\(AB\), \(AC\) are two equal line segments, meeting at an acute angle. \(X\) is a point such that \(AB\), \(AC\) subtend equal angles at \(X\). Find the locus of \(X\). (Angles at \(X\) are to be reckoned positive in whichever sense they are measured.)
Find the \emph{width} of a regular tetrahedron of side \(a\), where \emph{width} is defined as the least distance between a pair of distinct parallel planes, each of which 'touches' the tetrahedron, either at a vertex, along an edge, or on a face. Hence or otherwise show that it is impossible to put a regular tetrahedron of side greater than \(\sqrt{2}\) in a cube of side 1. Show, with a sketch, how a regular tetrahedron of side exactly \(\sqrt{2}\) may be put in such a cube (the boundary of the cube may be used).
Prove that the locus, if it exists, of the meets of perpendicular real tangents to the hyperbola \(x^2/a^2 - y^2/b^2 = 1\) is a circle. Show that it cannot cut the hyperbola again.
\(p(\phi)\) is the positive length of the projection of a fixed line-segment of length \(l\) on an axis at a variable direction \(\phi\). Prove that $$\int_{\phi=0}^{\phi=2\pi} p(\phi) d\phi = 4l.$$ Hence or otherwise prove that if a triangle \(ABC\) lies entirely within a triangle \(XYZ\), then $$\text{perimeter } \triangle ABC < \text{perimeter } \triangle XYZ.$$
\(P\) is a point in the plane of a triangle \(ABC\), not lying on any side of the triangle. The point \(P'\) is such that (i) the bisectors of the angles between \(AP\) and \(AP'\) bisect the angles between \(AB\) and \(AC\), and (ii) the bisectors of the angles between \(BP\) and \(BP'\) bisect the angles between \(BC\) and \(BA\). Prove that the bisectors of the angles between \(CP\) and \(CP'\) bisect the angles between \(CA\) and \(CB\). Find the locus of \(P'\), if \(P\) varies on the circumcircle of the triangle \(ABC\).
The diagonals \(AC\), \(BD\) of the cyclic quadrilateral \(ABCD\) meet in \(O\), and \(L\), \(M\) are the feet of the perpendiculars drawn from \(O\) to \(AB\), \(CD\). Prove that $$\frac{OL}{OM} = \frac{OA}{OD}.$$ Show that \(L\) and \(M\) are equidistant from the mid-point of \(AD\).
\(O\) is a point in the plane of a circle \(C\), lying outside \(C\). \(P\) is a variable point on \(C\), and \(P'\) is a point on \(OP\) such that \(OP' = \lambda OP\), where \(\lambda\) is a positive constant. Prove that the locus of \(P'\) is a circle \(C'\). If \(OP'\) meets \(C'\) again in \(Q'\), and if the circle through \(O\) and \(P\) orthogonal to \(C\) meets \(C\) again in \(R\), prove that \(OR\) passes through the point of \(C'\) diametrically opposite to \(Q'\).
\(ABC\) is a triangle, with vertices ordered in a counter-clockwise sense. Show that the resultant of counter-clockwise rotations of angles \(\pi-A\) about \(A\), \(\pi-B\) about \(B\), and \(\pi-C\) about \(C\) (in the order mentioned) is a translation of the plane in the direction \(AC\), through a distance equal to the perimeter of the triangle.
\(A\), \(B\), \(C\), \(D\) are the points \((r\cos\theta, r\sin\theta)\), for \(\theta = \alpha, \beta, \gamma, \delta\); and \(a\), \(b\), \(c\), \(d\) are the tangents to the circle \(x^2+y^2 = r^2\) at \(A\), \(B\), \(C\), \(D\). Show that the intersections \((ab)\), \((bc)\), \((cd)\), \((da)\) are concyclic if, and only if, \(\alpha+\gamma-\beta-\delta = k\pi\), where \(k\) is an integer. If this condition is satisfied, and if \(R\) is the radius of the circle circumscribing the quadrilateral \(abcd\), and \(p\) the distance of the centre of this circle from the origin, prove that, if \(k\) is even, \(p = R\).
Two circles \(C_1\), \(C_2\) meet in \(A\), \(B\). A parabola drawn through \(A\) again in \(P_1\), \(Q_1\) and meets \(C_2\) again in \(P_2\), \(Q_2\). The normals to the parabola at \(P_1\) and \(Q_1\) meet in \(R\). Prove that \(R\) is a normal to the parabola, and that if \(A_1\), \(B_1\) are the images of \(A\), \(B\) in the axis of the parabola, then the normals meet at \(R_2\).