geometry problems from British Math Olympiads (BrMO) [Round 1 : BMO1, Round 2: BMO2]
with aops links
British Mathematical Olympiad (ΒrΜΟ) 1993-2020
BMO Round 2 was named as FIST at years 1972-91
FIST= further international selection test
1965 - 1991
A pupil is swimming at the centre of a circular pond. At the edge of the pond there is a teacher, who wishes to catch the pupil, bur who cannot swim. The teacher can run four times as fast as the pupil can swim, but not as fast as the pupil can run. Can the pupil escape from the teacher? Justify your answer.
A chord of length $\sqrt3$ divides a circle of unit radius into two regions. Find the rectangle of maximum area which can be inscribed in the smaller of those two regions.
Let $A_1,A_2,..,A_n $ be the vertices of a regular polygon and $A_1A_2,..,A_nA_1 $ be its $n$ sides .If $\frac{1}{A_1A_2}-\frac{1}{A_1A_4}=\frac{1}{A_1A_3} $ then the value of $n$ is?
A square nut of side $a$ is to be turned by means of a spanner, the hole in which consists of a regular polygon of side $b$. Find the conditions on $a$ and $b$ for this to be possible.
In a triangle, the length of the altitude dropped to side $a$ is equal to $h_a$, and respectively to side $b$ is $h_b$. Prove that if $a> b$ then $a + h_a> b + h_b$.
If the circumcentere and the incentre of a triangle coincide, then the triangle is equilateral.
When it is possible to find points equidistant both from two given points $P$ and $Q$ from a given straight line $AB$? When it is possible, show how to costruct such points with ruler and compasses only.
The ''[i]distance[/i]'' $d$ between two points $(x_1,y_1), (x_2,y_2)$ in the $(x,y)$-plane is defined by $d=|x_2-x_1|+ |y_2-y_1|$. Using this notion of distance, find the locus of all points $(x,y)$ satisfying $x\ge 0, y\ge 0$ and which are equidistant from the origin and from a fixed point $(a,b)$ in the plane, where $a>b$. Distinguish the cases acoording to the signs of $a$ and $b$.
Two spheres of radii $a$ and $b$ are tangent to each other and a plane is tangent to these spheres at different points. Find the radius of the largest sphere wihch can pass between the first two spheres and the plane.
A chord of length $\ell$ divides the interior of a circle of radius $r$ into two regions, $D_1$ and $D_2$ . A circle $S$ of maximal radius is inscribed in $D_2$. The area of the part of $D_1$ outside $S$ is $A$. Show that $A$ is greatest when the area od $D_1$ exceeds that of $D_2$ and when $\ell=\frac{16 \pi r}{16+\pi^2}$
Find the lengths of the sides of a triangle if its altitudes have lengths $3, 4$ and $6$.
The faces of a tetrahedron $ABCD$ are formed by four congruent triangles. If $\alpha$ is the angle between a pair of opposite edges of the tetrahedron, show that $$\cos \alpha =\frac{\sin (B-C)}{\sin (B+C)}.$$
A long corridor of unit length has a right-angled corner in it. A rigid length of pipe (the thickness of which may be neglected) lies on, and is everywhere in contact with, the plane floor of the corridor. The length of the pipe (which may be curved) is defined as the straight-line distance of pipe between two ends. Find the maximum length of pipe subject to the condition that it can be moved along both arms of the corridor and round the corner without leaving contact with the floor.
A square has sides of length $r$.All four vertices of the square lie on the sides of a circumscribing triangle. The incircle of the triangle has radius $r$. Prove that $2r> x> r\sqrt2 $.
Describe a ruler-and-compass method of constructing an equilateral triangle the area of which equals to that of a given triangle.
You are probably aware of the remarkable fact that hte three points of intersection of htrre appropriate pairs of trisectors of the internal angles of any triangle $T$ form the vertices of an equilateral triangle $t$. Given a real constant $A>0$, consider the class of all triangles $T$ the area of which equals $A$. Find with this class, the maximum value of the area of the triangle $t$. Discuss whether there is minimu value.
The two base angles $B$ and $C$ of an isosceles triangle $ABC$ are equal to $50^o$. The point $D$ lies on $BC$, so that $\angle BAD=50^o$ and the point $E$ lies on $AC$ such that $\angle ABE=30^o$. Find $\angle BED$.
