009-10 BrMO Round 2 P2In triangle $ ABC$ the centroid is $ G$ and $ D$ is the midpoint of $ CA$. The line through $ G$ parallel to $ BC$ meets $ AB$ at $ E$. Prove that $ \angle AEC = \angle DGC$ if, and only if, $ \angle ACB = 90^{\circ}$

2010-11 BrMO Round 1 P3

Let $ABC$ be a triangle with $\angle CAB$ a right-angle. The point $L$ lies on the side $BC$ between $B$ and $C$. The circle $ABL$ meets the line $AC$ again at $M$ and the circle $CAL$ meets the line $AB$ again at $N$. Prove that $L, M$ and $N$ lie on a straight line.

2010-11 BrMO Round 1 P5

Circles $S_{1}$ and $S_{2}$ meet at $L$ and $M$. Let $P$ be a point on $S_{2}$. Let $PL$ and $PM$ meet $S_{1}$ again at $Q$ and $R$ respectively. The lines $QM$ and $RL$ meet at $K$. Show that, as $P$ varies on $S_{2}$, $K$ lies on a fixed circle.

2010-11 BrMO Round 2 P1Let $ABC$ be a triangle and $X$ be a point inside the triangle. The lines $AX,BX$ and $CX$ meet the circle $ABC$ again at $ P,Q$ and $R$ respectively. Choose a point $U$ on $XP$ which is between $X$ and $P$. Suppose that the lines through $U$ which are parallel to $AB$ and $CA$ meet $XQ$ and $XR$ at points $V$ and $W$ respectively. Prove that the points $R,W, V$ and $Q$ lie on a circle.

2011-12 BrMO Round 1 P3

Consider a circle $S$. The point $P$ lies outside $S$ and a line is drawn through $P$, cutting $S$ at distinct points $X$ and $Y$ . Circles $S_1$ and $S_2$ are drawn through $P$ which are tangent to $S$ at $X$ and $Y$ respectively.Prove that the difference of the radii of $S_1$ and $S_2$ is independent of the positions of $P$,$ X$ and $Y$

2011-12 BrMO Round 1 P6

Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A$, $B$ and $C$ are $D , E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds.

2011-12 BrMO Round 2 P1

The diagonals $AC$ and $BD$ of a cyclic quadrilateral meet at $E$. The midpoints of the sides $AB, BC, CD$ and $DA$ are $P, Q, R$ and $S$ respectively. Prove that the circles $EPS$ and $EQR$ have the same radius.

2012-13 BrMO Round 1 P2

Two circles $S$ and $T$ touch at $X$. They have a common tangent which meets $S$ at $A$ and $T$ at $B$. The points $A$ and $B$ are different. Let $AP$ be a diameter of $S$. Prove that $B, X$ and $P$ lie on a straight line.

A triangle has sides of length at most $2,3$ & $4$ respectively. Determine with proof the max. possible area of the triangle.

Let $ABC$ be a triangle. Let $S$ be the circle through $B$ tangent to $CA$ at $A$ and let $T$ be the circle through $C$ tangent to $AB$ at $A$. The circles $S$ and $T$ intersect at $A$ and $D$. Let $E$ be the point where the line $AD$ meets the circle $ABC$. Prove that $D$ is the midpoint of $AE$.

2012-13 BrMO Round 2 P2

The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.

2013-14 BrMO Round 1 P2

In the acute-angled triangle $ABC$, the foot of the perpendicular from $B$ to $CA$ is $E$. Let $l$ be the tangent to the circle $ABC$ at $B$. The foot of the perpendicular from $C$ to $l$ is $F$, Prove that $EF$ is parallel to $AB$.

2013-14 BrMO Round 1 P5

Let $ABC$ be an equilateral triangle, and let $P$ be a point inside this triangle. Let $D, E$ and $F$ be the feet of the perpendiculars from $P$ to the sides $BC, CA$ and $AB$ respectively. Prove that

(a) $AF + BD + CE = AE + BF + CD$ and

(b) $[APF] + [BPD] + [CPE] = [APE] + [BPF] + [CPD]$.

The area of triangle $XYZ$ is denoted $[XYZ]$.

2013-14 BrMO Round 2 P4

Let $ABC$ be a triangle with $P$ a point in it s interior. Let $AP$ meet the circumcircle of $ABC$ at $A'$. The points $B'$ and $C'$ defined similiarly. Let $O_A$ be the circumcenter of BCP. The circumcenters $O_B$, $O_C$ are defined similiarly. Let $O_A'$ be the circumcenter of $B'C'P$, $O_B'$ and $O_C$ define similiarly. Prove that lines $O_AO_A'$, $O_B'O_B$ and $O_C'O_C$ concur.

2014-15 BrMO Round 1 P5

Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.

2014-15 BrMO Round 2 P3

Two circles touch one another internally at $A$. A variable chord $PQ$ of the outer circle touches the inner circle. Prove that the locus of the incentre of triangle $APQ$ is another circle touching the given circles at $A$.

2015-16 BrMO Round 1 P2

Let $ABCD$ be a cyclic quadrilateral and let the lines $CD$ and $BA$ meet at $E$. The line through $D$ which is tangent to the circle $ADE$ meets the line $CB$ at $F$. Prove that triangle $CDF$ is isosceles.

2015-16 BrMO Round 1 P5

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to $BC, CA$ and $AB$ respectively. Let $P, Q, R$ and $S$ be the feet of the perpendiculars from $D$ to $BA, BE, CF$ and $CA$ respectively. Prove that $P, Q, R$ and $S$ are collinear.

Three circles of radius $r_{1},r_{2},r_{3}$ touch each other externally and share a common tangent. The tangent touches the three circles at $A,B,C$ with $B$ between $A$ and $C$. Show that:

$16(r_{1}+r_{2}+r_{3})\geq9(AB+BC+CA)$

Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA$ and $CB$ produced meet at $Q$. The midpoint of $AB$ is $E$. Prove that if $PQ$ is perpendicular to $AC$, then $PE$ is perpendicular to $BC$.

2016-17 BrMO Round 1 P5

Let $ABC$ be a triangle with $\angle A < \angle B < 90^{\circ}$ and let $\Gamma$ be the circle through $A, B$ and $C$. The tangents to $\Gamma$ at $A$ and $C$ meet at $P$. The line segments $AB$ and $PC$ produced meet at $Q$. It is given that $[ACP] = [ABC] = [BCQ]$.

Prove that $\angle BCA = 90^{\circ}$.

Here $[XYZ]$ denotes the area of triangle $XYZ$.

Consider a cyclic quadrilateral $ABCD$. The diagonals $AC$ and $BD$ meet at $P$, and the rays $AD$ and $BC$ meet at $Q$. the internal angle bisector of $\angle BQA$ meets $AC$ at $R$ and the internal angle bisector of $\angle APD$ meets $AD$ at $S$. Prove that $RS$ is parallel to $CD$.

2017-18 BrMO Round 1 P3

The triangle $ABC$ has $AB = CA$ and $BC$ is its longest side. The point $N$ is on the side $BC$ and $BN = AB$. The line perpendicular to $AB$ which passes through $N$ meets $AB$ at $M$. Prove that the line $MN$ divides both the area and the perimeter of triangle $ABC$ into equal parts.

2017-18 BrMO Round 2 P1

Consider triangle $ABC$. The midpoint of $AC$ is $M$. The circle tangent to $BC$ at $B$ and passing through $M$ meets the line $AB$ again at $P$. Prove that $AB \times BP = 2BM^2$.

2018-19 BrMO Round 1 P1

Let $\Gamma$ be a semicircle with diameter $AB$. The point $C$ lies on the diameter $AB$ and points $E$ and $D$ lie on the arc $BA$, with $E$ between $B$ and $D$. Let the tangents to $\Gamma$ at $D$ and $E$ meet at $F$. Suppose that $\angle ACD = \angle ECB$. Prove that $\angle EFD = \angle ACD + \angle ECB$.

2018-19 BrMO Round 2 P1

Let $ABC$ be a triangle. Let $L$ be the line through $B$ perpendicular to $AB$. The perpendicular from $A$ to $BC$ meets $L$ at the point $D$. The perpendicular bisector of $BC$ meets $L$ at the point $P$. Let $E$ be the foot of the perpendicular from $D$ to $AC$. Prove that triangle $BPE$ is isosceles.

source: bmos.ukmt.org.uk/home/bmolot.pdf

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