The complex numbers \(a, b, c\) satisfy the equations \[ a+b+c=3, \quad abc=2, \quad \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix} = 0. \] Find the cubic equation of which they are the roots, and hence or otherwise determine their values.
If \(f(x)\) and \(g(x)\) are two polynomials in \(x\) of degrees \(m\) and \(n\) respectively, \(m \ge n\), show that there exist unique polynomials \(q(x)\) and \(r(x)\) such that \[ f(x) = q(x)g(x) + r(x), \] where the degree of \(r(x)\) is less than \(n\). Deduce that the highest common factor of \(f(x)\) and \(g(x)\) is the same as that of \(g(x)\) and \(r(x)\). Find the highest common factor of \[ x^5-x+15, \quad x^4+5x^2+9. \]
Sum the series: \(\sin\theta - 2\cos 2\theta + 3\sin 3\theta - \dots - 2n\cos 2n\theta\).
If \(a_1, a_2, \dots, a_n\) are all positive, and \(s_r = a_1^r + a_2^r + \dots + a_n^r\), prove that \(ns_3 \ge s_1s_2\).
Find a pair of integers \(x, y\) such that \[ 11x^2 + 14(x+y)(y-11) + 616 < 0. \] (Hint. Consider the pole of \(x=0\) with respect to the related conic.)
Three circles touch one another (internally or externally), and the three points of contact are distinct. Show by inversion, or otherwise, that in general there are exactly two circles which touch each of the given three, and that these two do not intersect. What is the exceptional case?
The perpendiculars from the vertices \(A, B\) of a triangle \(ABC\) to the opposite sides meet in \(O\). Prove that any conic through \(A, B, C, O\) is a rectangular hyperbola. Deduce that the three altitudes of a triangle are concurrent.
In each of the following two cases, either prove the statement true or give a counter-example to show it is false:
Show that there are three normals from a general point to the parabola \(y^2=4ax\), and that the feet of the three normals are concyclic with the vertex of the parabola. Find the locus of points from which two of the normals coincide, and show that this locus meets the parabola in the points \((8a, \pm 4\sqrt{2a})\).
Three equal circular arcs, each of radius \(a\) and angle \(\beta (<2\pi/3)\), are joined together to form a plane convex figure with three vertices. The angle \(\beta\) is such that, as the figure rolls along a fixed line, its topmost point at any moment lies on a fixed parallel line. Sketch the figure, and describe the path of one of its vertices as it rolls.