British TST 1985 - 2015 (UK FST, NST) 62p

geometry problems from British IMO Team Selection Tests (TST) with aops links

    FIST= further international (older name of BMO2)

FST = first 

SIST= second international

NST = next

Not from Shortlist

 

1985 - 2015

1985 British FIST 2  p2

The in-circle of $\triangle ABC$, where $AB > AC$, touches $BC$ at $L$ and $LM$ is a diameter of the in-circle. $AM$ produced cuts $BC$ at $N$.

(i) Prove $NL = AB - AC$.

(ii) A circle $S$ of variable radius touches $BC$ at $M$. The tangents (other than $BC$) from $B$ and $C$ to $S$ intersect at $P$. $P$ moves as the radius of $S$ varies. Find the locus of $P$

1985 British FIST 2  p4

$A, B, C, D$ are points on a sphere of radius $1$. Given that $AB \cdot BC \cdot CA \cdot DA \cdot DB \cdot DC = \frac{512}{27}$, prove that $ABCD$ is a regular tetrahedron.

1986 British SIST  p1

$A, B, C, A', B', C'$ are six points on a circle such that the chords $AA', BB', CC'$ meet in a point. Prove that $AB \cdot B'C' \cdot CA' = A'B' \cdot BC \cdot C'A$.     (*)

Is it true, conversely, that if $A, B, C, A', B', C'$ are six points on a circle satisfying (*) then $AA', BB', CC'$ are concurrent? Justify your answer.

1987 British RST  p1  (Reading Selection Test)

$A, B, C$ are the angles of an acute-angled triangle. Prove that

$(\tan A + \tan B + \tan C)^{2} \geq (\sec A + 1)^{2} +(\sec B+1)^{2} + (\sec C + 1)^{2}$.

1987 British RST  p3  (Reading Selection Test)

$Q$ is a convex quadrilateral whose four vertices are on the circumference of a circle whose centre is $O$. The distance of any side of $Q$ from $O$ is half the length of the opposite side. Prove that the diagonals of $Q$ intersect orthogonally.

1988 British SIST  p1

The triangle $ABC$ is not equilateral. Show that the equations

$\frac{\sin (A- \theta)}{a^{2}} = \frac{\sin (B- \theta)}{b^{2}} = \frac{\sin (C- \theta)}{c^{2}}$

are satisfied by a unique number $\theta$ in the interval $0 < \theta < \frac{\pi}{6}$.

[It may help to work in terms of $a, b, c$ and the area $\triangle$ of the triangle.]

1988 British SIST  p4

The equilateral triangle $ABC$ is inscribed in a circle $K$. $T$ is a point of $K$ lying on the smaller arc $AB$. Prove that $AT+ BT = CT$.

A second circle $k$ touches $K$ at $T$ and lies inside $K$. The lengths of the tangents from $A, B, C$ to $k$ are $a, b, c$, respectively. Prove that $a+b=c$.

  

1989 British Geometry Test G1

$A, B$ are two points on a sphere whose centre is $O$ and whose radius is $r$. $AB$ subtends an angle $2 \theta$ at $O$. Two planes are drawn through $AB$, each plane inclined at an angle $\alpha$ to the plane $AOB$. $C_{1}$ and $C_{2}$ are the centres of the circles in which these planes cut the sphere. Find the length $C_{1}C_{2}$.

1989 British Geometry Test G2

$ABC$ is a triangle; $D, E, F$ are the feet of the perpendiculars from $A, B, C$ respectively to the opposite sides; $D', E', F'$ are the midpoints of $EF, FD, DE$ respectively.

The line $E'F'$ meets $CA$ at $Q$ and $AB$ at $R'$;

The line $F'D'$ meets $AB$ at $R$ and $BC$ at $P'$;

The line $D'E'$ meets $BC$ at $P$ and $CA$ at $Q'$. Prove that

(i) $QR' = RP' = PQ'$;

(ii) $P, P', Q, Q', R, R'$ lie on a circle.

