Prove that a sphere can be drawn to cut orthogonally three circles in space, each of which intersects each of the other two in two points.
If \({}_nC_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\) by the binomial theorem where \(n\) is a positive integer, prove that \[ \sum_{s=0}^{s=n-r} {}_rC_s \cdot {}_{n-r}C_s = {}_{2n-r}C_n. \]
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 \(m'P \cos^2\alpha/[m'(m+m'+m'') + mm'' \sin^2\alpha]\).