\(ABCD\) is a uniform plane quadrilateral lamina, whose diagonals intersect in \(E\). If the point \(H\) divides \(AC\) in the ratio \(AC+EC : AC+EA\), and \(K\) divides \(BD\) in the ratio \(BD+ED : BD+EB\), shew that the centre of gravity of the lamina bisects \(HK\).
Shew that the feet of the perpendiculars drawn from a point on the circumscribing circle to the three sides of a triangle lie on a straight line. Shew that the four circles which are circumscribed to the four triangles that can be formed by taking three out of four straight lines have a common point.
Explain how to find the highest common factor of two positive integers \(a\) and \(b\). Shew that if \(a\) and \(b\) have no common factor, the indeterminate equation \[ ax - by = 1 \] has an infinity of integral solutions; and that the general solution is \((x_0+mb, y_0+ma)\), where \((x_0, y_0)\) is a particular solution and \(m\) an arbitrary integer; and that there is one and only one solution \((x,y)\) such that \(ay+bx\) is numerically not greater than \(\frac{1}{2}(a^2+b^2)\).
The side of a hill is an inclined plane with slope of 1 in 30. A level railway running along the surface of the hill curves into it by a cutting which is 20 yards deep at the point at which the entrance to a tunnel is reached. The radius of the curve is 1000 yards. The cutting is 10 yards wide at the bottom, and the sides slope at an angle of 45\(^\circ\). Estimate the cubic contents of the earth excavated in making the cutting.
Prove that if \(a, b\), and \(c\) are positive integers the chance that \(a^2 + b^2 + c^2\) is divisible by 7 is one-seventh.
Two equal circular cylinders rest in parallel positions on a horizontal plane. An isosceles triangular prism, whose vertical angle is \(\alpha\), rests between them in a symmetrical position, its base being horizontal. If all the surfaces are equally rough, shew that equilibrium will be preserved if the coefficient of friction exceeds \(\tan \frac{1}{4}(\pi-\alpha)\).
Find the ratio of the volume of a regular tetrahedron to the volume of the regular tetrahedron formed by joining the central points of the faces of the former.
Prove the formula \[ f(x+h) - f(x) = hf'(x+\theta h), \] where \(0 < \theta < 1\). Deduce that a function of \(x\) whose differential coefficient is positive increases steadily with \(x\). Prove that the functions \[ \log(1+x) - \frac{x}{1+x}, \quad (2+x)\log(1+x) - 2x \] are positive for all positive values of \(x\); and shew how this sequence of functions may be continued further.
Prove that, if a series of polygons with a given number of sides are drawn with each side in a given direction, and all the angular points but one on specified straight lines, the locus of the last angular point is a straight line. Show how this may be a useful aid in drawing the reciprocal force diagram of a frame. Take as an example any case in which the lines of the force diagram cannot be drawn in succession without some artifice or auxiliary calculation or construction. (A case which may be taken is that of a King post or other roof truss, unsymmetrically loaded, for which the diagram is to be drawn without any preliminary calculation of the supporting forces.)
Prove that \[ \begin{vmatrix} \alpha^4-1 & \alpha^3 & \alpha \\ \beta^4-1 & \beta^3 & \beta \\ \gamma^4-1 & \gamma^3 & \gamma \end{vmatrix} = -(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)\alpha\beta\gamma\left(\beta\gamma+\gamma\alpha+\alpha\beta - \frac{1}{\beta\gamma} - \frac{1}{\gamma\alpha} - \frac{1}{\alpha\beta}\right). \]