Problems

1. Let \(a\), \(b\) and \(c\) be three non-zero complex numbers with the properties \(a + b + c = 0\) and \(a^2 + b^2 + c^2 = 0\). Show that \(a\), \(b\) and \(c\) cannot all be real. Show further that \(a\), \(b\) and \(c\) all have the same modulus.
2. Show that it is not possible to find three non-zero complex numbers \(a\), \(b\) and \(c\) with the properties \(a + b + c = 0\) and \(a^3 + b^3 + c^3 = 0\).
3. Show that if any four non-zero complex numbers \(a\), \(b\), \(c\) and \(d\) have the properties \(a + b + c + d = 0\) and \(a^3 + b^3 + c^3 + d^3 = 0\), then at least two of them must have the same modulus.
4. Show, by taking \(c = 1\), \(d = -2\) and \(e = 3\) that it is possible to find five real numbers \(a\), \(b\), \(c\), \(d\) and \(e\) with distinct magnitudes and with the properties \(a + b + c + d + e = 0\) and \(a^3 + b^3 + c^3 + d^3 + e^3 = 0\).

In this question, you need not consider issues of convergence. For positive integer \(n\) let \[\mathrm{f}(n) = \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \ldots\] and \[\mathrm{g}(n) = \frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} - \ldots\,.\]

1. Show, by considering a geometric series, that \(0 < \mathrm{f}(n) < \dfrac{1}{n}\).
2. Show, by comparing consecutive terms, that \(0 < \mathrm{g}(n) < \dfrac{1}{n+1}\).
3. Show, for positive integer \(n\), that \((2n)!\,\mathrm{e} - \mathrm{f}(2n)\) and \(\dfrac{(2n)!}{\mathrm{e}} + \mathrm{g}(2n)\) are both integers.
4. Show that if \(q\,\mathrm{e} = \dfrac{p}{\mathrm{e}}\) for some positive integers \(p\) and \(q\), then \(q\,\mathrm{f}(2n) + p\,\mathrm{g}(2n)\) is an integer for all positive integers \(n\).
5. Hence show that the number \(\mathrm{e}^2\) is irrational.

1. Show that if \[\frac{1}{x} + \frac{2}{y} = \frac{2}{7}\,,\] then \((2x - 7)(y - 7) = 49\). By considering the factors of \(49\), find all the pairs of positive integers \(x\) and \(y\) such that \[\frac{1}{x} + \frac{2}{y} = \frac{2}{7}\,.\]
2. Let \(p\) and \(q\) be prime numbers such that \[p^2 + pq + q^2 = n^2\] where \(n\) is a positive integer. Show that \[(p + q + n)(p + q - n) = pq\] and hence explain why \(p + q = n + 1\). Hence find the possible values of \(p\) and \(q\).
3. Let \(p\) and \(q\) be positive and \[p^3 + q^3 + 3pq^2 = n^3\,.\] Show that \(p + q - n < p\) and \(p + q - n < q\). Show that there are no prime numbers \(p\) and \(q\) such that \(p^3 + q^3 + 3pq^2\) is the cube of an integer.

1. By integrating one of the two terms in the integrand by parts, or otherwise, find \[\int \left(2\sqrt{1+x^3} + \frac{3x^3}{\sqrt{1+x^3}}\right)\,\mathrm{d}x\,.\]
2. Find \[\int (x^2+2)\frac{\sin x}{x^3}\,\mathrm{d}x\,.\]
3. 1. Sketch the graph with equation \(y = \dfrac{\mathrm{e}^x}{x}\), giving the coordinates of any stationary points.
   2. Find \(a\) if \[\int_a^{2a} \frac{\mathrm{e}^x}{x}\,\mathrm{d}x = \int_a^{2a} \frac{\mathrm{e}^x}{x^2}\,\mathrm{d}x\,.\]
   3. Show that it is not possible to find distinct integers \(m\) and \(n\) such that \[\int_m^n \frac{\mathrm{e}^x}{x}\,\mathrm{d}x = \int_m^n \frac{\mathrm{e}^x}{x^2}\,\mathrm{d}x\,.\]

Let \(\mathrm{h}(z) = nz^6 + z^5 + z + n\), where \(z\) is a complex number and \(n \geqslant 2\) is an integer.

1. Let \(w\) be a root of the equation \(\mathrm{h}(z) = 0\).
   1. Show that \(|w^5| = \sqrt{\dfrac{\mathrm{f}(w)}{\mathrm{g}(w)}}\), where \[\mathrm{f}(z) = n^2 + 2n\operatorname{Re}(z) + |z|^2 \quad \text{and} \quad \mathrm{g}(z) = n^2|z|^2 + 2n\operatorname{Re}(z) + 1.\]
   2. By considering \(\mathrm{f}(w) - \mathrm{g}(w)\), prove by contradiction that \(|w| \geqslant 1\).
   3. Show that \(|w| = 1\).
2. It is given that the equation \(\mathrm{h}(z) = 0\) has six distinct roots, none of which is purely real.
   1. Show that \(\mathrm{h}(z)\) can be written in the form \[\mathrm{h}(z) = n(z^2 - a_1 z + 1)(z^2 - a_2 z + 1)(z^2 - a_3 z + 1),\] where \(a_1\), \(a_2\) and \(a_3\) are real constants.
   2. Find \(a_1 + a_2 + a_3\) in terms of \(n\).
   3. By considering the coefficient of \(z^3\) in \(\mathrm{h}(z)\), find \(a_1 a_2 a_3\) in terms of \(n\).
   4. How many of the six roots of the equation \(\mathrm{h}(z) = 0\) have a negative real part? Justify your answer.

1. Suppose that there are three non-zero integers \(a\), \(b\) and \(c\) for which \(a^3 + 2b^3 + 4c^3 = 0\). Explain why there must exist an integer \(p\), with \(|p| < |a|\), such that \(4p^3 + b^3 + 2c^3 = 0\), and show further that there must exist integers \(p\), \(q\) and \(r\), with \(|p| < |a|\), \(|q| < |b|\) and \(|r| < |c|\), such that \(p^3 + 2q^3 + 4r^3 = 0\). Deduce that no such integers \(a\), \(b\) and \(c\) can exist.
2. Prove that there are no non-zero integers \(a\), \(b\) and \(c\) for which \(9a^3 + 10b^3 + 6c^3 = 0\).
3. By considering the expression \((3n \pm 1)^2\), prove that, unless an integer is a multiple of three, its square is one more than a multiple of \(3\). Deduce that the sum of the squares of two integers can only be a multiple of three if each of the integers is a multiple of three. Hence prove that there are no non-zero integers \(a\), \(b\) and \(c\) for which \(a^2 + b^2 = 3c^2\).
4. Prove that there are no non-zero integers \(a\), \(b\) and \(c\) for which \(a^2 + b^2 + c^2 = 4abc\).

1. Use De Moivre's theorem to prove that for any positive integer \(k > 1\), \[ \sin(k\theta) = \sin\theta\cos^{k-1}\theta \left( k - \binom{k}{3}(\sec^2\theta - 1) + \binom{k}{5}(\sec^2\theta - 1)^2 - \cdots \right) \] and find a similar expression for \(\cos(k\theta)\).
2. Let \(\theta = \cos^{-1}(\frac{1}{a})\), where \(\theta\) is measured in degrees, and \(a\) is an odd integer greater than \(1\). Suppose that there is a positive integer \(k\) such that \(\sin(k\theta) = 0\) and \(\sin(m\theta) \neq 0\) for all integers \(m\) with \(0 < m < k\). Show that it would be necessary to have \(k\) even and \(\cos(\frac{1}{2}k\theta) = 0\). Deduce that \(\theta\) is irrational.
3. Show that if \(\phi = \cot^{-1}(\frac{1}{b})\), where \(\phi\) is measured in degrees, and \(b\) is an even integer greater than \(1\), then \(\phi\) is irrational.

A sequence \(u_k\), for integer \(k \geqslant 1\), is defined as follows. \[ u_1 = 1 \] \[ u_{2k} = u_k \text{ for } k \geqslant 1 \] \[ u_{2k+1} = u_k + u_{k+1} \text{ for } k \geqslant 1 \]