Of a regular polygon of $2n$ sides, there are $n$ diagonals which pass through the centre of the inscribed circle. The angles which these diagonals subtend at two given points $A$ and $B$ on the circumference, are $a_1,a_2, a_, ..., a_n$ and $b_1,b_2, b_, ..., b_n$. Prove that $\sum_{i=1}^{n}\tan^2a_i=\sum_{i=1}^{n}\tan^2b_i$.
The triangle $ABC$ has angles $A,B$ and $C$, in desending order of magnitude. Circles are drawn such that each circle cuts each side of the triangle internally in two real distict points. The lower limit to the radii of such circles is the radius of the incircle of the triangle $ABC$. Show that the upper limit, is not $R$, the radius of the circumcircle, and fin this upper limit in terms of $R, A$, and $B$.
In a plane, two circles of unequal radii intersect at $A$ and $B$, and through an arbitrary point $P$ a straight line $L$ is to be constructed, so that the two circles intersect equal chords on $L$. By considering distances from the point of intersection of $L$ with $AB$ (produced if necessary), or otherwise:
a) find a rular and compass costruction for $L$, and
b) find the region of the plave in which $P$ must lie for the construction to be possible.
In a right circular cone, the semi-vertical angle of which is $\theta$, a cube is placed so that four of its vertices are on the base and four on the curved surface. Prove that as $\theta$ varies the maximum of the ratio for the volume of the cube to the volume of the cone occurs when $\sin \theta = \frac13$.
(i) Two fixed circles are touched by a variable circle at $P$ and $Q$. Prove that $PQ$ passes through one of two fixed points.
(ii) State a true theorem about ellipses or if you like about conics in general of which (i) is a particular case.
Prove that it is Impossible for all the faces of a convex polyhedron to be hexagons.
$X$ and $Y$ are the feet of the perpendiculars from $P$ to $CA$ and $CB$ respectively, where $P$ is in the plane of triangle $ABC$. $PX = PY$. The straight line through $P$ which is perpendicular to $AB$ cuts $XY$ at $Z$. Prove that $CZ$ bisects $AB$.
Three parallel lines $AD, BE, CF$ are drawn through the vertices of triangle $ABC$ meeting the opposite sides in $D,E,F$ respectively.The points $P,Q,R$ divide $AD, BE, CF$ respectively in the same ratio $k : 1$ and $P,Q,R$ are collinear. Find the value of $k$.
The interior of a wine glass is a right circular cone. The glass is half filled with water and then slowly tilted so that the water starts and continues to spill from a point $P$ on the rim. What fraction of the whole conical interior is occupied by water when the horizontal plane of the water level bisects the generator of the cone furthest from $P$?
Find, with proof, the length $d$ of the shortest straight line which bisects the area of an arbitrarily given triangle. Express $d$ in terms of the area $A$ of the triangle and one of its angles. Show that there is a shorter line (not straight) which bisects the area of the given triangle.
A sphere with centre $O$ and radius $r$ is cut in a circle $K$ by a horizontal plane distant $\frac12 r$ above $O$. The part of the sphere above the plane is removed and replaced by a right circular cone having $K$ as its base and having its vertex $V$ at a distance $2r$ vertically above $O$. $Q$ is a point on the sphere on the same horizontal level as $O$. The plane $OVQ$ cuts the circle $K$ in two points $X$ and $Y$, of which $Y$ is the further from $Q$. $P$ is a point of the cone lying on $VY$, whose position can be determined by the fact that the shortest path from $P$ to $Q$ over the surfaces of cone and sphere cuts the circle $K$ at an angle of $45^o$. Prove $VP =\sqrt3 r/ \sqrt{(1+ 1/\sqrt5)}$ .
[In a spherical triangle $ABC$ the sides are arcs of great circles (centre $O$) and the sides are measured by the angles they subtend at $O$. You may find these spherical triangle formulae useful :
$ \sin a/ \sin A = \sin b/ \sin B = \sin c/ \sin C$ , $ \cos a = \cos b \cos c + \sin b sin c \cos A$].