1989 British Geometry Test G3

$ABCD$ is a quadrilateral inscribed in a circle. $BD$ meets $AC$ at $X$ and (produced) passes through the point of intersection $T$ of the  tangents at $A$ and $C$. Prove that the perpendicular distances of $X$ from the sides $AB, BC, CD, DA$ of the quadrilateral are proportional to the lengths of those sides.

1990 British SIST p3

Given a set of points $P$ in the $xy$-plane, we define a set $P^{*}$ according to the following rule:

$(x^{*}, y^{*}) \in P^{*}$ if and only if $xx^{*} + yy^{*} \leq1$ for all $(x, y) \in P$.

Find all triangles $T$ such that $T^{*}$ is obtained from $T$ by a half-turn about the origin.

1991 British FST p1

$ABC$ 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        $sin (a)  \le \frac{1}{3}$

1991 British SIST p2

Vertex $A$ of triangle $ABC$ is equidistant from the circumcentre $O$ and the orthocentre $H$. Find all possible values of angle $A$.

1992 British FST p1

Circles $C_{1}, C_{2}$, with centres $O_{1}, O_{2}$ and radii $r_{1}, r_{2}$ respectively, are drawn so that the distance between their centres is given by $O_{1}O_{2} = \sqrt{r_{1}^{2}+r_{2}^{2}}$. The circles $C_{1}$ and $C_{2}$ intersect at $A$ and $B$. From the point $P$ on $C_{1}$ furthest from $O_{2}$ lines $PA, PB$ are drawn to intersect $C_{2}$ at $D, E$ respectively. Prove that $DE$ is the diameter of $C_{2}$ perpendicular to $O_{1}O_{2}$.

A point $Q$ (distinct from $P$) is now chosen on the major arc $AB$ of $C_{1}$. Lines $QA, QB$ are drawn to intersect $C_{2}$ at $F, G$ respectively. Prove that $FG$ is also a diameter of $C_{2}$. Locate, with proof, the point where $FB$ and $GA$ meet.

1993 British FST p3

Six rods $A_{1}A_{2}, A_{2}A_{3}, A_{3}A_{4}, A_{4}A_{5}, A_{5}A_{6}, A_{6}A_{1}$ of lengths $(a_{1}+a_{2}), (a_{2}+a_{3}), (a_{3}+a_{4}), (a_{4}+a_{5}), (a_{5}+a_{6}), (a_{6}+a_{1})$ respectively, where $a_{i} >0 (i = 1, 2, ..., 6)$, are freely jointed together to form a hexagon $A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}$. Prove that the hexagon can be adjusted so that it takes the shape of a convex hexagon cirscibing a circle.

Prove also that if the real numbers $a_{i} (i= 1, 2, ..., 6)$ are permuted amongst themselves in any way, the six new resulting rods have the same property and that whatever permutation is made the inscribed circle always has the same radius. Find the radius explicitly in terms of elementary symmetric functions of $a_{1}, a_{2},  a_{3}, a_{4}, a_{5}, a_{6}$.

Does there exist a convex hexagon, with side lengths $1, 2, 3, 4, 5, 6$ in some order, circumscribing a circle?

1994 British FST p2

Let $S$ be the circumcircle of an isosceles triangle $ABC$ with $AB = AC$. The point $P$ lies on the arc $BC$ of $S$ on the opposite side of $BC$ to $A$. Let $X$ be the point on $AP$ such that $AX= AC$ and $Y$ be the point on $BP$ such that $BY=BC$.

(i) Prove that $C, Y, P, X$ lie on a circle.

(ii) Prove that $YX$, the tangent at $B$ to the circumcircle $S$ and the line through $C$ perpendicular to $AB$ are concurrent.

1995 British FST p3

$ABC$ is a triangle. Its incircle touches the sides $BC, CA, AB$ at $X, Y, Z$ respectively. The points $A', B', C'$ are the feet of the perpendiculars from $X$ to $YZ$, from $Y$ to $ZX$ and from $Z$ to $XY$ respectively.