1. Show that, for every pair of consecutive terms of this sequence, except the first pair, the term with odd subscript is larger than the term with even subscript.
2. Suppose that two consecutive terms in this sequence have a common factor greater than one. Show that there are then two consecutive terms earlier in the sequence which have the same common factor. Deduce that any two consecutive terms in this sequence are co-prime (do not have a common factor greater than one).
3. Prove that it is not possible for two positive integers to appear consecutively in the same order in two different places in the sequence.
4. Suppose that \(a\) and \(b\) are two co-prime positive integers which do not occur consecutively in the sequence with \(b\) following \(a\). If \(a > b\), show that \(a-b\) and \(b\) are two co-prime positive integers which do not occur consecutively in the sequence with \(b\) following \(a-b\), and whose sum is smaller than \(a+b\). Find a similar result for \(a < b\).
5. For each integer \(n \geqslant 1\), define the function \(\mathrm{f}\) from the positive integers to the positive rational numbers by \(\mathrm{f}(n) = \dfrac{u_n}{u_{n+1}}\). Show that the range of \(\mathrm{f}\) is all the positive rational numbers, and that \(\mathrm{f}\) has an inverse.

Consider the following steps in a proof that \(\sqrt{2} + \sqrt{3}\) is irrational.

1. If an integer \(a\) is not divisible by 3, then \(a = 3k \pm 1\), for some integer \(k\). In both cases, \(a^2\) is one more than a multiple of 3.
2. Suppose that \(\sqrt{2} + \sqrt{3}\) is rational, and equal to \(\frac{a}{b}\), where \(a\) and \(b\) are positive integers with no common factor greater than one.
3. Then \(a^4 + b^4 = 10a^2b^2\).
4. So if \(a\) is divisible by 3, then \(b\) is divisible by 3.
5. Hence \(\sqrt{2} + \sqrt{3}\) is irrational.

1. Show clearly that steps 1, 3 and 4 are all valid and that the conclusion 5 follows from the previous steps of the argument.
2. Prove, by means of a similar method but using divisibility by 5 instead of 3, that \(\sqrt{6} + \sqrt{7}\) is irrational. Why can divisibility by 3 not be used in this case?

The sequence \(u_0, u_1, \ldots\) is said to be a constant sequence if \(u_n = u_{n+1}\) for \(n = 0, 1, 2, \ldots\). The sequence is said to be a sequence of period 2 if \(u_n = u_{n+2}\) for \(n = 0, 1, 2, \ldots\) and the sequence is not constant.

1. A sequence of real numbers is defined by \(u_0 = a\) and \(u_{n+1} = f(u_n)\) for \(n = 0, 1, 2, \ldots\), where $$f(x) = p + (x - p)x,$$ and \(p\) is a given real number. Find the values of \(a\) for which the sequence is constant. Show that the sequence has period 2 for some value of \(a\) if and only if \(p > 3\) or \(p < -1\).
2. A sequence of real numbers is defined by \(u_0 = a\) and \(u_{n+1} = f(u_n)\) for \(n = 0, 1, 2, \ldots\), where $$f(x) = q + (x - p)x,$$ and \(p\) and \(q\) are given real numbers. Show that there is no value of \(a\) for which the sequence is constant if and only if \(f(x) > x\) for all \(x\). Deduce that, if there is no value of \(a\) for which the sequence is constant, then there is no value of \(a\) for which the sequence has period 2. Is it true that, if there is no value of \(a\) for which the sequence has period 2, then there is no value of \(a\) for which the sequence is constant?

The \(n\)th degree polynomial P\((x)\) is said to be reflexive if:

1. P\((x)\) is of the form \(x^n - a_1x^{n-1} + a_2x^{n-2} - \cdots + (-1)^na_n\) where \(n \geq 1\);
2. \(a_1, a_2, \ldots, a_n\) are real;
3. the \(n\) (not necessarily distinct) roots of the equation P\((x) = 0\) are \(a_1, a_2, \ldots, a_n\).

1. Find all reflexive polynomials of degree less than or equal to 3.
2. For a reflexive polynomial with \(n > 3\), show that $$2a_2 = -a_2^2 - a_3^2 - \cdots - a_n^2.$$ Deduce that, if all the coefficients of a reflexive polynomial of degree \(n\) are integers and \(a_n \neq 0\), then \(n \leq 3\).
3. Determine all reflexive polynomials with integer coefficients.