The sides $BC, CA, AB$ of a triangle touch a circle at $X,Y,Z$ respectively. Prove that the centre of the circle lies on the straight line through the midpoints of $BC$ and of $AX$.
$A_1A_2A_3A_4A_5$ is a regular pentagon whose sides are each of length $2a$. For each $i = 1,2,...,5$,$K_i$ is the sphere with centre $A_i$ and radius $a$. The spheres $K_1,K_2,...K_5$ are all touched externally by each of two spheres $P_1$ and $P_2$ also of radius $a$. Determine with proof and without tables whether $P_1$ and $P_2$ have or have not a common point.
Determine with proof the point $P$ inside a given triangle $ABC$ for which the product $PL\cdot PM \cdot PN$ is a maximum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
\An altitude of a tetrahedron is a line through a vertex perpendicular to the opposite face. Prove that the four altitudes of a tetrahedron are concurrent if and only if each edge of the tetrahedron is perpendicular to its opposite edge.
Find all triangles $ABC$ for which $AB + AC = 2$ cm. and $AD + BC = \sqrt5$ cm. , where $AD$ is the altitude through $A$, meeting $BC$ at right angles in $D$.
From a point $O$ in $3$-D space, three given rays $OA$, $OB$, $OC$ emerge, the angles $BOC$, $COA$, $AOB$ being $\alpha$, $\beta$, $\gamma$ respectively ($0< \alpha,\beta,\gamma<\pi$). Prove that, given $2s > 0$, there are unique points $X, Y, Z$ on $OA$, $OB$, $OC$ respectively such that the triangles $YOZ, ZOX$ and $XOY$ have the same perimeter $2s$, and express OX in terms of $x$ and $\sin\frac12 \alpha$, $\sin\frac12 \beta$ and $\sin\frac12 \gamma$.
On the diameter $AB$ bounding a semi-circular region there are two points $P$ and $Q$, and on the semi-circular arc there are two points $R$ and $S$ such that $PQRS$ is a square. $C$ is a point on the semi-circular arc such that the areas of the triangle $ABC$ and the square $PQRS$ are equal. Prove that a straight line passing through one of the points $R$ and $S$ and through one of the points $A$ and $B$ cuts a side of the square at the incentre of the triangle.
$H$ is the orthocentre of triangle $ABC$. The midpoints of $BC, CA, AB$ are $A', B', C'$ respectively. A circle with centre $H$ cuts the sides of triangle $A'B'C'$ (produced if necessary) in six points, $D_1, D_2$ on $B’C'$,$E_1, E_2$ on $C'A'$ and $F_1, F_2$ on $A'B'$. Prove that $AD_1=AD_2=BE_1=BE_2=CF_=CF_2$.
$PQRS$ is a quadrilateral of area $A$. $O$ is a point inside it. Prove that if $2A = OP^2 + OQ^2 + OR^2 + OS^2$, then $PQRS$ is a square and $O$ is its centre.
A right circular cone stands on a horizontal base, radius $r$. Its vertex $V$ is at a distance $1$ from every point on the perimeter of the base. A plane section of the cone is an ellipse whose lowest point is $L$ and whose highest point is $H$. On the curved surface of the cone, to one side of the plane $VLH$, two routes from $L$ to $H$ are marked. $R_1$ is along the semi-perimeter of the ellipse and $R_2$ is the route of shortest length. Find the condition that $R_1$ and $R_2$ intersect between $L$ and $H$.
In the triangle $ABC$ with circumcentre $O$, $AB = AC$, $D$ is the midpoint of AB and $E$ is the centroid of triangle $ACD$. Prove that $OE$ is perpendicular to $CD$.
$P, Q, R$ are arbitrary points on the sides $BC, CA, AB$ respectively of triangle $ABC$ . Prove that the triangle whose vertices are the centres of the circles $AQR , BRP , CPQ$ is similar to triangle $ABC$ .
A plane cuts a right circular cone with vertex $V$ in an ellipse $E$ and meets the axis of the cone at $C$. $A$ is an extremity of the major axis of $E$ . Prove that the area of the curved surface of the slant cone with $V$ as vertex and $E$ as base is $\frac{VA}{VE } \times$ (area of $E$ ) .
