Establish the equations of a cycloid in the form \begin{align*} x &= a(\theta+\sin\theta) \\ y &= a(1-\cos\theta) \end{align*} A circle rolls a long distance along a straight line. Prove that the path described by a point on its circumference is longer than the path described by its centre in the ratio \(\frac{4}{\pi}\).
A variable point \(X\) is taken on the side \(BC\) of a quadrilateral \(ABCD\); and the line drawn through \(B\) parallel to \(AX\) cuts in \(P\) the line drawn through \(C\) parallel to \(DX\). Prove that the locus of \(P\) is a fixed line parallel to \(AD\).
Three tangents to a parabola whose focus is \(S\) form the triangle \(ABC\). Prove that the tangent to the parabola that is perpendicular to \(SA\) cuts \(BC\) in the same point as the line joining \(S\) to the centre of the circle \(ABC\).
Prove that \(a^2+b^2+c^2-bc-ca-ab\) is a factor of the expression \[ (b-c)^n+(c-a)^n+(a-b)^n \] if \(n\) is a positive integer which is not a multiple of 3.
If \((1+x)^n = c_0+c_1 x+\dots+c_n x^n\), prove that \[ \frac{c_0}{n+1}-\frac{c_1}{n+2}+\frac{c_2}{n+3}-\dots+(-1)^n\frac{c_n}{2n+1} = \frac{(n!)^2}{(2n+1)!}. \]
Prove that \[ 16\sin\frac{\pi}{30}\sin\frac{7\pi}{30}\sin\frac{11\pi}{30}\sin\frac{17\pi}{30} = 1. \]
\(ABC\) is a triangle inscribed in a circle whose centre is \(O\) and radius \(R\); and \(AO, BO, CO\) meet \(BC, CA, AB\) in \(D,E,F\). Prove that \[ \frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF} = \frac{2}{R}. \]
A fixed line cuts two perpendicular lines \(OA, OB\) in \(A, B\); a variable line cuts \(OA, OB\) in \(X, Y\), such that \(XA=YB\); and \(E, Z\) are the mid-points of \(AB, XY\). Prove that the locus of the point of intersection of the line drawn from \(E\) perpendicular to \(XY\) with the line drawn from \(Z\) perpendicular to \(AB\) is a fixed line through \(O\).
Through a fixed point \((h,k)\) a variable line is drawn cutting the parabola \(y^2=4ax\) in \(P, Q\); and chords \(PP', QQ'\) are drawn perpendicular to \(PQ\). Prove that the locus of the point of intersection of \(PQ\) with \(P'Q'\) is the rectangular hyperbola \[ y(x+2a-h)=2ak. \]
\(PQ\) is a variable chord of a given ellipse; and the circle whose diameter is \(PQ\) cuts the ellipse again in \(P', Q'\). Prove that, if \(PQ\) always touches a given concentric conic, \(P'Q'\) envelopes a conic which is similar to it.