1. Write down the most general polynomial of degree 4 that leaves a remainder of 1 when divided by any of \(x-1\,\), \(x-2\,\), \(x-3\,\) or \(x-4\,\).
2. The polynomial \(\P(x)\) has degree \(N\), where \(N\ge1\,\), and satisfies \[ \P(1) = \P(2) = \cdots = \P(N) =1\,. \] Show that \(\P(N+1) \ne 1\,\). Given that \(\P(N+1)= 2\,\), find \(\P(N+r)\) where \(r\) is a positive integer. Find a positive integer \(r\), independent of \(N,\) such that \(\P(N+r) = N+r\,\).
3. The polynomial \({\rm S}(x)\) has degree 4. It has integer coefficients and the coefficient of \(x^4\) is 1. It satisfies \[ {\rm S}(a) = {\rm S}(b) = {\rm S}(c) = {\rm S}(d) = 2001\,, \] where \(a\), \(b\), \(c\) and \(d\) are distinct (not necessarily positive) integers.
   • Show that there is no integer \(e\) such that \({\rm S}(e) = 2018\,\).
   • Find the number of ways the (distinct) integers \(a\), \(b\), \(c\) and \(d\) can be chosen such that \({\rm S}(0) = 2017\) and \(a < b< c< d\,.\)

In this question, you may assume that, if a continuous function takes both positive and negative values in an interval, then it takes the value \(0\) at some point in that interval.

1. The function \(\f\) is continuous and \(\f(x)\) is non-zero for some value of \(x\) in the interval \(0\le x \le 1\). Prove by contradiction, or otherwise, that if \[ \int_0^1 \f(x) \d x = 0\,, \] then \(\f(x)\) takes both positive and negative values in the interval \(0\le x\le 1\).
2. The function \(\g\) is continuous and \[ \int_0^1 \g(x) \, \d x = 1\,, \quad \int_0^1 x\g(x) \, \d x = \alpha\, , \quad \int_0^1 x^2\g(x) \, \d x = \alpha^2\,. \tag{\(*\)} \] Show, by considering \[ \int_0^1 (x - \alpha)^2 \g(x) \, \d x \,, \] that \(\g(x)=0\) for some value of \(x\) in the interval \(0\le x\le 1\). Find a function of the form \(\g(x) = a+bx\) that satisfies the conditions \((*)\) and verify that \(\g(x)=0\) for some value of \(x\) in the interval \(0\le x \le 1\).
3. The function \(\h\) has a continuous derivative \(\h'\) and \[ \h(0) = 0\,, \quad \h(1) = 1\,, \quad \int_0^1 \h(x) \, \d x = \beta\,, \quad \int_0^1 x \h(x) \, \d x = \tfrac{1}{2}\beta (2 - \beta) \,. \] Use the result in part (ii) to show that \(\h^\prime(x)=0\) for some value of \(x\) in the interval \(0\le x\le 1\).

For each non-negative integer \(n\), the polynomial \(\f_n\) is defined by \[ \f_n(x) = 1 + x + \frac{x^2}{2!} + \frac {x^3}{3!} + \cdots + \frac{x^n}{n!} \]

1. Show that \(\f'_{n}(x) = \f_{n-1}(x)\,\) (for \(n\ge1\)).
2. Show that, if \(a\) is a real root of the equation \[\f_n(x)=0\,,\tag{\(*\)}\] then \(a<0\).
3. Let \(a\) and \(b\) be distinct real roots of \((*)\), for \(n\ge2\). Show that \(\f_n'(a)\, \f_n'(b)>0\,\) and use a sketch to deduce that \(\f_n(c)=0\) for some number \(c\) between \(a\) and \(b\). Deduce that \((*)\) has at most one real root. How many real roots does \((*)\) have if \(n\) is odd? How many real roots does \((*)\) have if \(n\) is even?

1. Given that \[ \int \frac {x^3-2}{(x+1)^2}\, \e ^x \d x = \frac{\P(x)}{Q(x)}\,\e^x + \text{constant} \,, \] where \(\P(x)\)and \(Q(x)\) are polynomials, show that \(Q(x)\) has a factor of \(x + 1\). Show also that the degree of \(\P(x)\) is exactly one more than the degree of \(Q(x)\), and find \(\P(x)\) in the case \(Q(x) =x+1\).
2. Show that there are no polynomials \(\P(x)\) and \(Q(x)\) such that \[ \int \frac 1 {x+1} \, \, \e^x \d x = \frac{\P(x)}{Q(x)}\,\e^x +\text{constant} \,. \] You need consider only the case when \(\P(x)\) and \(Q(x)\) have no common factors.