$ABCD$ is a quadrilateral which has an inscribed circle. With the side $AB$ is associated $$u_{AB} = p_1 \sin \angle DAB + p_2 \sin \angle ABC$$ where $p_1, p_2$ are the perpendiculars from $A, B$ respectively to the opposite side $CD$ . Define $u_{BC},u_{CD},u_{DA}$ likewise, using in each case perpendiculars to the opposite side.
Show that $u_{AB}=u_{BC}=u_{CD}=u_{DA}$
Two circles $S_1$ and $S_2$ each touch a straight line $p$ at the same point $P$. All points of $S_2$ , except $P$, are in the interior of $S_1$ . A straight line $q$ (i) is perpendicular to $p$, (ii) touches $S_2$ at $R$, (iii) cuts $p$ at $L$ and (iv) cuts $S_1$ at $N$ and $M$ , where $M$ is between $L$ and $R$.
(a) Prove that $RP$ bisects angle $MPN$.
(b) If $MP$ bisects angle $RPL$, find, with proof, the ratio of the areas of $S_1$ and $S_2$ .
A cylindrical container has height $6$ cm and radius $4$ cm . It rests on a circular hoop which has also radius $4$ cm and the hoop is fixed in a horizontal plane. The container rests with its axis horizontal and with each of its circular rims touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop at two points. Find, with proof, the locus of the centre of one of the cylinder's circular ends.
A circle $S$ of radius $R$ has two parallel tangents $t_1$ , $t_2$ . A circle $S_1$ of radius $r_1$ touches $S$ and $t_1$, a circle $S_2$ of radius $r_2$ touches $S$ and $t_2$, also $S_1$ touches $S_2$ and all the circle contacts are external. Calculate $R$ in terms of $r_1$ and $r_2$ .
$AB, AC, AD$ are three edges of a cube. $AC$ is produced to $E$ so that $AE = 2AC$ and $AD$ is produced to $F$ so that $AF = 3AD$. Prove that the area of the section of the cube by any plane parallel to BCD is equal to the area of the section of tetrahedron $ABEF$ by the same plane.
In a triangle $ABC$, $\angle BAC = 100^o$ and $AB=AC$. A point $D$ is chosen on the side $AC$ so that $\angle ABD=\angle CBD$. Prove that $AD + DB = BC$.
A line parallel to the side $BC$ of an acute-angled triangle $ABC$ cuts the side $AB$ at $F$ and the side $AC$ at $E$ . Prove that the circles on $BE$ and $CF$ as diameters intersect on the altitude of the triangle drawn from $A$ perpendicular to $BC$.
The triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of angle $BAC$ (which is not a right angle) are $P$ and Q. Prove that $PQ$ passes through the middle point of $BC$.
Points $P, Q$ lie on the sides $AB, AC$ respectively of triangle $ABC$ and are distinct from $A$. The lengths $AP, AQ$ are denoted by $x, y$ respectively, with the convention that $x > 0$ if $P$ is on the same side of $A$ as $B$, and $x < 0$ on the opposite side, similarly for y. Show that $PQ$ passes through the centroid of the triangle if and only if
$$3xy = bx + cy$$ where $b= AC$, $c = AB$.
$OA, OB, OC$ are mutually perpendicular lines. Express the area of triangle $ABC$ in terms of the areas of triangles $OBC, OCA, OAB$.
$ABCD$ is a square and $P$ is a point on the line $AB$. Find the maximum and minimum values of the ratio $PC/PD$, showing that these occur for the points $P$ given by $AP \times BP = AB^2$.
The angles $A, B, C, D$ of a convex quadrilateral satisfy the relation $\cos A + \cos B + \cos c + \cos D= 0$. Prove that $ABCD$ is a trapezium or is cyclic.
The diagonals of a convex quadrilateral $ABCD$ intersect at $O$. The centroids of triangles $AOD$ and $BOC$ are $P$ and $Q$. The orthocentres of triangles $AOB$ and $COD$ are $R$ and $S$. Prove that $PQ$ is perpendicular to $RS$.
$ABCD$ is a quadrilateral inscribed in a circle of radius $r$. The diagonals $AC,BD$ meet at $E$.