(i) Prove that $AA', BB', CC'$ are concurrent.

(ii) Proved that  area $\{\triangle ABC \} \geq 16 \cdot$ area $\{ \triangle A'B'C' \}$

1996 British FST p1

Two circles $\Gamma_{1}$ and $\Gamma_{2}$ intersect at $D$ and $E$. The point $B$ on $\Gamma_{1}$ and the point $C$ on $\Gamma_{2}$ are such that $D$ is the midpoint of $BC$. The tangent to $\Gamma_{1}$ at $B$ and the tangent to $\Gamma_{2}$ at $C$ meet at $A$. Prove that $DA \cdot DE = DC^{2}$.

2001 British TST1 p1

Let O be the circumcentre and H the orthocentre of the acute-angled triangle ABC. Show that the minimum value of OP + HP, as P varies over the perimeter of the triangle, is exactly the circumradius of ABC.

2002 British FST  p1

We are given a circle $\Gamma$ and a straight line $\ell$ which is tangent to the circle at the point B. Through any point A on $\Gamma$  we drop the perpendicular AP to $\ell$  where $P \in \ell$  Let M be the point which is the reflection of $P$ in the line $AB$. Determine the locus of M as A varies on $\Gamma$ .

2002 British NST  p1

Let $ABC$ be a triangle and $l$ the line through $C$ which is parallel to $AB$. The internal bisector of angle $A$ meets the side $BC$ at $D$ and the line $\ell$ at $E$. The internal bisector of angle $B$ meets the side $AC$ at $F$ and the line $\ell$ at $G$. Suppose that $GF = DE$. Show that $AC = BC$.

2003 British FST  p1

Consider triangle $ABC$. Let $U$, $V$, $W$ be points such that $U$ is on the line $BC$, $V$ is on the line $CA$, $W$ is on the line $AB$. It is given that $AU$, $BV$ and $CW$ are concurrent at a point $P$. Also $AU$ is a median of the triangle, $BV$ is an altitude and $CW$ is the internal angle bisector of $\angle BCA$. Suppose that $P$ lies on the perpendicular bisector of at least one of the sides of the triangle $ABC$. Prove that the triangle $ABC$ is equilateral.

2005 British NST1 p1

An acute triangle $ABC$ is given. Find the locus of points $M$ in the interior of $ABC$ such that $AB - FG = \frac{MF \cdot AG + MG \cdot BF}{CM}$ where $F$ and $G$ are the feet of the perpendiculars from $M$ to $BC$ and $AC$ respectively.

2005 British NST2 p1

Let $n \geq 2$ be a natural number. A pyramid $P$ has base $A_1A_2...A_{2n}$ and apex $O$. The polygon $A_1A_2...A_{2n}$  is regular and the point $C$ is its centre. The line $OC$ is perpendicular to the plane of the base of $P$. A sphere passes through $O$ and meets each of the line segments $OA_i$ internally. For each $i = 1, 2,..., 2n$, let $X_i$ be the point (other than $O$) where the sphere meets $OA_i$. Prove that $OX_1+OX_3 + ... + OX_{2n-1} =  OX_2+OX_4 + ... + OX_{2n}$

2006 British FST1  p1

Let $E$ be the intersection of the diagonals of the cyclic quadrilateral $ABCD$. Let $F$ and $G$ denote the respective midpoints of the sides $AB$ and $CD$. Prove that the line through $G$ which is perpendicular to $AC$, the line through $F$ which is perpendicular to $BD$ and the line through $E$ which is perpendicular to $AD$ are concurrent.

2006 Brithish FST2  p2
Let $ABCD$ be a cyclic quadrilateral, and $P$ a point in its interior such that $\angle BPC= \angle PAB+\angle PDC$. Let $E$, $F$ and $G$ be the orthogonal projections of $P$ on the sides $AB$, $DA$ and $CD$, respectively. Show that triangles $BPC$ and $EFG$ are similar.

