If a (commutative) ring has multiplicative identity 1, the element \(x\) is said to have order \(n\) if \(n\) is the least positive integer for which \(x^n = 1\). Show, by considering the elements \(-1\) and \(1 + u + u^4\), that, if a ring has an element \(u\) of order 5, then it has either an element of order 2, or one of order 3. [Note: It is possible to have \(1 + 1 = 0\) in a ring.]
The sequence \(a_0, a_1, \ldots, a_{n-1}\) is such that, for each \(i\) \((0 \leq i \leq n-1)\), \(a_i\) is the number of \(i\)'s in the sequence. (Thus for \(n = 4\) we might have \(a_0, a_1, a_2, a_3 = 1, 2, 1, 0\).) If \(n \geq 7\), show that the sequence can only be $$n-4, 2, 1, 0, 0, \ldots, 0, 1, 0, 0, 0.$$ [Hint: Show that the sum of all the terms is \(n\), and that there are \(n - a_0 - 1\) non-zero terms other than \(a_0\), which sum to \(n - a_0\).]
The famous Four Colour Theorem (still unproved) asserts that the regions of any geographical map in the plane may be coloured using only four colours in such a way that regions which touch along an edge are distinctly coloured. In the case when there is a path composed of edges which includes every vertex just once, show that it is possible to colour the map in such a way that two colours are used for the portion enclosed by the path, and two for the remainder. [You may suppose that the regions of the map are straight-edged polygons, whose edges and vertices are called the edges and vertices of the map.]
If \(A\), \(B\), \(C\) are numbers such that \(A t^2 + 2Bt + C \geq 0\) for all real \(t\), show that \(B^2 \leq AC\). By considering \((f(x) + g(x))^2\), show that $$\left(\int_a^b f(x)g(x)dx\right)^2 \leq \int_a^b (f(x))^2 dx \int_a^b (g(x))^2 dx$$ for any continuous functions defined on the interval \([a, b]\). Obtain the inequality $$\int_0^{\pi/2} \sin^4 x \, dx \leq \frac{1}{8}\sqrt{\pi}.$$
The one-player game of Topswaps is played as follows. The player holds a pack of \(n\) cards, numbered from 1 to \(n\) in a random order. If the top card is numbered \(k\), he calls \(k\), reverses the order of the top \(k\) cards, and continues. Show that the pack eventually reaches a constant state in which the top card is numbered 1. [Hint: if \(k > 1\), and, from some point onwards, no card numbered higher than \(k\) is called, then \(k\) is called at most once thereafter.]
Let \(n\) be an odd number such that some power of 2 leaves remainder 1 on division by \(n\). Show, by considering the sequence of remainders of \(1, 2, 2^2, \ldots\) on division by \(n\), that there is a number \(m < n\) such that \(2^k - 1\) is divisible by \(n\) if and only if \(k\) is divisible by \(m\). If \(2^n - 1\) is divisible by \(n\), show that \(2^m - 1\) is divisible by \(m\). Deduce that for no number \(n\) greater than 1 is \(2^n - 1\) divisible by \(n\).
\(ABC\) is a triangle, whose angles are \(3\alpha, 3\beta, 3\gamma\). Points \(P, Q, R\) interior to the triangle are such that \begin{align} \angle PBC &= \beta, \quad \angle PCB = \gamma,\\ \angle PBC &= \beta, \quad \angle CRQ = \frac{1}{3}\pi + \beta,\\ \angle PCQ &= \gamma, \quad \angle BPR = \frac{1}{3}\pi + \gamma. \end{align} The points \(H\), on \(AC\), and \(K\), on \(AB\), are such that \(\angle QHC = \frac{1}{3}\pi + \beta\), \(\angle RKB = \frac{1}{3}\pi + \gamma\). Prove (i) that the triangle \(PQR\) is equilateral, (ii) that \(A, K, R, Q, H\) lie on a circle, and (iii) that \(AR, AQ\) trisect the angle \(A\).
The number of hours of sleep of a group of patients was recorded. On a subsequent night the patients were each given a sleeping pill and the number of hours of sleep was again recorded. The results were as follows:
If \(x_1, x_2, \ldots, x_n\) is a random sample from the uniform distribution with density function \(f(x) = 1/\theta\), \(0 < x < \theta\), where \(\theta\) is an unknown parameter:
Solution:
Ten different numbers are chosen at random from the integers 1 to 100. If the largest of these is divisible by 32 find an expression for the probability that the smallest is divisible by 24.
Solution: The largest can be, 32, 64, 96. \begin{array}{c|c|c} \text{largest} & \text{smallest} & \text{number in between} & \text{choices} \\ \hline 32 & 24 & 7 & \binom{7}{8} = 0 \\ 64 & 24 & 39 & \binom{39}{8} \\ 64 & 48 & 15 & \binom{15}{8} \\ 96 & 24 & 71 & \binom{71}{8} \\ 96 & 48 & 47 & \binom{47}{8} \\ 96 & 72 & 23 & \binom{23}{8} \\ \end{array} Therefore the probability that both the largest number is divisible by 32 and the smallest by 24 can be given by: The probability the largest is divisible by 32 is \begin{align*} && \mathbb{P}(\text{largest divisible by }32) &= \frac{\binom{31}{9} + \binom{64}{9} + \binom{96}{9}}{\binom{100}{100}} \\ \Rightarrow && \mathbb{P}(\text{smallest divisible by }24|\text{largest divisible by }32) &= \frac{\binom{39}{8}+\binom{15}{8}+\binom{71}{8}+\binom{47}{8}+\binom{23}{8}}{\binom{31}{9} + \binom{63}{9} + \binom{95}{9}} \end{align*}