Prove that, if \(a,b,c,d\) are four unequal positive quantities, \[ 4\Sigma a^4 > \Sigma a \cdot \Sigma a^3 > (\Sigma a^2)^2 > 16abcd. \]
The feet of three vertical flagstaffs, of heights \(\alpha, \beta, \gamma\), stand at the angular points \(ABC\) of a triangle on a horizontal plane. Prove that the inclination to the horizontal of the plane through the tops of the flagstaffs is \[ \tan^{-1}\left[ \text{cosec } A \left\{ \frac{(\alpha-\beta)^2}{c^2} + \frac{(\alpha-\gamma)^2}{b^2} - \frac{2(\alpha-\beta)(\alpha-\gamma)}{bc}\cos A \right\}^{\frac{1}{2}} \right]. \]
\(\theta, \phi\) are the two unequal values of \(x\) which satisfy the equation \[ \sin^3\alpha \text{ cosec } x + \cos^3\alpha \sec x = 1 \] and which do not differ by a multiple of \(\pi\). Prove that \(\theta+\phi+2\alpha = (2n+1)\pi\), and \(2\cos\frac{1}{2}(\theta-\phi) = \sin 2\alpha\).
Sum to infinity \[ \frac{1}{1^4 \cdot 2^4} + \frac{1}{2^4 \cdot 3^4} + \frac{1}{3^4 \cdot 4^4} + \dots. \]
A conic is inscribed in a triangle \(ABC\) touching \(BC\) at \(P\). The middle points of the sides are \(D, E, F\) and \(O\) is the centre of the conic. Prove that \(AP, DO\) and \(EF\) are concurrent.
Six equal uniform rods, each of weight \(w\), freely jointed at their ends form a regular hexagon \(ABCDEF\). It is suspended from \(A\) and the regular hexagonal form is maintained by two equal light rods jointed to \(B, E\) and \(C, F\). Prove that the thrust in each is \(3w\).
A particle is projected with a given velocity from a point \(P\) to pass through another given point \(Q\) at horizontal and vertical distances \(a\) and \(b\) respectively from \(P\). Prove that if \(H\) is the difference in the greatest heights and \(R\) the difference in the ranges on the horizontal plane through \(P\) for the two possible paths, then \(H/R = \frac{1}{2}a/b\).
Three particles of masses \(m, m', m''\) are attached to the points \(A, B, C\) of an inextensible string. They are laid on a smooth horizontal table, with the portions of the string between the particles taut and the angle \(ABC\) obtuse, and equal to \(\pi-\alpha\). A blow \(P\) is applied to \(C\) parallel to and in the direction \(AB\). Prove that \(m\) begins to move with velocity \(m'P \cos^2\alpha/[m'(m+m'+m'') + mm'' \sin^2\alpha]\).
A smooth wedge of mass \(M\) and angle \(\alpha\) rests on a smooth horizontal plane on which it is free to slide. Another wedge of the same mass and angle is placed on the face of the first, so that the upper face of the second is horizontal. A smooth particle of mass \(m\) is placed on the upper face of the second, and the whole system is allowed to move freely under gravity. Prove that, if the two wedges can only move in directions perpendicular to their edges, \(m\) descends vertically with acceleration \[ \frac{2(m+M)\sin^2\alpha}{M+(2m+M)\sin^2\alpha}g. \]
Prove that if three forces acting upon a rigid body are in equilibrium, their lines of action must all lie in the same plane, and must either pass through a point or be parallel. A uniform rod of length \(2a\) and weight \(W\) hangs in an oblique position, supported by an inextensible string of length \(2l\) (\(l>a\)) whose ends are fastened to the ends of the rod and which passes over a smooth peg, and a weight \(w\) is attached to the rod at a distance \(d\) from its middle point. Prove that the lengths of the string on the two sides of the peg are \[ l\left(1-\frac{d}{ak}\right), \quad l\left(1+\frac{d}{ak}\right), \] where \[ k=1+\frac{W}{w}. \]