Alternative formulation.
A point $P$ is in the interior of the cyclic quadrilateral $ABCD$ and has the property $\angle BPC = \angle BAP+\angle PDC$. The feet of the perpendiculars from $P$ to $AB$, $AD$ and $DC$ are respectively denoted $E$, $F$ and $G$. Show that $\triangle FEG$ and $\triangle PBC$ are similar.

2006 British NST1 p2 (Ireland 2002)

Let $ABC$ be a triangle with integer side lengths, and let its incircle touch $BC$ at $D$ and $AC$ at $E$. If $|AD^2-BE^2| \le 2$, show that $AC=BC$.

2006 British NST2 p3

Triangle $ABC$ has incentre $I$. The line $AI$ meets the circumcircle of $ABC$ again at $D$. The feet of the perpendiculars from $I$ to $BD$ and $CD$ are denoted $E$ and $F$ respectively. Given that $IE + IF = \frac12 AD$, determine the possible values of $\angle BAC$.

2007 British FST1 p2

Let triangle $ABC$ have a right angle at $A$. Let $D$ denote the foot of the perpendicular from $A$ to $BC$. Let $I$ and $J$ be the respective incentres of triangles $ABD$ and $ADC$. Draw the line $IJ$ to meet $AB$ at $E$ and $AC$ and $F$, Show that $A$ is the circumcentre of triangle $EDF$.

2007 British FST2 p3

Let $A$ be a point exterior to a circle $\Gamma$. Two lines through $A$ meet $\Gamma$ at $B,C$ and $D,E$ respectively, with $D$ between $A$ and $E$. Draw the line through $D$ which is parallel to $AC$, and let it meet $\Gamma$ again at $F$. Suppose that $AF$ meets $\Gamma$ again at $G$, and that $EG$ meets $AC$ at $M$. Prove that  $\frac{1}{AM} =\frac{1}{AB} + \frac{1}{AC}$.

2007 British NST1 p1

Triangle $ABC$ has circumcentre $O$ and centroid $M$. The lines $OM$ and $AM$ are perpendicular. Let $AM$ meet the circumcircle of $ABC$ again at $K$. Lines $CK$ and $AB$ intersect at $D$ and $BK$ and $AC$ intersect at $E$.Prove that the circumcentre of triangle $ADE$ lies on the circumcircle of $ABC$.

2007 British NST2 p2

Let $ABC$ a triangle, and let $\Gamma_b$ and $\Gamma_c$ the excircles relatives to the sides $AC$ and $AB$, respectively. $\Gamma_b$ touches $AC$ at $P$ and the extensions of the sides $AB,AC$ at the points $D,E$, respectively. $\Gamma_c$ touches $AB$ at $Q$ and the extensions of the sides $AC,BC$ at the points $F,G$, respectively. The line $FG$ intersects line $DE$ at the point $X$, $GQ$ intersects $PE$ at $Y$. Prove that $X,A,Y$ are collinear.

2007 British NST3 p1

Let ABC be a triangle with AB ^ AC. The incircle I of ABC touches the sides BC, CA, AB at the points D, E, F, respectively. Let AD intersect Z at D and P. Let Q be the intersection of the line EF and the line passing through P and perpendicular to AD, and let X, Y be intersections of the line AQ and DE, DF, respectively. Show that the point A is the midpoint of XY.

2007 British NST4 p1

Consider triangle $ABC$. $B'$ is on the line $AC$ and the line $BB'$ passes through the incentre $I$. The point $C'$ is similarly defined. The line $B'C'$ meets the circumcircle of triangle $ABC$ at $M$ and $N$. Prove that the circumradius of triangle $MIN$ is twice the circumradius of triangle $ABC$.

2008 British FST1 p1

A triangle $ABC$ is given. A circle $\Gamma$ passes through $A$ and is tangent to the side $BC$ at a point $P$. The circle $\Gamma$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. Prove that the (minor) arcs $MP$ and $NP$ are equal if and only if $\Gamma$  is tangent to the circumcircle of $ABC$ at $A$.

