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.
Prove that, if the circle of curvature at any point \(P\) on the cardioide \(r=a(1+\cos\theta)\), which has its cusp at \(O\) and \(OA\) for its axis, cuts the curve again in \(Q\), then \[ \frac{1}{OQ} - \frac{9}{OP} = -\frac{8}{OA}. \]
Three rods \(BC, CA, AB\), of which the weights are \(p,q,r\), form a triangle \(ABC\) which is suspended by a string attached to the vertex \(A\). Prove that the angle \(\theta\) at which \(BC\) is inclined to the vertical in the position of equilibrium is given by \[ (2p+q+r)\cot\theta = (p+q)\cot C \sim (p+r)\cot B. \]
\((n+1)\) bricks of the same size are piled one above another in a vertical plane so that they rest, each one overlapping the one below by as much as possible. Prove that, if \(2a\) is the length of a brick, the lowest but one overlaps the lowest by a length \(a/n\). Shew also that, if each brick overlaps the one next below by a length \(a/n\), the greatest number of bricks that may be piled up is \((2n-1)\).
A string, of which one end is attached to a mass \(m\) lying on a smooth table, passes over the edge of the table, and after passing over a smooth fixed pulley close to the table and on a level with it has its other end attached to a mass \(m'\); between the table and the pulley the string hangs in a loop and supports a smooth ring of mass \(M\). The string lies in a vertical plane perpendicular to the edge of the table. Find the motion and the tension of the string, and shew that the mass \(m'\) will remain at rest if \(M=4mm'/(2m-m')\).
A smooth sphere of mass \(M\) is suspended from a fixed point by an inelastic string, and another sphere of mass \(m\) impinges directly on it with velocity \(v\) in a direction making an acute angle \(\alpha\) with the vertical. Shew that the loss of energy due to the impact is \[ \frac{1}{2}\frac{mM(1-e^2)v^2}{M+m\sin^2\alpha}, \] where \(e\) is the coefficient of elasticity.