Show that \(n\) coplanar lines in 'general position' (i.e. no two lines parallel, no three lines concurrent) divide the plane into \(\frac{1}{2}(n^2+n+2)\) regions. Show, also, that the regions may be coloured, each either red or blue, in such a way that no two regions whose boundaries have a line segment in common have the same colour. Find the number of regions into which \(n\) planes, in general position, divide three-dimensional space.
Show that the operation of matrix multiplication on the set \(M_2\) of real \(2 \times 2\) matrices is associative but not commutative. Let \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\), \(O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\) and let \(A\), \(B\) be members of \(M_2\). Prove that
Let \(N = \{1, 2, 3, \ldots\}\) and let \(F\) be the set of all real-valued functions \(f\) on \(N\) such that \(f(1) = 1\). For members \(f\), \(g\) of \(F\), define the function \(f * g\) on \(N\) by \[(f * g)(n) = \sum_{d|n} f(d) g(n/d)\] for all \(n\) of \(N\), where the summation is over all divisors \(d\) of \(n\) (including 1 and \(n\)). Prove that \((F, *)\) is an abelian (i.e. commutative) group, with identity element \(e\) given by \[e(n) = \begin{cases} 1 & (n = 1),\\ 0 & (n > 1). \end{cases}\] [N.B. You may assume, without proof, that each element of \(F\) has an inverse.] Let \(s\), \(\mu\) be the elements of \(F\) given by \[s(n) = 1 \quad \text{(all \(n\) of \(N\))},\] \[\mu(n) = \begin{cases} 1 & (n = 1),\\ (-1)^k & \text{(if \(n = p_1 \ldots p_k\), a product of \(k\) distinct primes)},\\ 0 & \text{(otherwise)}. \end{cases}\] Prove that \(s*\mu = e\) and deduce that, if \(g\) belongs to \(F\) and if \(f(n) = \sum_{d|n} g(d)\) (all \(n\) of \(N\)), then \(g(n) = \sum_{d|n} \mu(d)f(n/d)\) (all \(n\) of \(N\)), where the summation is taken over all divisors \(d\) of \(n\) in both cases.
For any real number \(x\), \([x]\) denotes the greatest integer not exceeding \(x\). Evaluate, for positive integers \(n\), \(r\), the number of multiples of \(r\) in the set \(\{1, 2, \ldots, n\}\). Positive integers \(a_1, \ldots, a_k\) satisfy \(a_i \leq N\) (\(i = 1, \ldots, k\)) and L.C.M.\((a_i, a_j) > N\) whenever \(i \neq j\). (L.C.M.\((a_i, a_j)\) is the least common multiple of \(a_i\) and \(a_j\).) By showing that \[\sum_{i=1}^{k} [N/a_i] \leq N,\] prove that \[\sum_{i=1}^{k} \frac{1}{a_i} \leq 2.\] Deduce that, for any real \(x > 2\), \[\sum_{\sqrt{x} < p \leq x} \frac{1}{p} \leq 2,\] where the summation is over prime numbers \(p\) in the given range, and deduce that if \(r\) is a positive integer and \(N = 2^{2^r}\), then \[\sum_{1 < p \leq N} \frac{1}{p} \leq 2r.\]
The churches of St Aldate, St Buryan and St Cett stand on the flat East Anglian plane, and their tall steeples (all of different heights) can be seen for miles around. As part of the university rag, three parties of students each choose a different patron saint and set out to push a bed along the path from which the two steeples belonging to the churches of the other two saints have the same apparent height. Show that the students will go round in circles and that the centres of the three circles will be collinear. Is it possible for the heights to be such that the three circles meet at a point? Is it possible for the heights to be such that none of the circles intersect? If two of the steeples were of the same height would you advise the rag committee to run this stunt? In each case give your reasons.
Let \(P\) and \(Q\) be points on the same side of a line \(l\). Let \(Q'\) be the reflection of \(Q\) in \(l\). Show that if \(O\) is a point on the line \(l\) then \(PO + OQ = PO + OQ'\). Deduce that \(PO + OQ\) is a minimum when \(O\) is so chosen that \(OP\) makes the same angle with \(l\) as does \(OQ\). Suppose now that \(ABC\) is an equilateral triangle and \(P\), \(Q\), \(R\) and \(T\) are points on \(BC\), \(CA\), \(AB\) and \(AB\) respectively. (Note that this means \(P\) lies between \(B\) and \(C\) inclusive; similar restrictions apply to \(Q\), \(R\) and \(T\).)
Each day a factory produces \(x_1\) tons of product \(A\), \(x_2\) tons of product \(B\), \(x_3\) tons of product \(C\) and \(x_4\) tons of waste \(D\). The nature of the process is such that \[x_1 + x_2 + x_3 = \lambda_1,\] \[x_4 - \log_e x_2 = \lambda_2,\] and \(x_1, x_2, x_3, x_4 \geq 0\). For what range of values of the constants \(\lambda_1\) and \(\lambda_2\) is it possible to find \(x_1, x_2, x_3\) and \(x_4\) satisfying the conditions above? The daily profit of the factory is \(2 \tan^{-1} x_1 + x_2\) (in thousands of pounds). Show how to choose \(x_1, x_2, x_3\) and \(x_4\) to maximise this profit.
Suppose \(x\) is a continuous function with continuous derivative satisfying \[\dot{x}(t) + x(t) = 0 \quad \text{for } |x(t)| \leq 1,\] \[\dot{x}(t) + 4x(t) = 0 \quad \text{for } 1 < |x(t)|,\] \[x(0) = 0, \quad \dot{x}(0) = v.\] Giving an account of your reasoning but without necessarily resorting to detailed calculation, show that \(x\) is periodic for all choices of \(v\). Give a rough sketch of how the period varies with \(v\), indicating the main features of your sketch and explaining why they occur (again exact numerical detail is not required). How would your various conclusions be altered (if at all) for the general initial conditions \(x(0) = u\), \(\dot{x}(0) = v\)?
Show that if \(m\) and \(n\) are integers with \(m \geq n \geq 1\), then \(1/m! \leq n^{n-m}/n!\). Deduce that, if \(n \geq 3\), then \[0 < n! \sum_{r=n+1}^{\infty} \frac{1}{r!} < 1.\] Shoe that if \[e = \sum_{r=0}^{\infty} \frac{1}{r!}\] then \(n!e\) is not an integer, and so \(e\) is irrational (ie \(e\) is not of the form \(p/q\) for any integers \(p\) and \(q\)). Show that if \[\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\] then (i) \(\cosh 1\) is irrational, (ii) \(\cosh \sqrt{2}\) is irrational.
A businessman puts money into one deal each year. At the end of each year, the deal has either fallen through, in which case he loses his entire outlay, or has been successful, in which case he recovers twice his outlay. The outcomes of different deals are independent, each one having a chance \(p > \frac{1}{2}\) of success, and the businessman can choose how much money he wishes to risk each year, within the limits of his current fortune. A possible strategy for the businessman is to put his entire current fortune into each deal. If his initial fortune is \(X_0\), show that his fortune \(X_n\) after \(n\) years' dealing has expectation \((2p)^nX_0\), but that he is certain to lose all his money eventually. A more conservative strategy is for him to invest a fixed proportion \(s\) of his current fortune in each deal, where \(0 < s < 1\). Show that the expected value of \(X_n\) is now only \([1+s(2p-1)]^nX_0\), but that, by choosing \(s\) suitably, he can guarantee an eventual compound growth rate of \(\alpha = 2p^p(1-p)^{1-p}\), in the sense that \(\frac{1}{n} \log_e (X_n/X_0)\) tends to \(\log_e \alpha\) as \(n \to \infty\). [Hint. Show that, with an appropriate definition of \(Y_n\), \(X_{n+1}\) can be expressed in the form \(X_{n+1} = X_n (1+s)^{Y_n}(1-s)^{1-Y_n}\). You may assume the 'law of large numbers' in the form: if \(Y_1, Y_2, \ldots\) are independent and identically distributed random variables with mean \(\beta\), then \(\frac{1}{n} \sum_{j=1}^n Y_j \to \beta\) as \(n \to \infty\).]