Problems

1. [(i)] Prove that \(n^5 - n\) is divisible by 30 for every integer \(n\).
2. [(ii)] Suppose that \(m_1\) is a positive integer divisible by 11. Prove that the integer \(m_2\) obtained by reversing the digits of \(m_1\) is also divisible by 11.

\(C\) is a circle of radius \(r\). Determine the length \(l\) of the side of a regular \(n\)-sided polygon inscribed in \(C\). Suppose that \(P_1\) and \(P_2\) are two \(n\)-sided polygons inscribed in \(C\), and that the lengths of the sides of \(P_1\) are the same as the lengths of the sides of \(P_2\), perhaps in a different order. Deduce that \(P_1\) and \(P_2\) have the same area. Show that, if \(P_1\) is not regular, then a polygon \(P_3\) with \(n\) sides can be inscribed in \(C\) in such a way that \(P_3\) has greater area than \(P_1\), and \(P_3\) has more edges of length \(l\) than \(P_1\) has. Hence prove that of all \(n\)-sided polygons which can be inscribed in \(C\), a regular polygon has the largest area.

Let \(m\) and \(n\) be integers with \(0 \leq m \leq n\). The function \(f_{n,m}(x)\), defined for \(|x| \neq 1\), is given by \[f_{n,m}(x) = \begin{cases} \frac{(x^n-1)(x^{n-1}-1)\ldots(x^{n-m+1}-1)}{(x^m-1)(x^{m-1}-1)\ldots(x-1)} & \text{if } m > 0, \\ 1 & \text{if } m = 0\end{cases}\] Prove that for \(0 < m < n\), \[f_{n,m}(x) = f_{n-1, m-1}(x) + x^m f_{n-1, m}(x).\] Show that \(f_{n,m}(x)\) can be expressed as a polynomial of degree \(m(n-m)\) for \(|x| \neq 1\), and that the value of this polynomial when \(x = 1\) is equal to the binomial coefficient \(\displaystyle \binom{n}{m}\).

Which of the following assertions hold for each positive integer \(n\)? Justify your answer with proofs or counter-examples as appropriate.

1. [(i)] If \(G\) is a group, the equation \(x^n = e\) has at least one solution in \(G\). [\(e\) denotes the identity element of \(G\).]
2. [(ii)] If \(G\) is a group, the equation \(x^n = e\) has at most \(n\) solutions in \(G\).
3. [(iii)] There exists a group with exactly \(n\) elements.
4. [(iv)] There exist infinitely many groups with exactly \(n\) elements, no two of which are isomorphic.

If \[y = \sin^{-1}x\] show that \[(1-x^2)y'' = xy',\] and hence using Leibniz' Theorem evaluate \(y^{(n)}(0)\). Write down the MacLaurin series for \(\sin^{-1}x\). By considering the series expansions of the two functions term by term, show that \[\sin^{-1}x < \frac{x}{1-x^2} \quad \text{for } 0 < x < 1.\]

On a tropical island, there are only two species of animal. Both species feed on the abundant supplies of vegetation and species \(A\) also feeds on species \(B\), which reproduce at a higher rate than species \(A\). The numbers of individuals, \(N_A\) and \(N_B\), can be regarded as continuous functions of time, satisfying the differential equations \[\frac{dN_A}{dt} = 2N_A + N_B,\] \[\frac{dN_B}{dt} = 4N_B - 2N_A.\] At \(t = 0\), there are \(N_0\) animals in species \(A\) and \(4N_0\) in species \(B\). By obtaining a second order equation, or otherwise, find the numbers in each species for \(0 \leq t < \alpha\), where \(\tan \alpha = -2\) and \(\frac{\pi}{2} < \alpha < \pi\). What do you expect to happen to the species for \(t > \alpha\) ?

Every packet of Munchmix cereal contains a degree certificate for one of the \(N\) degrees of the University of Camford. In true egalitarian spirit, all degrees are equally likely, and the contents of different packets are independent.

1. [(i)] If \(T\) packets of Munchmix are bought, show that the expected number of different degrees thereby acquired is \[N\left[1 - \frac{(N-1)^T}{N^T}\right].\] [You may find it helpful to consider the random variables \(X_j\) \((1 \leq j \leq N)\) defined by \(X_j = 1\) if the \(j\)th degree is acquired, and \(X_j = 0\) otherwise.]
2. [(ii)] By considering the number of packets needed to acquire a new degree when you already have \(r\) degrees, or otherwise, show that the expected number of packets needed to gain a full set of \(N\) degrees is \[N(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{N}).\]

The King of Smorgasbrod proposes to raise lots of money by fining those who sell underweight kippers. The weight of a kipper is normally distributed with mean 200 grams and standard deviation 10 grams. Kippers are packed in cartons of 625 and vast quantities of them are consumed. The Efficient Extortion Committee has produced three possible schemes for determining the fines.

1. Weigh the entire carton, and fine the vendor 1500 crowns if the average weight of a kipper is less than 199 grams.
2. Weigh 25 randomly selected kippers and fine the vendor 100 crowns if the average weight of a kipper is less than 198 grams.
3. Remove kippers one at a time and at random from the carton until an over-weight kipper has been found, and fine the vendor \(3n(n-1)\) crowns, where \(n\) is the number of kippers removed.

Which of the EEC's schemes should the avaricious king select?

Two identical snowploughs plough the same stretch of road in the same direction. The first starts at \(t = 0\) when the depth of snow is \(d\) metres, and the second starts from the same point \(\tau\) seconds later. Snow falls at a constant rate of \(k\) metres/second. It may be assumed that each snowplough moves at a speed equal to \(b/z\) metres/second, where \(z\) is the depth of snow it is ploughing, and that it clears all the snow. Show that:

1. [(i)] the time taken for the first snowplough to travel \(x\) metres is \[(e^{kx/b}-1)d/k \text{ seconds};\]
2. [(ii)] at time \(t > \tau\), the second snowplough has moved \(y\) metres, where \(t\) satisfies \[b \frac{dt}{dy} = kt - d (e^{ky/b} - 1);\]
3. [(iii)] the snowploughs collide when they have moved a distance \(b\tau/d\) metres.

The body of a skater may be represented by a uniform cylinder of mass \(M\) and radius \(a\), with two uniform thin rods of mass \(m\) and length \(2b\), representing his arms, hinged on the circumference of the cylinder at opposite ends of a diameter. Starting from first principles, find the moment of inertia of his body about the axis of the cylinder when his arms are out-stretched and when they lie by his sides. The skater stands upright and spins with angular velocity \(\omega_1\) about a vertical axis with his arms out-stretched. Find his angular velocity \(\omega_2\) when he lowers his arms to his sides and show that the work he needs to do in this process is \[2mb \left[ \frac{\omega_1^2(a + \frac{2b}{3})(Ma^2 + 4m[\frac{1}{3}b^2 + (b+a)^2]))}{a^2(M+4m)} - g \right]\]