Prove that if $AC$ is perpendicular to $BD$, then $EA^2 + ED^2 + EC^2 + ED^2 = 4r^2$.$(*)$
Is it true that if $(*)$ holds then $AC$ is perpendicular to $BD$? Give a reason for your answer.
A ladder of length $\ell$ rests against a vertical wall. Suppose that there is a rung on the ladder which has the same distance $d$ from both the wall and the (horizontal) ground. Find explicitly in terms of $\ell$ and $d$, the height $h$ from the ground that the ladder reaches up the wall.
1992 - 2019
Round 1 also known as BMO1
Let $ABCDE$ be a pentagon inscribed in a circle. Suppose that $AC$,$BD$,$CE$,$DA$ and $EB$ are parallel to $DE$, $EA$, $AB$, $BC$ and $CD$, respectively. Does it follow that the pentagon has to be regular? Justify your claim.
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.
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 2 P1
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.
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 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 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 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 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 2 P3
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 $\angle BPC + \angle BAC = 180$.
.
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 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 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 2 P2
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 2 P1
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 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 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 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 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 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 $BC$.
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 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 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.
2019-20 BrMO Round 2 P2
Describe all collections S of at least four points in the plane such that no three points are collinear and such that every triangle formed by three points in S has the same circumradius.
2020-21 BrMO Round 2 P3
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$.
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$.
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$
1997-98 BrMO Round 1 P3
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.
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
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 $a^{2}+b^{2}+c^{2}=8R^{2}$
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 $\angle CAD=60$.
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 $S$.
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.
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 $CP = CQ$.
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
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 $AP=PQ$
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.
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}$ .
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
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 $[XYZ]$.
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.
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.
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 $XYZ$.
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.
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 $\angle EFD = \angle ACD + \angle ECB$.
2019-20 BrMO Round 1 P3
Two circles $S_1$ and $S_2$ are tangent at $P$. A common tangent, not through $P$, touches $S_1$ at $A$ and $S_2$ at $B$. Points $C$ and $D$, on $S_1$ and $S_2$ respectively, are outside the triangle $APB$ and are such that $P$ is on the line $CD$. Prove that $AC$ is perpendicular to $BD$.
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$.
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 $MCN$.
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.
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$
$ 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
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 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.
2000-01 BrMO Round 1 P2Circle $S$ 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 $a^{2}+b^{2}+c^{2}=8R^{2}$
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 $\angle CAD=60$.
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 $S$.
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.
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 $CP = CQ$.
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.
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 $AP=PQ$
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.
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}$ .
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 $BC=2AF$.
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
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 maximum 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 $AE$.
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.
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 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.
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 maximum 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 $AE$.
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 $[XYZ]$.
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.
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.
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 $XYZ$.
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.
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 $\angle EFD = \angle ACD + \angle ECB$.
2019-20 BrMO Round 1 P3
Two circles $S_1$ and $S_2$ are tangent at $P$. A common tangent, not through $P$, touches $S_1$ at $A$ and $S_2$ at $B$. Points $C$ and $D$, on $S_1$ and $S_2$ respectively, are outside the triangle $APB$ and are such that $P$ is on the line $CD$. Prove that $AC$ is perpendicular to $BD$.
1992 - 2019
Round 2 also known as BMO2
The circumcircle of the triangle $ABC$ has a radius $R$ satisfying $AB^2 + AC^2 = BC^2 - R^2$. Prove that the angles of the triangle are uniquely determined, and state the values for the angles.
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 $\displaystyle PA \frac{\text{sin} \, \angle BAC}{\text{sin} \, \alpha} = PB \frac{\text{sin} \, \angle ABC}{\text{sin} \, \beta} =PC \frac{\text{sin} \, \angle ACB}{\text{sin} \, \gamma}$
$ 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
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 $\angle AFC=\angle ADB$.
Prove that if $\angle DAC=\angle ABE$ ,if and only if $\angle AFC=\angle ADB$.
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 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.
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$.
1999-2000 BrMO Round 2 P1
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.
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 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 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 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 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 2 P3
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 $\angle BPC + \angle BAC = 180$.
.
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 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 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 2 P2
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 2 P1
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 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 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 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 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 $A$.
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 $BC$.
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 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 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.
2019-20 BrMO Round 2 P2
Describe all collections S of at least four points in the plane such that no three points are collinear and such that every triangle formed by three points in S has the same circumradius.