2008 British NST1 p2

Let $A$ be a point in the interior of triangle $ABC$. The line $AX$ meets the side$BC$ at $A_1$. Points $B_1$ and $C_1$ are similarly defined. Let $R_1, R_2$ and $R_3$ be the respective radii of circles $XBC, XCA$ and $XAB$, and $R$ be the circumradius of triangle $ABC$. Prove that

$$\frac{XA_1}{AA_1} R_1+\frac{XB_1}{BB_1} R_2 +\frac{XC_1}{CC_1} R_3 \ge R$$

2008 British NST3 p2

Let $ABCD$ be a parallelogram but not a rhombus. Let $E$ be the foot of the perpendicular from $B$ to $AC$. The line through $E$ which is perpendicular to $BD$ intersects $BC$ at $F$ and $AB$ at $G$. Show that $EF = EG$ if, and only if, $ABCD$ is a rectangle.

2008 British NST4 p3

A point $X$ lies in the plane of triangle $ABC$. A circle $\Gamma$ passing through $X$ meets $XA,XB$ and $XC$ again at $P, Q$ and $R$ respectively. Let $\Gamma$ meet the circles $BXC, AXC$ and $AXB$ again at $T, L$ and $M$ respectively. Prove that $PT, QL$ and $RM$ are concurrent.

2009 British FST1 p2

Each point of the plane is painted one of three colours. Show that there exists a triangle in the plane such that the following three conditions are satisfied:

(a) The three vertices have the same colour.

(b) The radius of the circumcircle of the triangle is 2009.

(c) One angle of the triangle is either two or three times larger than one of the other two angles of the triangle. 

2009 British FST1 p3   ( also here )

Let the quadrilateral $ABCD$ be inscribed in a circle with center $O$. Suppose that $\angle B$ and $\angle C$ are obtuse, and let lines $AB$ and $CD$ intersect at $E$. Let $P$ and $R$ be the foot of the perpendiculars dropped from $E$ to the lines $BC$ and $AD$ respectively. Extend $EP$ to hit $AD$ at $Q$, and let $ER$ hit $BC$ at $S$. Finally, let $K$ be the midpoint of $QS$. Prove that $E, K, O$ are collinear.

2009 British FST2 p2

Triangle $ABC$ has a right-angle at $C$, and the point $M$ on $AB$ is strictly between $A$ and $B$. Let $S, S_1$ and $S_2$ denote the circumcentres of $\vartriangle ABC, \vartriangle  AMC$ and $\vartriangle MBC$ respectively.

(a) Show that the points $M, C, S, S_1$ and $S_2$ lie on a circle.

(b) For which position of $M$ does this circle have the least radius?

2009 British NST1 p1

The convex quadrilateral $ABCD$ has the property that $|AB| = |AC| = |BD|$. Let $P$ be the intersection point of its diagonals. Let $O$ and $I$ respectively be the circumcentre and incentre of triangle $ABP$. Show that if $O\ne I$, then $OI$ and $CD$ are perpendicular.

2009 British NST2 p3

Let ABC be a triangle. Its incircle is tangent to AB at E, while the excircle opposite A is tangent to AB at F. Let D be the point on BC for which the incircles of triangles ABD and ACD have equal radii. The lines DE and DB meet the circumcircle of triangle ADF for a second time at X and Y. Show that XY || AB if, and only if, AB = AC.

2009 British NST3 p

Let ABC be an acute-angled triangle, and M be a point in its plane distinct from the vertices. Show that the vector equation a/ MA MA + b/ MB  MB + c/ MC MC = 0

holds if, and only if, M is the orthocentre of ABC.

2009 British NST4 p1

The triangle $ABC$ has a right angle at $C$. Let $P$ be a point inside $ABC$ such that $|AP| = |AC|$. Let $M$ be the midpoint of the hypotenuse $AB$ and $T$ be the foot of the altitude dropped from $C$. Prove that $PM$ is a bisector of $\angle BPT$ if, and only if, $\angle A = 60^o$.

