Let \(x\) be a positive non-zero integer. \(S^1(x)\) will denote the sum of the digits of \(x\) when written in the scale of 10 (e.g. \(S^1(193) = 1+9+3 = 13\)). For \(i = 1, 2, \ldots\) we define \(S^{i+1}(x) = S^1(S^i(x))\). Show that \(x - S^1(x)\) is divisible by 9 for all \(i\). Denote by \(\mathcal{S}(x)\) the set \(\{S^1(x), \ldots, S^n(x)\}\) where \(n\) is the least integer such that \(S^n(x) \leq 9\) (e.g. \(\mathcal{S}(193) = \{13, 4\}\)). Show that if the smallest element of \(\mathcal{S}(x^2)\) is not a square, then it is 7, and the smallest element in \(\mathcal{S}(x)\) is 4 or 5. Deduce that if \begin{equation*} 0 < x^2 < 1000, \end{equation*} and no element of \(\mathcal{S}(x^2)\) is a square, then \(x = 4\) or 5.
Show that if \(a, b, c, d \in \mathbb{Q}\), the rational numbers, and \(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} = 0\), then \(a = b = c = d = 0\). Let \(V\) be the set of all real numbers of the form \(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\), where \begin{equation*} a, b, c, d \in \mathbb{Q}. \end{equation*} Show that if \(\alpha, \beta \in V\), then \(\alpha\beta \in V\). Show that if \(\alpha_0, \ldots, \alpha_4 \in V\), then there exist \begin{equation*} t_0, \ldots, t_4 \in \mathbb{Q}, \end{equation*} not all zero, such that \(t_0\alpha_0 + \ldots + t_4\alpha_4 = 0\). Deduce that if \(\alpha \in V\), then there is a polynomial, with rational coefficients, of degree at most 4 of which \(\alpha\) is a root. Find a polynomial with rational coefficients of degree 4 of which \(\alpha = \sqrt{2} + \sqrt{3}\) is a root. From this polynomial deduce the existence of an element \(\beta \in V\) such that \(\alpha\beta = 1\) and express \(\beta\) as a sum of powers of \(\alpha\). Is it true that for every non-zero \(\alpha \in V\), there is an element \(\beta \in V\) such that \(\alpha\beta = 1\)? [You may assume that \(\sqrt{2}\), \(\sqrt{3}\) and \(\sqrt{6}\) are not rational and that any set of simultaneous equations with rational coefficients in which there are more unknowns than equations has a non-zero rational solution.]
Given a sequence \(u_0, u_1, u_2, \ldots\) we define a new sequence \(u'_0, u'_1, u'_2, \ldots\) by \begin{equation*} u'_n = \sum_{i=0}^{n} (-1)^i \binom{n}{i} u_{n-i}, \end{equation*} where \(\binom{n}{i}\) denotes the usual binomial coefficient. Applying the same process to the sequence \(u'_0, u'_1, u'_2, \ldots\) we obtain a sequence \(u''_0, u''_1, u''_2, \ldots\). Show that \(u''_n = \sum_{i=0}^{n} c_{n,i} u_i\) for some coefficients \(c_{n,i}\), and find these coefficients. If the sequence \(u_0, u_1, u_2, \ldots\) satisfies the recurrence relation \(u_n = nu_{n-1}\), show that \(u'_0, u'_1, u'_2, \ldots\) satisfies \(u'_n = nu'_{n-1}+(-1)^n u_0\).
Two lines in the plane are perpendicular. An ellipse in the plane moves so that it always touches both lines. Describe the locus of the centre of the ellipse.
\(P, Q, R\) are any points on the sides \(BC, CA, AB\) respectively of the triangle \(ABC\). Prove that the circles \(AQR, BRP, CPQ\) meet at a point. Using a special case of this theorem, or otherwise, describe how to construct a point \(L\) such that the angles \(LBC, LCA, LAB\) are equal.
Prove that a curve in the plane has constant curvature \(c \neq 0\) if and only if it is a circle (or portion thereof).
Let \(f_n(x) = (x^2-1)^n\) and let \(\phi_n(x) = \frac{d^n}{dx^n} \{f_n(x)\}\). Use Leibniz' theorem on the differentiation of products to show that \begin{equation*} \frac{d^r}{dx^r} \{f_n(x)\} \end{equation*} vanishes at \(x = 1\) and \(x = -1\) for all values of \(r < n\). Hence show that \(\int_{-1}^{1} x^k\phi_n(x)dx = 0\) for all \(k < n\), and deduce that if \(m \neq n\) then \begin{equation*} \int_{-1}^{1} \phi_m(x)\phi_n(x)dx = 0. \end{equation*}
Sketch the curve \(y^2 = x^3(1-x^2)\). From your sketch, estimate the number of times the line \(y = ax\) cuts the curve for various values of the constant \(a\). Find the range of values of \(a\) for which the line \(y = ax\) cuts the curve in exactly one point other than the origin.
A man tosses a coin until he tosses a head for the \(n\)th time. The number of tosses he makes is denoted by \(N\). Show that the probability that \(N < 2n-1\) is \(\frac{1}{2}\), and find the expected value of \(N\).
At a certain university, two lecturers (\(A\) and \(B\)) each gave parallel courses in first-year analysis and in second-year algebra, on the same syllabus in each case, and students were free to choose which lecturer they followed. One day a count was made, and attendances were found to be as in the following table: