What is the order of the smallest non-commutative group? Prove that there is, up to isomorphism, only one such group of that order. Carefully justify your answers.
(i) \(X, Y\) and \(Z\) are positive numbers. Prove that \[(Y+Z-X)(Z+X-Y)(X+Y-Z) \leq XYZ.\] (ii) \(z_1, z_2...z_n\) are complex numbers and \(\omega = (z_1 + z_2... + z_n)/n\). Prove that, for any complex number \(z\), \[\sum_{i=1}^{n} |z-z_i|^2 \geq n|z-\omega|^2.\]
A Euclidean motion \(M\) of the plane is a transformation of the plane onto itself of the form of a rotation or reflection followed by a translation (so that distance and angle are preserved). Some construction is given whereby from any set of \(n\) distinct points \(A_1, ..., A_n\) a point \(P\) (respectively a line \(l\)) is obtained. The construction is called geometric for \(A_1, ..., A_n\) if (i) \(P\) (respectively \(l\)) does not depend on the order of the points \(A_1, ..., A_n\), and (ii) for any Euclidean motion \(M\) the given construction applied to the points \(M(A_1), ..., M(A_n)\) yields \(M(P)\) (respectively \(M(l)\)). Determine (a) all geometric point constructions, and (b) all geometric line constructions, for a pair of distinct points \(A, B\). By considering symmetric configurations, or otherwise, show that there are no geometric line constructions for \(n\) points if \(n \geq 3\). Exhibit, however, a geometric point construction for \(n\) points.
A set of points in the plane is \(k\)-distant if the distances \(d(A_i, A_j)\) (\(i \neq j\)) take precisely \(k\) distinct values. Thus the 1-distant sets consist of two points or are the vertices of an equilateral triangle. Given an equilateral triangle \(ABC\) determine all possible 2-distant sets containing \(A, B, C\). Deduce that a 2-distant set has at most 5 points.
Let \(f\) be a continuous function on \([0, \infty)\) which is increasing (that is, if \(x \leq y\) then \(f(x) \leq f(y)\)). For \(s \geq 0\) define \(F(s) = \int_0^s f(x)dx\). Show that for \(s \geq 0, t \geq 0, 0 < \lambda \leq 1\), \[F(\lambda s + (1 - \lambda)t) \leq \lambda F(s) + (1 - \lambda)F(t).\] Suppose \(g\) is a continuous increasing function on \([0, \infty)\) such that \(g(f(x)) = x\) and \(f(g(y)) = y\), and hence \(f(0) = g(0) = 0\). For \(t \geq 0\), define \(G(t) = \int_0^t g(y)dy\). Demonstrate by means of a diagram that for \(s \geq 0\) and \(t \geq 0\), \[F(s) + G(t) \geq st.\] Show that, for non-negative \(a\) and \(b\), \[a^{\frac{1}{3}}b^{\frac{2}{3}} \leq \frac{1}{3}a + \frac{2}{3}b \leq \log(e^a + 2e^b) - \log3.\]
(i) By considering \(A(1 + \eta - x^2)^n\) for suitable values of \(A, \eta\) and \(n\), show that, given \(\epsilon > 0\) and \(0 < \beta < \alpha < 1\), we can find a polynomial \(P(x)\) such that \[P(x) \geq 1 \text{ for } |x| \leq \beta,\] \[0 \leq P(x) \leq 1 \text{ for } \beta \leq |x| \leq \alpha,\] \[0 \leq P(x) \leq \epsilon \text{ for } \alpha \leq |x| \leq 1.\] (ii) Show that, given \(\epsilon > 0\), there is a polynomial \(Q(x)\) such that \[|Q(x)| \leq \epsilon \text{ for } -1 \leq x \leq -\epsilon,\] \[-\epsilon \leq Q(x) \leq 1+\epsilon \text{ for } -\epsilon \leq x \leq \epsilon,\] \[|Q(x)-1| \leq \epsilon \text{ for } \epsilon \leq x \leq 1.\] (iii) Show that, given \(\epsilon > 0\) and \(0 < a \leq 1\), there is a polynomial \(R(x)\) such that \[|R(x)| \leq \epsilon \text{ for } a \leq |x| \leq 1,\] \[|R(x)-(1-a^{-1}|x|)| \leq \epsilon \text{ for } |x| \leq a.\]
A mouse \(M\) is running at a constant speed \((U, 0)\) along the line \(y = 0\). At \(t = 0\), the mouse is at position \((a, 0)\), where \(a > 0\), and a cat \(C\) is at \((0, b)\). The cat starts running at constant speed \(V\) in a direction which is always towards the mouse. If \(O\) is the origin and \(\psi\) the acute angle \(OMC\), show that \[\frac{d}{dt}(\cot \psi) = \frac{U}{y},\] where \((x, y)\) is the position of \(C\) at any time \(t\). If \(b \ll a\), show that the path of \(C\) is given approximately, for \(t > 0\), by an equation of form \[x = Ay^{1-\lambda} + B,\] where \(A\) and \(B\) are constants to be found and \(\lambda = U/V\), provided \(\lambda > 1\). Find the approximate equation of the path when \(\lambda = 1\).
Craps is played between a gambler and a banker as follows. On each throw, the gambler throws two dice. If his first throw is 7 or 11 he wins and if it is 2, 3 or 12 he loses. If his first throw is none of these he throws repeatedly until either he again throws the same as his first throw, in which case he wins, or he throws a 7, in which case he loses. What is the probability that he wins?
The author of a scientific paper claims to have done the following experiment 3600 times. The subject wrote down a number, then a die was thrown and the number shown on the die compared with the prediction. He claims that the results were as shown.
Let \(\mathbf{r}\) denote the position vector of a particle relative to a point \(O\) on the earth's surface. In a certain approximation the effects of the earth's rotation are described by the equation \[\ddot{\mathbf{r}}+2\mathbf{\omega} \wedge \dot{\mathbf{r}} = \mathbf{g},\] where \(\mathbf{g}\) is the acceleration due to gravity, pointing vertically downwards, and \(\mathbf{\omega}\) is another constant vector (pointing in the direction of the earth's axis of rotation and equal in magnitude to its angular velocity). If the particle is projected from \(O\), with velocity \(\mathbf{v}\), at time \(t = 0\), show that \[\dot{\mathbf{r}}+2\mathbf{\omega} \wedge \mathbf{r} = \mathbf{g}t+\mathbf{v}.\] Deduce that \[\mathbf{r} = \frac{1}{2}\mathbf{g}t^2 + \mathbf{v}t - \frac{1}{3}\mathbf{\omega} \wedge \mathbf{g}t^3 - \mathbf{\omega} \wedge \mathbf{v}t^2\] if terms of order \(\omega^2\) may be neglected. The flight ends when the particle hits the horizontal plane through \(O\). Continuing to neglect terms of order \(\omega^2\) show that the time of flight is \[2g^{-2}\mathbf{g}.\mathbf{v}(1 + 2g^{-2}\mathbf{\omega}.(\mathbf{v} \wedge \mathbf{g})).\]