\(X\), \(Y\), \(Z\) are the centres of squares described externally on the sides of a triangle. Prove that \(AX\), \(YZ\) are perpendicular and of equal length.
Prove that the circumcircles of the four triangles formed by the sets of four lines in general position in a plane meet in a point. Four circles \(C_i\) \((i = 1, 2, 3, 4)\) in a plane pass through a point \(A\). The circles \(C_5\), \(C_6\) meet again in \(A_{jk}\), and \(C_{jk}\) is the circle through \(A_{jk}\), \(A_{kl}\), \(A_{lj}\). Prove that the circles \(C_{234}\), \(C_{341}\), \(C_{412}\), \(C_{123}\) meet in a point.
Prove that the orthocentre of the triangle formed by the points \((a\cos\alpha, a\sin\alpha)\), \((a\cos\beta, a\sin\beta)\), \((a\cos\gamma, a\sin\gamma)\) is the point \((a(\cos\alpha + \cos\beta + \cos\gamma), a(\sin\alpha + \sin\beta + \sin\gamma))\). Show that the centre of the nine-point circle of a triangle lies within the triangle if and only if the difference between the greatest and least angles of the triangle is less than a right angle.
Find the locus of intersection of perpendicular normals to the parabola \(y^2 = 4ax\). Sketch this curve in relation to the parabola.
\(APQ\) is a variable chord of the conic \(S = 0\), passing through the fixed point \(A\) (not on \(S\)) and meeting \(S\) in \(P\), \(Q\). Show that the locus of the point dividing \(PQ\) in the ratio \(\lambda:1\) is the curve \begin{align} (\lambda + 1)^2(S - S_0)^2 = (\lambda - 1)^2(S_0^2 - SS_{00}), \end{align} where \(S = ax^2 + 2hxy + \ldots\), \(S_0 = ax_0 x_0 + h(xy_0 + x_0y) + \ldots\), \(S_{00} = ax_0^2 + 2hx_0y_0 + \ldots\), and \((x_0, y_0)\) are the coordinates of \(A\). Interpret this equation (i) when \(\lambda = 1\), (ii) when \(\lambda = 0\), (iii) when \(\lambda = -1\).
\(ABC\) is an acute-angled triangle and \(BC\) is its shortest side. The altitude from \(A\) to \(BC\) is of length \(h\), and the circumradius is \(R\). Prove that \(2R^2 < h^2\).
Prove that the area of a sphere \(S\) between two parallel planes \(\pi\), \(\pi'\) both of which meet \(S\) depends only on the radius of \(S\) and the distance between \(\pi\) and \(\pi'\). A circular disc \(D\) of radius 1 may be covered by \(r\) planks each of width \(w\). By considering the areas above the planks on the sphere of which \(D\) is the equator, show that \(rw \geq 2\), and that if \(rw = 2\) then the planks must be placed parallel to one another.
\(E\) is the elliptical billiard table whose boundary is \begin{align} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \end{align} A ball \(B\) leaves the focus \((ae, 0)\) of \(E\) and bounces perfectly elastically whenever it hits the boundary. Describe the path of \(B\), and show that it eventually approximates to the major axis of the ellipse.
A string of length \(\pi\) is attached to the point \((-1, 0)\) of the circle \(x^2 + y^2 = 1\), and is wrapped round the circle so that its other end is at the point \((1, 0)\). The string is unwound, being kept taut, and is wound up again the other way; if \(S\) is the path of the end of the string, show that \(S\) has length \(2\pi^2\), and find the area enclosed by \(S\).
Describe the curve \begin{align} (x^2 + y^2)^2 - 4x^2 = a \end{align} for \(a = -6, -4, -2, 0, 2, 4\). (Accurate diagrams are not necessary.)