A spherically symmetric cloud of stars contracts under the action of its own gravitational field according to the equation \begin{align} \frac{d^2r}{dt^2} = -\frac{kM}{r^2}, \end{align} where \(r\) is the radius of a sphere concentric with the centre of the cloud which contains a total mass \(M\) of stars, and \(k\) is a constant. If the cloud is initially at rest with uniform density, prove that the density remains uniform for all subsequent times.
Show that the simultaneous equations \begin{align} x + y + z &= 3, \\ x^2 + y^2 + z^2 &= 3, \\ x^3 + y^3 + z^3 &= c, \end{align} have no solution in real numbers \(x\), \(y\), \(z\) unless \(c = 3\).
In the gambling game of toss-penny, after each toss either \(A\) gives \(B\) one penny, these two outcomes being equally likely, and the game continues until either \(A\) or \(B\) is exhausted of pennies. If \(P(m, n)\) is the probability that \(A\) will win, given that \(A\) has \(m\) pennies and \(B\) \(n\) pennies at the start, show (by considering the position after the first toss) that $$P(m, n) = \frac{1}{2}P(m-1, n+1) + \frac{1}{2}P(m+1, n-1).$$ Hence find the value of \(P(m, n)\).
The complex numbers \(a\), \(b\), \(c\) are represented in the Argand diagram by the points \(A\), \(B\), \(C\). Show that \(ABC\) is an equilateral triangle if and only if \(a\), \(b\), \(c\) are not all equal and $$a^2 + b^2 + c^2 - bc - ca - ab = 0.$$ Three equilateral triangles \(XYZ\), \(YZX\), \(ZXY\) are drawn outwards from the sides of a triangle \(XYZ\). Show that the triangles \(XYZ\), \(X'Y'Z'\) have a common centre of gravity.
Show that if \(k\) is an integer greater than or equal to \(0\) then $$\sum_{n=0}^{\infty} \frac{n^k}{n!} = n_k e,$$ where \(n_k\) is an integer and \(e = \sum_{n=0}^{\infty} \frac{1}{n!}\). Show also that \(n_0 = 1\), and that $$n_k = \sum_{r=0}^{k-1} {}^k C_r n_r,$$ where \({}^k C_r\) is the coefficient of \(x^r\) in \((1+x)^k\) for \(k = 1, 2, \ldots\). (In this question both \(0^0\) and \(0!\) are to be taken to be \(1\).)
Show that if the distance between the points \(A\) and \(B\) is greater than \(d\), then the two spheres of radius \(\frac{1}{2}d\) with centres at \(A\) and \(B\) have no point in common. Hence show that if \(P_1, \ldots, P_n\) are \(n\) points contained in a cube of side \(X\), such that the distance between each pair of these points is at least \(1\), then $$n < \frac{6(X+1)^3}{\pi}.$$
All three angles of the triangle \(ABC\) are less than \(120^\circ\). Show that the minimal value of \(PA + PB + PC\) (as \(P\) varies in the plane of the triangle) is attained at a point \(P'\) having the property that $$\overrightarrow{B P'} C = \overrightarrow{C P'} A = \overrightarrow{A P'} B = 120^\circ;$$ and prove that there is just one point in the plane with this property.
The circle \(c\) has radius \(a\) and centre \(A\), and the point \(B\) is distance \(b\) from \(A\). \(P\) is the foot of the perpendicular from \(B\) to a tangent of \(C\). Show that the locus of \(P\) as the tangent varies is a curve touching \(C\) at two points, and find the area enclosed by this curve.
(i) \(a\), \(b\), \(c\), \(d\) are positive numbers, \(c\) and \(d\) not being equal. Find the limit of $$\frac{(a^x - b^x)}{(c^x - d^x)}$$ as \(x\) tends to \(0\). (ii) If \(a\), \(b\), \(c\) are positive numbers, show that \(a^{\log_b c} = c^{\log_b a}\).
Let \(f_n(x)\) denote, for each integer \(n\) greater than or equal to \(0\), the function $$e^{-x} - 1 + x - \frac{x^2}{2!} + \ldots + (-1)^{n+1} \frac{x^n}{n!}.$$ By integrating \(f_n(t)\) over a suitable range and using induction on \(n\), show that if \(x\) is positive then \(f_n(x)\) is positive or negative according as \(n\) is odd or even.