2020-21 BrMO Round 2 P3
Let $ABC$ be a triangle with $AB>AC$. Its circumcircle is $\Gamma$ and its incentre is $I$. Let $D$ be the contact point of the incircle of $ABC$ with $BC$. Let $K$ be the point on $\Gamma$ such that $\angle AKI$ is a right angle. Prove that $AI$ and $KD$ meet on $\Gamma$.
1972 - 1991
BMO2 also known as FIST
When $k = 1$ find all points $P$ in space such that $a \cdot PA^2 + b \cdot PB^2 + c\cdot PC^2 = k abc$ , where $a,b,c$ are the lengths of the sides $BC,CA,AB$ of triangle $ABC$, and prove your result. What is the effect of altering $k$?
A polygonal line is a continuous line $A_1A_2A_3...A_{n+1}$, where, for $r = 1$ to $n$, $A_rA_{r+1}$ is a straight line segment. In a square of side $50$, a polygonal line $L$ is constructed in such a way that the distance of any point inside the square from $L$ (i.e. from the nearest point of $L$) is less than $1$. Prove that the length of $L$ is greater than $1248$.
(USSR)
1975 British FIST p4 (easier version of 1974 ILL p17)The diagram illustrates a configuration of $12$ circles. The set S of $12$ circles contains three subsets $S_3, S_4, S_5$ each having $4$ circles and such that each of the $4$ circles of $S_r$ touches $r$ circles of $S$.
Prove that such a configuration of $12$ circles exists on the surface of a sphere with all the $12$ circles having equal radii.
A ’figure-of-eight' curve, $S$, consists of two touching circles of equal radii. Show that a pair of two distinct congruent hexagons (not necessarily convex) exists with the following properties:
(a) All the vertices of the hexagons lie on $S$.
(b) Neither hexagon has all its vertices on one circle.
(c) Neither hexagon can be obtained from the other by a single translation a single rotation or a single reflection.
Through a point $P$ in the interior of a fixed triangle $ABC$ lines $PL, PM,PN$ are drawn parallel to the medians through $A,B,C$ respectively to meet $BC, CA, AB$ at $L,M,N$ respectively. Prove that $$\frac{BL}{BC} + \frac{CM}{CA}+ \frac{AN}{AB}$$ is constant (independent of $P$).
A plane convex pentagon $ABCDE$ is said to have the "unit triangle property" if the area of each of the triangles $ABC, BCD, CDE, DEA, EAB$ is unity. Show that all plane convex pentagons with the unit triangle property have the same area and that there is an infinite number of such pentagons no two of which are congruent.
$VLMN$ and $VABC$ are tetrahedra with $A,B,C$ on $VL,VM,VN$, produced as necessary. The in-centre of triangle $LMN$ coincides with the centroid of triangle $ABC$.
(i) Determine $VA,VB,VC$ in terms of the sides of triangle $LMN$ and $VL,VM,VN$.
(ii) Determine the condition that the tetrahedra have equal volumes.
(iii) If the tetrahedra have unequal volumes, determine , with proof, which has the greater volume.
An axis of a solid is, for the purposes of this question, defined to be a atraight line joining two points on the surface of the solid and such that the solid, when rotated about this line through an angle which is greater than $0^o$ and less than $360^o$, coincides with itself. How many axes has a cube? Draw three diagrams to show the three different types of axis and state the minimum angle of rotation for each type.
(No fornal proofs are required.)
$ ABC$ is a triangle. The internal bisector of the angle $A$ meets the circumcircle again at $P$. $Q$ and $R$ are similarly defined. Prove that $AP + BQ + CR > AB + BC + CA$.
Two points $A,B$ and a line $k$ are given in a plane. Locate, with proof, the point $P$ of the plane for which $PA^2+ PB^2+ PN^2$ is a minimum, where $N$ is the foot of the perpendicular from $P$ to $k$.
Give a generalisation without proof for three points $A, B, C$ and $PA^2+ PB^2+ PC^2+PN^2$ a minimum.
Consider tlie three escribed circles of the triangle $ABC$, that is, the three distinct circles each of which touches one side of triangle $ABC$ internally and the other two externally. Each pair of escribed circles has just one common tangent which is not a side of triangle $ABC$, and the three such common tangents form a triangle $T$.
