A triangle moves so that each of two sides passes through a fixed point. Prove that its base touches a fixed circle.
\(TPT'\) is the tangent to a hyperbola, whose centre is \(C\), meeting the asymptotes in \(T\) and \(T'\). \(PQ\) parallel to \(CT\) meets the directrix in \(Q\). Prove that \(T'Q\) is parallel to \(ST\) where \(S\) is the focus inside the branch of the curve on which \(P\) lies.
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 \(3w\).