2010 British FST1 p2

Let $ABCD$ be a trapezium with $AB$ parallel to $DC$ and $|AB| > |CD|$. Let $E$ and $F$ be points on the segments $AB$ and $DC$ respectively, such that $AE: EB = DF: FC$. Let $K$ and $L$ be points on the segment $EF$ such that $\angle BKA = \angle  BCD$ and $\angle DLC = \angle ABC$. Show that $K, L, B$ and $C$ are concyclic.

2010 British FST2 p2

Let S be a set of 1953 points in the plane. Every two points of S are at least distance 1 apart. Prove that S contains a subset T of 217 points, every two at least distance $\sqrt3$ apart.

2010 British NST1 p3

Let $ABCD$ be a cyclic trapezium with $AD \parallel BC$ and $|AD| < |BC|$. The circle is called $\Gamma$ , and has centre $O$. Let $P$ be a variable point on the part of the ray $BC$ that is beyond $C$. It is given that $PA$ is not tangent to $\Gamma$  . The circle with diameter $PD$ meets $\Gamma$ again at $E$. Let $M$ be the intersection of the lines $BC$ and $DE$, and $N$ be the second point of intersection of the line $PA$ and $\Gamma$. Prove that the lines $MN$ pass through a fixed point as $P$ varies.

2010 British NST2 p2

Let $I$ be the incentre of triangle $ABC$. The incircle touches $AB$ and $BC$ at $X$ and $Y$ respectively. The line $XI$ meets the incircle again at $M$. Let $X'$ be the point of intersection of $AB$ and $CM$. The point $L$ on the segment $X'C$ is such that $X'L = CM$. Prove that $A, L$ and $Y$ are collinear if, and only if, $| AB| = |AC|$.

2010 British NST4 p2

The triangle $ABC$ is not isosceles. Let the inscribed circle $\Gamma$ have centre $I$ and touch the sides at $A_1, B_1$ and $C_1$ in the natural notation. Let $AA_1$ meet $\Gamma$ again at $A_2$, and define $B_2$ in similar fashion. The points $A_3$ on $B_1C_1$ and $B_3$ on $A_1C_1$ are such that $A_1A_3$ and $B_1B_3$ are angle bisectors in triangle $A_1B_1C_1$. 

Prove the following statements:

(a) $A_2A_3$ bisects $\angle B_1A_2C_1$.

(b) Let $P$ and $Q$ be the intersection points of the circumcircles of triangles $A_1A_2A_3$ and $B_1B_2B_3$, then $I$ lies on the line $PQ$.

2011 British FST2 p2

Let $ABCD$ be a cyclic quadrilateral, whose circumcircle has centre $O$. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $AD$. Show that if the area of the quadrilateral $ABCD$ is four times the area of the triangle $OEF$, then one of $BC$ and $DC$ is a diameter.

2011 British NST1 p1

Circles $\Gamma_1$ and $\Gamma_2$ meet at $M$ and $N$. Let $A$ be on $\Gamma_1$ and $D$ on $\Gamma_2$. The lines $AM$ and $AN$ meet $\Gamma_2$ again at $B$ and $C$ respectively; the line $DM$ and $DN$ meet $\Gamma_1$ again at $E$ and $F$, respectively. Assume that $M, N, F, A, E$ are in cyclic order around $\Gamma_1$, and that $AB$ and $DE$ are congruent. Prove that $A, F, C$ and $D$ lie on a circle whose centre does not depend on the position of $A$ and $D$ on the circles.

2011 British NST2 p2

Let $ABC$ be a triangle with a right angle at $C$. Let $CN$ be an altitude. A circle $\Gamma$ is tangent to the line segments $BN, CN$, and the circumcircle of $ABC$. If $D$ is where $\Gamma$ kisses $BN$, prove that $CD$ bisects $BCN$.

