A particle of unit mass falls from a position of unstable equilibrium at the top of a rough sphere of radius \(a\). Show that the equations of motion may be written \(a\omega\frac{d\omega}{d\theta} = g\sin\theta - \mu R\), \(R = g\cos\theta - a\omega^2\), where \(\theta\) is the inclination to the upward vertical of the line from the particle to the centre of the sphere, \(\omega = \dot{\theta}\), and \(R\) is the reaction of the sphere on the particle. Show that if \(\mu = 0\), the particle leaves the sphere at \(\theta = \alpha\), where \(\cos\alpha = \frac{2}{3}\). Now suppose \(\mu\) is positive but small. Solve the first equation approximately by giving \(R(\theta)\) the value it has in the solution for \(\mu = 0\). Hence obtain an improved formula for \(R(\theta)\), and by regarding the required value of \(\theta\) as \(\alpha\) plus a small correction, show that the particle leaves the sphere where \(\theta = \alpha + \mu\left(2-\frac{4\alpha}{3\sin\alpha}\right)\) approximately. [Use the facts that, if \(x\) is small, \(\sin x\) and \(\cos x\) can be approximated by \(x\) and \(1\), respectively.]
Two uniform rough cylinders, each with radius \(a\), lie touching one another on a rough horizontal table. A third identical cylinder lies on these two. The end faces of all three cylinders are coplanar. The coefficient of friction for all pairs of surfaces in contact has the same value, \(\mu\). Find the least value of \(\mu\) for which the cylinders can be in equilibrium.
Prove that, if \(a\) and \(b\) are integers, then \(6a + 5b\) is divisible by 13 if and only if \(3a - 4b\) is. Determine all positive integers \(k\) such that if \(a\) and \(b\) are integers then \(6a + 5b\) is divisible by \(k\) if and only if \(3a - 4b\) is.
Suppose that of the 6 people at a party at least two out of every three know each other, and that all acquaintanceships are mutual. Prove that there are at least 3 people who all know each other. Does this assertion hold if the party consists of only 5 people?
Let \(n\) be an integer and let \(p\) be a prime. Prove that the exponent of \(p\) in the prime factorization of \(n!\) is given by \(\frac{n-s}{p-1}\), where \(s\) is the sum of the digits of \(n\) when written to the base \(p\). How many zeros are at the end of 1000!, when written to the base 60? [You are reminded that every integer \(n\) can be written as \(n = p_1^{a_1} \cdot \ldots \cdot p_k^{a_k}\), where \(p_1 < p_2 < \ldots < p_k\) are primes and \(a_1, \ldots, a_k\) are integers. The exponent of \(p\) in the prime factorization of \(n\) is \(a_i\) if \(p = p_i\) for some \(i\), \(1 \leq i \leq k\); otherwise it is zero.]
\(ABCDE\) is a regular pentagon inscribed in a circle, and \(A'\) is the other extremity of the diameter through \(A\). Prove that \[DC^2 - A'C^2 = \frac{1}{4}A'A^2.\]
A circle touches the ellipse \(x^2/a^2 + y^2/b^2 = 1\) at its intersections with the line \(x = c\). Find its centre and radius. Interpret your results when \(c\) is formally put equal to (i) \(a\), (ii) a value strictly between \(a\) and \(a/e\), (iii) \(a/e\), where \(e\) is the eccentricity of the ellipse.
The Cartesian coordinates of a particle \(P\) at time \(t\) are \((x(t), y(t))\), where \[x = u(1+t), \quad (u > 0),\] \[\frac{dy}{dx} = \frac{y}{x} + \frac{x}{(x^2+y^2)^{\frac{1}{2}}}.\] Initially the particle is on the \(x\) axis; if \(O\) is the origin \((0, 0)\), prove that the slope of \(OP\) increases with time, and show that \(4y = 3x\) after a time \[t = \sqrt{2}\exp(15/32) - 1.\]
By applying the Taylor expansion to the function \(f(x) \equiv (x^2-1)^n\), or otherwise, prove that for all \(x\), and \(h \neq 0\), \[\left[\frac{(x^2-1) + 2hx + h^2}{h}\right]^n = \sum_{r=0}^{2n} \frac{h^{r-n}}{r!}\left(\frac{d}{dx}\right)^r[(x^2-1)^n].\] Write \((x^2-1)/h\) for \(h\) on each side of the above equation, and show that for \(1 \leq m \leq n\), \[\frac{1}{(n-m)!}\left(\frac{d}{dx}\right)^{n-m}[(x^2-1)^n] = \frac{1}{(n+m)!}(x^2-1)^m\left(\frac{d}{dx}\right)^{n+m}[(x^2-1)^n].\] Deduce that \[y(x) = \left(\frac{d}{dx}\right)^n[(x^2-1)^n]\] satisfies the differential equation \[\frac{d}{dx}\left[(x^2-1)\frac{dy}{dx}\right] - n(n+1)y = 0.\]
For each integer \(n \geq 1\), write \(t_n\) for the number of ways of placing \(n\) people into groups (so that \(t_1 = 1\), \(t_2 = 2\), \(t_3 = 5\), etc.). Defining \(t_0 = 1\), show that \[t_{n+1} = \sum_{r=0}^{n} \binom{n}{r}t_{n-r},\] for \(n \geq 0\), and hence show that \(t_n/n!\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(\exp(\exp x - 1)\), for each \(n \geq 1\).