$O$ is the circumcentre of triangle $ABC$. Prove that $OA$ is perpendicular to a side of $T$.
$\ell, m, m$ are three lines in space. Neither $\ell$ nor $m$ is perpendicular to $n$. Points $P$ and $Q$ vary on $\ell$ and $m$ respectively in such a way that $PQ$ is perpendicular to $n$. The plane through $P$ perpendicular to $m$ meets $n$ at $R$ and the plane through $Q$ perpendicular to $\ell$ meets $n$ at $S$. Prove that $RS$ is of constant length.
The triangle $ABC$ is right-angled at $C$ . Find all the points $D$ in the plane satisfying the conditions $$AD \cdot BC = AC\cdot BD = \frac{1}{\sqrt2} AB \cdot CD$$
$ABCD$ is a tetrahedron with $DA = DB = DC = d$ and $AB = BC = CA = e$ . $M$ and $N$ are the midpoints of $AB$ and $CD$ . A plane $\pi$ passes through $MN$ and cuts $AD$ and $BC$ at $P$ and $Q$ respectively.
(i) Prove that $AP/AD = BQ/BC$ (= $t$, say) .
(ii) Determine with proof that value of $t$ , expressed in terms of $d$ and $e$ , which minimises the area of the quadrilateral $MQNP$ .
1985 British FIST p1 (also here)
$O$ is a point outside a circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$ with $A,C$ the midpoints of $OB,OD$ respectively. Also the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC$ .
$ABCD$ is a tetrahedron which has a circumsphere passing through $A, B, C, D$ and an in-sphere touching each triangular face at an interior point of that face. The two spheres have the same centre $O$ . $H$ is the orthocentre of triangle $ABC$ and $H'$ is the foot of the perpendicular from $D$ on to the plane of that triangle. Prove that $AB = CD$, $AC = BD$, $AD = BC$ and that $OH = OH'$ .
$C_1$ and $C_2$ are two circles. $A_1, A_2$ are fixed points on $C_1, C_2$ respectively.
$A_1P_1,A_2P_2$ are parallel chords of $C_1, C_2$ . Find the locus of the midpoint of $P_1P_2.$
$ABC$ is an equilateral triangle. The circle $\Gamma_1$ has centre A and radius $AB$. $\Gamma_2$ is the circle on $AB$ as diameter. A circle with centre $P$ on $AC$ touches $\Gamma_1$ internally at $C$ and $\Gamma_2$ externally at $Q$. Show that $AP/AC = 4/5$ and calculate the ratio $AQ/AC$.
$L$ and $M$ are two skew lines in space, i.e. they neither meet nor are parallel. $A, B$ are the points on $L, M$ respectively such that $AB$ is perpendicular to both $L$ and $M$. Points $P$ on $L, Q$ on $M$ vary so that $P \ne A$, $Q \ne B$, $PQ$ is of constant length.
Show that the centre of the sphere through $A, B, P, Q$ lies on a fixed circle with centre the midpoint of $AB$.
$M$ is a point on the side $AC$ of triangle $ABC$ such that triangles $BAM, BMC$ have inscribed circles of equal radius. Find the length of $BM$ in terms of the lengths $a,b,c$ of the sides of triangle $ABC$ .
Let $l$ denote the length of the smallest diagonal of all rectangles inscribed in a triangle $T$ . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of $T$ .) Determine the maximum value of $\frac{l^2}{S(T)}$ taken over all triangles
($S(T )$ denotes the area of triangle $T$ ).
$I$ is the centre of the circle inscribed to triangle $ABC$, $J$ is the centre of the exscribed circle which touches $AB$ and $AC$ produced beyond $B$ and $C$ respectively. Prove that $AI \cdot AJ = AB \cdot AC$ and that $AI \cdot BJ \cdot CJ = AJ \cdot BL \cdot CI$.
$ ABC$ is a right triangle in $ C$,and $ a$ is the measure of the angle between the median that pass trough $ A$ and the hypotenuse. Prove that $ sin (a) \le \frac{1}{3}$
https://bmos.ukmt.org.uk/home/bmo.shtml
No comments:
Post a Comment