2011 British NST4 p3

Let ABC be a scalene triangle. Let l_A be the tangent to the nine-point circle at the foot of the perpendicular from A to BC, and let l'_A be the tangent to the nine-point circle from the midpoint of BC. The lines I_A and l'_A intersect at A'; we define B' and C' similarly. Show that the lines AA', BB' and CC' are concurrent.

2012 British FST1 p1

Let $ABCD$ be a cyclic quadrilateral so that $BC$ and $AD$ meet at a point $P.$ Consider a point $Q,$ different from $B,$ on the line $BP$ such that $P Q = BP,$ and construct the parallelograms $CAQR$ and $DBCS.$ Prove that the points $C, Q, R, S$ are concyclic.

2013 British NST2 p2

A triangle $ABC$ is given. Let $D, E$ and $F$ be points on the lines $BC, CA$ and $AB$ (respectively) such that $AF = EF$ and $BF = DF$. Prove that the orthocentre of triangle $ABC$ lies on the circle $DCE$.

2013 British NST3 p4

Circles $\Omega$ and $\omega$ are tangent at a point $P$, and $\omega$  lies inside  $\Omega$. A chord $AB$ of  $\Omega$ is tangent to $\omega$  at $C$. The line $PC$ meets  $\Omega$ again at $Q$. Chords $QR$ and $QS$ of  $\Omega$ are tangent to $\omega$ . Let $I, X$ and $Y$ be the incentres of triangles $ABP, ABR$ and $ABS$ respectively. Prove that $\angle PXI + \angle IYP = 90^o$.

2014 British FST1 p3

Let$ABC$ be an acute triangle with orthocentre $H$. The external bisector of angle $\angle CHB$ intersects $AB$ and $AC$ at $D$ and $E$ respectively.The internal bisector of $\angle CAB$ intersects the circumcircle of triangle angle $ADE$ again at $K$. Show that $HK$ passes through the midpoint of $BC$

2014 FST2 p1

Let $ABC$ be a triangle. For a point $P$ of the plane, let $A'$ be the foot of the perpendicular dropped from $P$ to $BC$. Points $B'$ and $C'$ are defined analogously. Find the locus of points $P$ in the plane such that $PA' \cdot PA=PB' \cdot PB=PC' \cdot PC$

2014 British NST2 p2

In triangle $ABC$, the excentres are $I_{a},I_{b},I_{c}$ in the natural notation.The excircle opposite 

$A$ touches $AB$ and $AC$ at $P$ and $Q$ respectively.The line $PQ$ intersects the lines $BI_{a}$ and $CI_{a}$ at $D$ and $E$ respectively.Let $A_{1}$ be the intersection of $DC$ and $BE$.The points $B_{1}$ and $C_{1}$ are defined analogously.Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent. 

2015 British FST1 p3

Let $\vartriangle ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_2 = CQ_1$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circumcircles of  $\vartriangle P_1P_2R$ and  $\vartriangle  Q_1Q_2R$ meet again at S, inside triangle  $\vartriangle P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $\angle P_1RS$ and $\angle Q_1RM$ are equal.

2015 British FST2 p1

An acute-angled triangle $\vartriangle ABC$ is given, and $A_1,B_1, C_1$ are the midpoints of sides $BC,CA, AB$ respectively. The internal angle bisector of $\angle AC_1C$ meets $AC$ at $L$, and the internal angle bisector of $\angle CC_1B$ meets $BC$ at $K$. The line $LK$ intersects $B_1C_1$ at $A_2$, and $A_1C_1$ at $B_2$. Prove that the lines $AA_2, BB_2, CC_1$ are concurrent.

2015 British NST3 p1 (Rioplatese 2006)

The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.

2015 British NST4 p1

In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ be the midpoint of $BC$, and $E$ the reflection of $C$ in line $DM$. Prove that $\angle{AEB} = \angle{ACD}$. 

source: https://bmos.ukmt.org.uk/home/selection-tests.shtml