Prove by induction or otherwise that if \(r\) is a positive integer then the sum of the infinite series \[ \frac{1^r}{1!} + \frac{2^r}{2!} + \frac{3^r}{3!} + \dots \] is an integral multiple of \(e\).
If three angles be such that the sum of their cosines is zero and the sum of their sines is zero, prove that any two of them differ by \(2r\pi \pm \frac{2\pi}{3}\), where \(r\) is an integer, and that the sum of the squares of their cosines is equal to the sum of the squares of their sines.
Two isosceles triangles have the same inscribed circle and the same circumscribed circle: prove that their vertical angles \(V_1\) and \(V_2\) satisfy the equation \[ \sin\frac{1}{2}V_1 + \sin\frac{1}{2}V_2 = 1. \]
From the focus \(S\) of an ellipse a perpendicular \(SY\) is drawn to a tangent and produced to \(Z\) so that \(SZ=2SY\); shew that the square of the tangent from \(Z\) to the director circle is equal to \(2SY^2\).
From a point \(T\) a perpendicular \(TL\) is drawn on its polar with respect to a parabola; prove that when \(T\) moves on a line parallel to the axis the locus of \(L\) is a line through the focus.
Given two vertices of a triangle and its area, shew that the locus of its orthocentre is two parabolas.
On the sides of a triangle \(ABC\) equilateral triangles \(BPC, CQA,\) and \(ARB\) are described externally; shew that the lines \(AP, BQ, CR\) are of equal length.
Two uniform rods, each of weight \(W\) and length \(a\), are freely jointed at \(A\), and each passes over a smooth peg at the same level. From \(A\) a weight \(W'\) is suspended. Shew that in the position of equilibrium the inclination \(\theta\) of the rods to the horizon is given by \[ \cos^3\theta = c(2W+W')/2Wa, \] \(c\) being the distance between the pegs.
Two equal particles are connected by a light string which is slung over the top of a smooth vertical circle: verify that the position of equilibrium is unstable. (It may be supposed that both particles rest on the circle, so that the length of the string is less than one-half of the circumference of the circle.)
Two particles are projected at the same instant from the same point under gravity; shew that the line joining them remains constant in direction and that its length increases uniformly with the time.