\(P\) 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 \(R_2\).
Prove that, if \(a^2t^4 = b^4\), an infinite number of triangles can be inscribed in an ellipse \(x^2/a^2 + y^2/b^2 = 1\) whose sides touch the parabola \(y^2 = 4cx\).
A convex polyhedron \(S\) is such that each vertex is the intersection of \(k\) faces with \(p_1, p_2, \ldots, p_k\) sides, the numbers \(p_1, p_2, \ldots, p_k\) being the same for all vertices of \(S\). If \(V\) is the total number of vertices of \(S\), prove that $$\sum_{r=1}^{k} \left(\frac{1}{2} - \frac{1}{p_r}\right) = 1 - \frac{2}{V}.$$ Show that (i) if \(k = 3\), and \(p_2 \neq p_3\), then \(p_1\) is even; (ii) if \(k = 4\), \(p_1 = p_3 > 3\), \(p_4 > 3\) then \(p_2 = p_4\). Hence or otherwise determine all possible values of \(p_1, p_2, \ldots, p_k\) for \(k = 4\) and indicate the number of vertices and faces of each kind of the corresponding polyhedron. [It is not necessary to show that polyhedra corresponding to these values of \(p_1, p_2\) exist.]
Sketch the three curves $$xy^2 = (a-x)^2(1-x)$$ for the following three values of the parameter \(a\): $$a = \frac{1}{2}, 1, 2.$$
Prove that the surface area of the spheroid, formed by rotating the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ about the \(x\)-axis, is $$2\pi b^2\left[1 + \frac{a^2}{bc}\sin^{-1}\frac{c}{a}\right],$$ where \(c^2 = a^2 - b^2\).