Prove that, if \(a, b, c\) are the sides of a triangle of area \(\Delta\), \[ \begin{vmatrix} (b-c)^2 & b^2 & c^2 & 1 \\ a^2 & (c-a)^2 & c^2 & 1 \\ a^2 & b^2 & (a-b)^2 & 1 \\ 1 & 1 & 1 & 0 \end{vmatrix} = -16\Delta^2. \]
The sides of a triangle \(ABC\) are cut by a conic in points \(A_1\) and \(A_2\), \(B_1\) and \(B_2\), \(C_1\) and \(C_2\) respectively; prove that \[ AB_1.AB_2.BC_1.BC_2.CA_1.CA_2 = AC_1.AC_2.BA_1.BA_2.CB_1.CB_2. \]
A family of conics touching the sides of a given triangle have their axes parallel to a given straight line; prove that their foci lie on a cubic curve. Shew that this curve passes through the angular points of the triangle and through the centres of the inscribed and escribed circles of the triangle.
Lines drawn from the vertices \(A, B, C\) of a triangle \(ABC\) to a point \(O\) within the triangle are produced to meet the opposite sides in \(D, E, F\) respectively. Shew that the least value of \(\lambda^2\frac{OD}{OA} + \mu^2\frac{OE}{OB} + \nu^2\frac{OF}{OC}\) is \(\mu\nu+\nu\lambda+\lambda\mu - \frac{1}{2}(\lambda^2+\mu^2+\nu^2)\).
Prove that, if \(a\tan\phi = b\tan\theta\), \[ a\left\{\sin\theta\cos\phi + \int_0^\phi \sin\phi\operatorname{cosec}\theta d\phi\right\} = b\left\{\sin\phi\cos\theta + \int_0^\theta \sin\theta\operatorname{cosec}\phi d\theta\right\}. \]
A tetrahedron is formed of six light rods jointed together, and the middle points of a pair of opposite rods are connected by a tight string. Shew that the stresses in the remaining rods are proportional to their lengths, and that if the rods are all of equal length, these stresses are \(\frac{\sqrt{2}}{4} \times \text{tension of the string}\).
The bottom of a rectangular box without a lid is a square of side \(2a\), and its height is \(2b\). It is half filled with water and rests on a perfectly rough plane inclined at an angle \(\theta\) to the horizontal, with two of the sides of the base horizontal. Shew that if \(b>a\tan\theta\) and the weight and thickness of the material of the box are neglected, the greatest inclination of the plane consistent with equilibrium is given by \[ a^2\tan^3\theta + (2a^2+3b^2)\tan\theta - 6ab=0. \]
A shell is fired vertically upwards with initial velocity \(u\); when it comes instantaneously to rest it bursts and fragments are projected in all directions with velocity \(v\); shew that they will fall to the ground within a circle of radius \[ \frac{v\sqrt{u^2+v^2}}{g}. \]
If \(\alpha\) stands for the fifth root of 2, and \(x = \alpha+\alpha^4\), prove that \[ x^5=10x^2+10x+6. \]
Solve the equations: