• # Paper 1, Section I, 2H

Write

$P=\left\{\mathbf{x} \in \mathbb{R}^{n}: x_{j} \geqslant 0 \text { for all } 1 \leqslant j \leqslant n\right\}$

and suppose that $K$ is a non-empty, closed, convex and bounded subset of $\mathbb{R}^{n}$ with $K \cap \operatorname{Int} P \neq \emptyset$. By taking logarithms, or otherwise, show that there is a unique $\mathbf{x}^{*} \in K \cap P$ such that

$\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} x_{j}^{*}$

for all $\mathbf{x} \in K \cap P$.

Show that $\sum_{j=1}^{n} \frac{x_{j}}{x_{j}^{*}} \leqslant n$ for all $\mathbf{x} \in K \cap P$.

Identify the point $\mathbf{x}^{*}$ in the case that $K$ has the property

$\left(x_{1}, x_{2}, \ldots, x_{n-1}, x_{n}\right) \in K \Rightarrow\left(x_{2}, x_{3}, \ldots, x_{n}, x_{1}\right) \in K$

Show that, given any $\mathbf{a} \in \operatorname{Int} P$, we can find a set $K$, as above, with $\mathbf{x}^{*}=\mathbf{a}$.

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• # Paper 2, Section I, $2 H$

Let $\Omega$ be a non-empty bounded open set in $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$ and let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be a continuous function. Give a proof or a counterexample for each of the following assertions.

(i) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})>0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(ii) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})<0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(iii) If $\phi$ is four times differentiable on $\Omega$ with

$\frac{\partial^{4} \phi}{\partial x^{4}}(\mathbf{x})+\frac{\partial^{4} \phi}{\partial y^{4}}(\mathbf{x})>0$

for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(iv) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})=0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

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• # Paper 2, Section II, H

Let $r:[-1,1] \rightarrow \mathbb{R}$ be a continuous function with $r(x)>0$ for all but finitely many values of $x$.

(a) Show that

$\langle u, v\rangle=\int_{-1}^{1} u(x) v(x) r(x) d x$

defines an inner product on $C([-1,1])$.

(b) Show that for each $n$ there exists a polynomial $P_{n}$ of degree exactly $n$ which is orthogonal, with respect to the inner product $(*)$, to all polynomials of lower degree.

(c) Show that $P_{n}$ has $n$ simple zeros $\omega_{1}(n), \omega_{2}(n), \ldots, \omega_{n}(n)$ on $[-1,1]$.

(d) Show that for each $n$ there exist unique real numbers $A_{j}(n), 1 \leqslant j \leqslant n$, such that whenever $Q$ is a polynomial of degree at most $2 n-1$,

$\int_{-1}^{1} Q(x) r(x) d x=\sum_{j=1}^{n} A_{j}(n) Q\left(\omega_{j}(n)\right)$

(e) Show that

$\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right) \rightarrow \int_{-1}^{1} f(x) r(x) d x$

as $n \rightarrow \infty$ for all $f \in C([-1,1])$.

(f) If $R>1, K>0, a_{m}$ is real with $\left|a_{m}\right| \leqslant K R^{-m}$ and $f(x)=\sum_{m=1}^{\infty} a_{m} x^{m}$, show that

$\left|\int_{-1}^{1} f(x) r(x) d x-\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right)\right| \leqslant \frac{2 K R^{-2 n+1}}{R-1} \int_{-1}^{1} r(x) d x$

(g) If $r(x)=\left(1-x^{2}\right)^{1 / 2}$ and $P_{n}(0)=1$, identify $P_{n}$ (giving brief reasons) and the $\omega_{j}(n)$. [Hint: A change of variable may be useful.]

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• # Paper 3 , Section I, $2 H$

State Runge's theorem on the approximation of analytic functions by polynomials.

Let $\Omega=\{z \in \mathbb{C}, \operatorname{Re} z>0, \operatorname{Im} z>0\}$. Establish whether the following statements are true or false by giving a proof or a counterexample in each case.

(i) If $f: \Omega \rightarrow \mathbb{C}$ is the uniform limit of a sequence of polynomials $P_{n}$, then $f$ is a polynomial.

(ii) If $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there exists a sequence of polynomials $P_{n}$ such that for each integer $r \geqslant 0$ and each $z \in \Omega$ we have $P_{n}^{(r)}(z) \rightarrow f^{(r)}(z)$.

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• # Paper 4, Section I, 2H

(a) State Brouwer's fixed-point theorem in 2 dimensions.

(b) State an equivalent theorem on retraction and explain (without detailed calculations) why it is equivalent.

(c) Suppose that $A$ is a $3 \times 3$ real matrix with strictly positive entries. By defining an appropriate function $f: \triangle \rightarrow \triangle$, where

$\triangle=\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=1, x_{1}, x_{2}, x_{3} \geqslant 0\right\}$

show that $A$ has a strictly positive eigenvalue.

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• # Paper 4, Section II, H

Let $x$ be irrational with $n$th continued fraction convergent

$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}}$

Show that

$\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$

and deduce that

$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{q_{n} q_{n+1}} .$

[You may quote the result that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1} \cdot$ ]

We say that $y$ is a quadratic irrational if it is an irrational root of a quadratic equation with integer coefficients. Show that if $y$ is a quadratic irrational, we can find an $M>0$ such that

$\left|\frac{p}{q}-y\right| \geqslant \frac{M}{q^{2}}$

for all integers $p$ and $q$ with $q>0$.

Using the hypotheses and notation of the first paragraph, show that if the sequence $\left(a_{n}\right)$ is unbounded, $x$ cannot be a quadratic irrational.

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• # Paper 1, Section I, $\mathbf{2 H}$

Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a continuous map never taking the value 0 and satisfying $\gamma(0)=\gamma(1)$. Define the degree (or winding number) $w(\gamma ; 0)$ of $\gamma$ about 0 . Prove the following.

(i) If $\delta:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ is a continuous map satisfying $\delta(0)=\delta(1)$, then the winding number of the product $\gamma \delta$ is given by $w(\gamma \delta ; 0)=w(\gamma ; 0)+w(\delta ; 0)$.

(ii) If $\sigma:[0,1] \rightarrow \mathbb{C}$ is continuous, $\sigma(0)=\sigma(1)$ and $|\sigma(t)|<|\gamma(t)|$ for each $0 \leqslant t \leqslant 1$, then $w(\gamma+\sigma ; 0)=w(\gamma ; 0)$.

(iii) Let $D=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and let $f: D \rightarrow \mathbb{C}$ be a continuous function with $f(z) \neq 0$ whenever $|z|=1$. Define $\alpha:[0,1] \rightarrow \mathbb{C}$ by $\alpha(t)=f\left(e^{2 \pi i t}\right)$. Then if $w(\alpha ; 0) \neq 0$, there must exist some $z \in D$, such that $f(z)=0$. [It may help to define $F(s, t):=f\left(s e^{2 \pi i t}\right)$. Homotopy invariance of the winding number may be assumed.]

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• # Paper 2, Section I, $2 \mathrm{H}$

Show that every Legendre polynomial $p_{n}$ has $n$ distinct roots in $[-1,1]$, where $n$ is the degree of $p_{n}$.

Let $x_{1}, \ldots, x_{n}$ be distinct numbers in $[-1,1]$. Show that there are unique real numbers $A_{1}, \ldots, A_{n}$ such that the formula

$\int_{-1}^{1} P(t) d t=\sum_{i=1}^{n} A_{i} P\left(x_{i}\right)$

holds for every polynomial $P$ of degree less than $n$.

Now suppose that the above formula in fact holds for every polynomial $P$ of degree less than $2 n$. Show that then $x_{1}, \ldots, x_{n}$ are the roots of $p_{n}$. Show also that $\sum_{i=1}^{n} A_{i}=2$ and that all $A_{i}$ are positive.

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• # Paper 2, Section II, H

Let $T$ be a (closed) triangle in $\mathbb{R}^{2}$ with edges $I, J, K$. Let $A, B, C$, be closed subsets of $T$, such that $I \subset A, J \subset B, K \subset C$ and $T=A \cup B \cup C$. Prove that $A \cap B \cap C$ is non-empty.

Deduce that there is no continuous map $f: D \rightarrow \partial D$ such that $f(p)=p$ for all $p \in \partial D$, where $D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leqslant 1\right\}$ is the closed unit disc and $\partial D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is its boundary.

Let now $\alpha, \beta, \gamma \subset \partial D$ be three closed arcs, each arc making an angle of $2 \pi / 3$ (in radians) in $\partial D$ and $\alpha \cup \beta \cup \gamma=\partial D$. Let $P, Q$ and $R$ be open subsets of $D$, such that $\alpha \subset P$, $\beta \subset Q$ and $\gamma \subset R$. Suppose that $P \cup Q \cup R=D$. Show that $P \cap Q \cap R$ is non-empty. [You may assume that for each closed bounded subset $K \subset \mathbb{R}^{2}, d(x, K)=\min \{\|x-y\|: y \in K\}$ defines a continuous function on $\mathbb{R}^{2}$.]

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• # Paper 3, Section I, H

State Runge's theorem about the uniform approximation of holomorphic functions by polynomials.

Explicitly construct, with a brief justification, a sequence of polynomials which converges uniformly to $1 / z$ on the semicircle $\{z:|z|=1, \operatorname{Re}(z) \leqslant 0\}$.

Does there exist a sequence of polynomials converging uniformly to $1 / z$ on $\{z:|z|=1, z \neq 1\}$ ? Give a justification.

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• # Paper 4, Section I, $2 \mathrm{H}$

Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category theorem.

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(n x) \rightarrow 0$ as $n \rightarrow \infty$ for every fixed $x \geqslant 1$. Show that $f(t) \rightarrow 0$ as $t \rightarrow \infty$.

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• # Paper 4, Section II, H

(a) State Liouville's theorem on the approximation of algebraic numbers by rationals.

(b) Let $\left(a_{n}\right)_{n=0}^{\infty}$ be a sequence of positive integers and let

$\alpha=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$

be the value of the associated continued fraction.

(i) Prove that the $n$th convergent $p_{n} / q_{n}$ satisfies

$\left|\alpha-\frac{p_{n}}{q_{n}}\right| \leqslant\left|\alpha-\frac{p}{q}\right|$

for all the rational numbers $\frac{p}{q}$ such that $0.

(ii) Show that if the sequence $\left(a_{n}\right)$ is bounded, then one can choose $c>0$ (depending only on $\alpha$ ), so that for every rational number $\frac{a}{b}$,

$\left|\alpha-\frac{a}{b}\right|>\frac{c}{b^{2}}$

(iii) Show that if the sequence $\left(a_{n}\right)$ is unbounded, then for each $c>0$ there exist infinitely many rational numbers $\frac{a}{b}$ such that

$\left|\alpha-\frac{a}{b}\right|<\frac{c}{b^{2}}$

[You may assume without proof the relation

$\left.\left(\begin{array}{ll} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{array}\right)=\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n+1} \end{array}\right)\left(\begin{array}{cc} a_{n+1} & 1 \\ 1 & 0 \end{array}\right), \quad n=1,2, \ldots .\right]$

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• # Paper 1, Section I, H

Let $T_{n}$ be the $n$th Chebychev polynomial. Suppose that $\gamma_{i}>0$ for all $i$ and that $\sum_{i=1}^{\infty} \gamma_{i}$ converges. Explain why $f=\sum_{i=1}^{\infty} \gamma_{i} T_{3^{i}}$ is a well defined continuous function on $[-1,1]$.

Show that, if we take $P_{n}=\sum_{i=1}^{n} \gamma_{i} T_{3^{i}}$, we can find points $x_{k}$ with

$-1 \leqslant x_{0}

such that $f\left(x_{k}\right)-P_{n}\left(x_{k}\right)=(-1)^{k+1} \sum_{i=n+1}^{\infty} \gamma_{i}$ for each $k=0,1, \ldots, 3^{n+1}$.

Suppose that $\delta_{n}$ is a decreasing sequence of positive numbers and that $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$. Stating clearly any theorem that you use, show that there exists a continuous function $f$ with

$\sup _{t \in[-1,1]}|f(t)-P(t)| \geqslant \delta_{n}$

for all polynomials $P$ of degree at most $n$ and all $n \geqslant 1$.

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• # Paper 2, Section I, H

Let $\mathcal{K}$ be the collection of non-empty closed bounded subsets of $\mathbb{R}^{n}$.

(a) Show that, if $A, B \in \mathcal{K}$ and we write

$A+B=\{a+b: a \in A, b \in B\}$

then $A+B \in \mathcal{K}$.

(b) Show that, if $K_{n} \in \mathcal{K}$, and

$K_{1} \supseteq K_{2} \supseteq K_{3} \supseteq \cdots$

then $K:=\bigcap_{n=1}^{\infty} K_{n} \in \mathcal{K}$.

(c) Assuming the result that

$\rho(A, B)=\sup _{a \in A} \inf _{b \in B}|a-b|+\sup _{b \in B} \inf _{a \in A}|a-b|$

defines a metric on $\mathcal{K}$ (the Hausdorff metric), show that if $K_{n}$ and $K$ are as in part (b), then $\rho\left(K_{n}, K\right) \rightarrow 0$ as $n \rightarrow \infty$.

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• # Paper 2, Section II, H

Throughout this question $I$ denotes the closed interval $[-1,1]$.

(a) For $n \in \mathbb{N}$, consider the $2 n+1$ points $r / n \in I$ with $r \in \mathbb{Z}$ and $-n \leqslant r \leqslant n$. Show that, if we colour them red or green in such a way that $-1$ and 1 are coloured differently, there must be two neighbouring points of different colours.

(b) Deduce from part (a) that, if $I=A \cup B$ with $A$ and $B$ closed, $-1 \in A$ and $1 \in B$, then $A \cap B \neq \emptyset$.

(c) Deduce from part (b) that there does not exist a continuous function $f: I \rightarrow \mathbb{R}$ with $f(t) \in\{-1,1\}$ for all $t \in I$ and $f(-1)=-1, f(1)=1$.

(d) Deduce from part (c) that if $f: I \rightarrow I$ is continuous then there exists an $x \in I$ with $f(x)=x$.

(e) Deduce the conclusion of part (c) from the conclusion of part (d).

(f) Deduce the conclusion of part (b) from the conclusion of part (c).

(g) Suppose that we replace $I$ wherever it occurs by the unit circle

$C=\{z \in \mathbb{C}|| z \mid=1\}$

Which of the conclusions of parts (b), (c) and (d) remain true? Give reasons.

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• # Paper 3, Section I, H

State Nash's theorem for a non zero-sum game in the case of two players with two choices.

The role playing game Tixerb involves two players. Before the game begins, each player $i$ chooses a $p_{i}$ with $0 \leqslant p_{i} \leqslant 1$ which they announce. They may change their choice as many times as they wish, but, once the game begins, no further changes are allowed. When the game starts, player $i$ becomes a Dark Lord with probability $p_{i}$ and a harmless peasant with probability $1-p_{i}$. If one player is a Dark Lord and the other a peasant the Lord gets 2 points and the peasant $-2$. If both are peasants they get 1 point each, if both Lords they get $-U$ each. Show that there exists a $U_{0}$, to be found, such that, if $U>U_{0}$ there will be three choices of $\left(p_{1}, p_{2}\right)$ for which neither player can increase the expected value of their outcome by changing their choice unilaterally, but, if $U_{0}>U$, there will only be one. Find the appropriate $\left(p_{1}, p_{2}\right)$ in each case.

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• # Paper 4, Section I, H

Show that $\pi$ is irrational. [Hint: consider the functions $f_{n}:[0, \pi] \rightarrow \mathbb{R}$ given by $\left.f_{n}(x)=x^{n}(\pi-x)^{n} \sin x .\right]$

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• # Paper 4, Section II, H

(a) Suppose that $K \subset \mathbb{C}$ is a non-empty subset of the square $\{x+i y: x, y \in(-1,1)\}$ and $f$ is analytic in the larger square $\{x+i y: x, y \in(-1-\delta, 1+\delta)\}$ for some $\delta>0$. Show that $f$ can be uniformly approximated on $K$ by polynomials.

(b) Let $K$ be a closed non-empty proper subset of $\mathbb{C}$. Let $\Lambda$ be the set of $\lambda \in \mathbb{C} \backslash K$ such that $g_{\lambda}(z)=(z-\lambda)^{-1}$ can be approximated uniformly on $K$ by polynomials and let $\Gamma=\mathbb{C} \backslash(K \cup \Lambda)$. Show that $\Lambda$ and $\Gamma$ are open. Is it always true that $\Lambda$ is non-empty? Is it always true that, if $K$ is bounded, then $\Gamma$ is empty? Give reasons.

[No form of Runge's theorem may be used without proof.]

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• # Paper 1, Section I, $2 F$

State and prove Sperner's lemma concerning colourings of points in a triangular grid.

Suppose that $\triangle$ is a non-degenerate closed triangle with closed edges $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$. Show that we cannot find closed sets $A_{j}$ with $A_{j} \supseteq \alpha_{j}$, for $j=1,2,3$, such that

$\bigcup_{j=1}^{3} A_{j}=\Delta, \text { but } \bigcap_{j=1}^{3} A_{j}=\emptyset$

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• # Paper 2, Section I, $2 F$

For $\mathbf{x} \in \mathbb{R}^{n}$ we write $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. Define

$P:=\left\{\mathbf{x} \in \mathbb{R}^{n}: x_{j} \geqslant 0 \text { for } 1 \leqslant j \leqslant n\right\} .$

(a) Suppose that $L$ is a convex subset of $P$, that $(1,1, \ldots, 1) \in L$ and that $\prod_{j=1}^{n} x_{j} \leqslant 1$ for all $\mathbf{x} \in L$. Show that $\sum_{j=1}^{n} x_{j} \leqslant n$ for all $\mathbf{x} \in L$.

(b) Suppose that $K$ is a non-empty closed bounded convex subset of $P$. Show that there is a $\mathbf{u} \in K$ such that $\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} u_{j}$ for all $\mathbf{x} \in K$. If $u_{j} \neq 0$ for each $j$ with $1 \leqslant j \leqslant n$, show that

$\sum_{j=1}^{n} \frac{x_{j}}{u_{j}} \leqslant n$

for all $\mathbf{x} \in K$, and that $\mathbf{u}$ is unique.

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• # Paper 2, Section II, F

(a) Give Bernstein's probabilistic proof of Weierstrass's theorem.

(b) Are the following statements true or false? Justify your answer in each case.

(i) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging pointwise to $f$ on $\mathbb{R}$.

(ii) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $\mathbb{R}$.

(iii) If $f:(0,1] \rightarrow \mathbb{R}$ is continuous and bounded, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $(0,1]$.

(iv) If $f:[0,1] \rightarrow \mathbb{R}$ is continuous and $x_{1}, x_{2}, \ldots, x_{m}$ are distinct points in $[0,1]$, then there exists a sequence of polynomials $P_{n}$ with $P_{n}\left(x_{j}\right)=f\left(x_{j}\right)$, for $j=1, \ldots, m$, converging uniformly to $f$ on $[0,1]$.

(v) If $f:[0,1] \rightarrow \mathbb{R}$ is $m$ times continuously differentiable, then there exists a sequence of polynomials $P_{n}$ such that $P_{n}^{(r)} \rightarrow f^{(r)}$ uniformly on $[0,1]$ for each $r=0, \ldots, m$.

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• # Paper 3, Section I, $2 F$

State a version of the Baire category theorem and use it to prove the following result:

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic, but not a polynomial, then there exists a point $z_{0} \in \mathbb{C}$ such that each coefficient of the Taylor series of $f$ at $z_{0}$ is non-zero.

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• # Paper 4, Section I, $2 F$

Let $0 \leqslant \alpha<1$ and $A>0$. If we have an infinite sequence of integers $m_{n}$ with $1 \leqslant m_{n} \leqslant A n^{\alpha}$, show that

$\sum_{n=1}^{\infty} \frac{m_{n}}{n !}$

is irrational.

Does the result remain true if the $m_{n}$ are not restricted to integer values? Justify your answer.

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• # Paper 4, Section II, F

We work in $\mathbb{C}$. Consider

$K=\{z:|z-2| \leqslant 1\} \cup\{z:|z+2| \leqslant 1\}$

and

$\Omega=\{z:|z-2|<3 / 2\} \cup\{z:|z+2|<3 / 2\}$

Show that if $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there is a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow f(z)$ uniformly on $K$.

Show that there is a sequence of polynomials $P_{n}$ such that $P_{n}(z) \rightarrow 0$ uniformly for $|z-2| \leqslant 1$ and $P_{n}(z) \rightarrow 1$ uniformly for $|z+2| \leqslant 1$.

Give two disjoint non-empty bounded closed sets $K_{1}$ and $K_{2}$ such that there does not exist a sequence of polynomials $Q_{n}$ with $Q_{n}(z) \rightarrow 0$ uniformly on $K_{1}$ and $Q_{n}(z) \rightarrow 1$ uniformly on $K_{2}$. Justify your answer.

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• # Paper 1, Section I, 2F

State Liouville's theorem on the approximation of algebraic numbers by rationals.

Suppose that we have a sequence $\zeta_{n}$ with $\zeta_{n} \in\{0,1\}$. State and prove a necessary and sufficient condition on the $\zeta_{n}$ for

$\sum_{n=0}^{\infty} \zeta_{n} 10^{-n !}$

to be transcendental.

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• # Paper 2, Section I, F

Are the following statements true or false? Give reasons, quoting any theorems that you need.

(i) There is a sequence of polynomials $P_{n}$ with $P_{n}(t) \rightarrow \sin t$ uniformly on $\mathbb{R}$ as $n \rightarrow \infty$.

(ii) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there is a sequence of polynomials $Q_{n}$ with $Q_{n}(t) \rightarrow f(t)$ for each $t \in \mathbb{R}$ as $n \rightarrow \infty$.

(iii) If $g:[1, \infty) \rightarrow \mathbb{R}$ is continuous with $g(t) \rightarrow 0$ as $t \rightarrow \infty$, then there is a sequence of polynomials $R_{n}$ with $R_{n}(1 / t) \rightarrow g(t)$ uniformly on $[1, \infty)$ as $n \rightarrow \infty$.

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• # Paper 2, Section II, F

State and prove Baire's category theorem for complete metric spaces. Give an example to show that it may fail if the metric space is not complete.

Let $f_{n}:[0,1] \rightarrow \mathbb{R}$ be a sequence of continuous functions such that $f_{n}(x)$ converges for all $x \in[0,1]$. Show that if $\epsilon>0$ is fixed we can find an $N \geqslant 0$ and a non-empty open interval $J \subseteq[0,1]$ such that $\left|f_{n}(x)-f_{m}(x)\right| \leqslant \epsilon$ for all $x \in J$ and all $n, m \geqslant N$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be defined by

$g(x)= \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational. }\end{cases}$

Show that we cannot find continuous functions $g_{n}:[0,1] \rightarrow \mathbb{R}$ with $g_{n}(x) \rightarrow g(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty .$

Define a sequence of continuous functions $h_{n}:[0,1] \rightarrow \mathbb{R}$ and a discontinuous function $h:[0,1] \rightarrow \mathbb{R}$ with $h_{n}(x) \rightarrow h(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty$.

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• # Paper 3, Section I, 2F

(a) Suppose that $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is a continuous function such that there exists a $K>0$ with $\|g(\mathbf{x})-\mathbf{x}\| \leqslant K$ for all $\mathbf{x} \in \mathbb{R}^{2}$. By constructing a suitable map $f$ from the closed unit disc into itself, show that there exists a $\mathbf{t} \in \mathbb{R}^{2}$ with $g(\mathbf{t})=\mathbf{0}$.

(b) Show that $g$ is surjective.

(c) Show that the result of part (b) may be false if we drop the condition that $g$ is continuous.

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• # Paper 4, Section I, F

If $x \in(0,1]$, set

$x=\frac{1}{N(x)+T(x)}$

where $N(x)$ is an integer and $1>T(x) \geqslant 0$. Let $N(0)=T(0)=0$.

If $x$ is also irrational, write down the continued fraction expansion in terms of $N T^{j}(x)\left(\right.$ where $\left.N T^{0}(x)=N(x)\right)$.

Let $X$ be a random variable taking values in $[0,1]$ with probability density function

$f(x)=\frac{1}{(\log 2)(1+x)}$

Show that $T(X)$ has the same distribution as $X$.

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• # Paper 4, Section II, 11F

(a) Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is continuous with $\gamma(0)=\gamma(1)$ and $\gamma(t) \neq 0$ for all $t \in[0,1]$. Show that if $\gamma(0)=|\gamma(0)| \exp \left(i \theta_{0}\right)$ (with $\theta_{0}$ real) we can define a continuous function $\theta:[0,1] \rightarrow \mathbb{R}$ such that $\theta(0)=\theta_{0}$ and $\gamma(t)=|\gamma(t)| \exp (i \theta(t))$. Hence define the winding number $w(\gamma)=w(0, \gamma)$ of $\gamma$ around 0 .

(b) Show that $w(\gamma)$ can take any integer value.

(c) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition, and $\left(\gamma_{1} \times \gamma_{2}\right)(t)=\gamma_{1}(t) \gamma_{2}(t)$, show that

$w\left(\gamma_{1} \times \gamma_{2}\right)=w\left(\gamma_{1}\right)+w\left(\gamma_{2}\right)$

(d) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition and $\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)\right|$ for all $t \in[0,1]$, show that

$w\left(\gamma_{1}\right)=w\left(\gamma_{2}\right)$

(e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.

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• # Paper 1, Section I, H

By considering the function $\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by

$R\left(a_{0}, \ldots, a_{n}\right)=\sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right|$

or otherwise, show that there exist $K_{n}>0$ and $\delta_{n}>0$ such that

$K_{n} \sum_{j=0}^{n}\left|a_{j}\right| \geqslant \sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right| \geqslant \delta_{n} \sum_{j=0}^{n}\left|a_{j}\right|$

for all $a_{j} \in \mathbb{R}, 0 \leqslant j \leqslant n$.

Show, quoting carefully any theorems you use, that we must have $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

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• # Paper 2, Section I, H

Define what it means for a subset $E$ of $\mathbb{R}^{n}$ to be convex. Which of the following statements about a convex set $E$ in $\mathbb{R}^{n}$ (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.

(i) The closure of $E$ is convex.

(ii) The interior of $E$ is convex.

(iii) If $\alpha: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is linear, then $\alpha(E)$ is convex.

(iv) If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous, then $f(E)$ is convex.

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• # Paper 2, Section II, H

Prove Bernstein's theorem, which states that if $f:[0,1] \rightarrow \mathbb{R}$ is continuous and

$f_{m}(t)=\sum_{r=0}^{m}\left(\begin{array}{c} m \\ r \end{array}\right) f(r / m) t^{r}(1-t)^{m-r}$

then $f_{m}(t) \rightarrow f(t)$ uniformly on $[0,1]$. [Theorems from probability theory may be used without proof provided they are clearly stated.]

Deduce Weierstrass's theorem on polynomial approximation for any closed interval.

Proving any results on Chebyshev polynomials that you need, show that, if $g:[0, \pi] \rightarrow \mathbb{R}$ is continuous and $\epsilon>0$, then we can find an $N \geqslant 0$ and $a_{j} \in \mathbb{R}$, for $0 \leqslant j \leqslant N$, such that

$\left|g(t)-\sum_{j=0}^{N} a_{j} \cos j t\right| \leqslant \epsilon$

for all $t \in[0, \pi]$. Deduce that $\int_{0}^{\pi} g(t) \cos n t d t \rightarrow 0$ as $n \rightarrow \infty$.

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• # Paper 3, Section I, $2 \mathrm{H}$

In the game of 'Chicken', $A$ and $B$ drive fast cars directly at each other. If they both swerve, they both lose 10 status points; if neither swerves, they both lose 100 status points. If one swerves and the other does not, the swerver loses 20 status points and the non-swerver gains 40 status points. Find all the pairs of probabilistic strategies such that, if one driver maintains their strategy, it is not in the interest of the other to change theirs.

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• # Paper 4, Section I, H

Let $a_{0}, a_{1}, a_{2}, \ldots$ be integers such that there exists an $M$ with $M \geqslant\left|a_{n}\right|$ for all $n$. Show that, if infinitely many of the $a_{n}$ are non-zero, then $\sum_{n=0}^{\infty} \frac{a_{n}}{n !}$ is an irrational number.

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• # Paper 4, Section II, H

Explain briefly how a positive irrational number $x$ gives rise to a continued fraction

$a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$

with the $a_{j}$ non-negative integers and $a_{j} \geqslant 1$ for $j \geqslant 1$.

Show that, if we write

$\left(\begin{array}{ll} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$

then

$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}$

for $n \geqslant 0$.

Use the observation [which need not be proved] that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ to show that

$\left|p_{n} / q_{n}-x\right| \leqslant 1 / q_{n} q_{n+1}$

Show that $q_{n} \geqslant F_{n}$ where $F_{n}$ is the $n$th Fibonacci number (thus $F_{0}=F_{1}=1$, $\left.F_{n+2}=F_{n+1}+F_{n}\right)$, and conclude that

$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{F_{n} F_{n+1}}$

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• # Paper 1, Section I, I

Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$. Let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be continuous with $\phi$ twice differentiable on $\Omega$.

(i) Why does $\phi$ have a maximum on $\bar{\Omega}$ ?

(ii) If $\epsilon>0$ and $\nabla^{2} \phi \geqslant \epsilon$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

(iii) If $\nabla^{2} \phi \geqslant 0$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

(iv) If $\nabla^{2} \phi=0$ on $\Omega$ and $\phi=0$ on $\partial \Omega$, show that $\phi=0$ on $\bar{\Omega}$.

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• # Paper 2, Section I, I

Let $x_{1}, x_{2}, \ldots, x_{n}$ be the roots of the Legendre polynomial of degree $n$. Let $A_{1}$, $A_{2}, \ldots, A_{n}$ be chosen so that

$\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$

for all polynomials $p$ of degree $n-1$ or less. Assuming any results about Legendre polynomials that you need, prove the following results:

(i) $\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$ for all polynomials $p$ of degree $2 n-1$ or less;

(ii) $A_{j} \geqslant 0$ for all $1 \leqslant j \leqslant n$;

(iii) $\sum_{j=1}^{n} A_{j}=2$.

Now consider $Q_{n}(f)=\sum_{j=1}^{n} A_{j} f\left(x_{j}\right)$. Show that

$Q_{n}(f) \rightarrow \int_{-1}^{1} f(t) d t$

as $n \rightarrow \infty$ for all continuous functions $f$.

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• # Paper 2, Section II,

State and prove Sperner's lemma concerning the colouring of triangles.

Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.

State Brouwer's fixed point theorem for a disc and sketch a proof of it.

Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function such that for some $K>0$ we have $\|g(x)-x\| \leqslant K$ for all $x \in \mathbb{R}^{2}$. Show that $g$ is surjective.

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• # Paper 3, Section I, $2 I$

Let $K$ be a compact subset of $\mathbb{C}$ with path-connected complement. If $w \notin K$ and $\epsilon>0$, show that there is a polynomial $P$ such that

$\left|P(z)-\frac{1}{w-z}\right| \leqslant \epsilon$

for all $z \in K$.

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• # Paper 3, Section II, I

Let $\alpha>0$. By considering the set $E_{m}$ consisting of those $f \in C([0,1])$ for which there exists an $x \in[0,1]$ with $|f(x+h)-f(x)| \leqslant m|h|^{\alpha}$ for all $x+h \in[0,1]$, or otherwise, give a Baire category proof of the existence of continuous functions $f$ on $[0,1]$ such that

$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$

at each $x \in[0,1]$.

Are the following statements true? Give reasons.

(i) There exists an $f \in C([0,1])$ such that

$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$

for each $x \in[0,1]$ and each $\alpha>0$.

(ii) There exists an $f \in C([0,1])$ such that

$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$

for each $x \in[0,1]$ and each $\alpha \geqslant 0$.

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• # Paper 4, Section I, $2 I$

Let $\mathcal{K}$ be the set of all non-empty compact subsets of $m$-dimensional Euclidean space $\mathbb{R}^{m}$. Define the Hausdorff metric on $\mathcal{K}$, and prove that it is a metric.

Let $K_{1} \supseteq K_{2} \supseteq \ldots$ be a sequence in $\mathcal{K}$. Show that $K=\bigcap_{n=1}^{\infty} K_{n}$ is also in $\mathcal{K}$ and that $K_{n} \rightarrow K$ as $n \rightarrow \infty$ in the Hausdorff metric.

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• # Paper 1, Section I, $2 \mathrm{G}$

(i) State Brouwer's fixed point theorem in the plane and an equivalent theorem concerning mapping a triangle $T$ to its boundary $\partial T$.

(ii) Let $A$ be a $3 \times 3$ matrix with positive real entries. Use the theorems you stated in (i) to prove that $A$ has an eigenvector with positive entries.

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• # Paper 2, Section I, G

State Chebyshev's equal ripple criterion.

Let

$h(t)=\prod_{\ell=1}^{n}\left(t-\cos \frac{(2 \ell-1) \pi}{2 n}\right)$

Show that if $q(t)=\sum_{j=0}^{n} a_{j} t^{j}$ where $a_{0}, \ldots, a_{n}$ are real constants with $\left|a_{n}\right| \geqslant 1$, then

$\sup _{t \in[-1,1]}|h(t)| \leqslant \sup _{t \in[-1,1]}|q(t)|$

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• # Paper 2, Section II, G

Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a continuous map never taking the value 0 and satisfying $\gamma(0)=\gamma(1)$. Define the degree (or winding number) $w(\gamma ; 0)$ of $\gamma$ about 0 . Prove the following:

(i) $w(1 / \gamma ; 0)=w\left(\gamma^{-} ; 0\right)$, where $\gamma^{-}(t)=\gamma(1-t)$.

(ii) If $\sigma:[0,1] \rightarrow \mathbb{C}$ is continuous, $\sigma(0)=\sigma(1)$ and $|\sigma(t)|<|\gamma(t)|$ for each $0 \leqslant t \leqslant 1$, then $w(\gamma+\sigma ; 0)=w(\gamma ; 0)$.

(iii) If $\gamma_{m}:[0,1] \rightarrow \mathbb{C}, m=1,2, \ldots$, are continuous maps with $\gamma_{m}(0)=\gamma_{m}(1)$, which converge to $\gamma$ uniformly on $[0,1]$ as $m \rightarrow \infty$, then $w\left(\gamma_{m} ; 0\right)=w(\gamma ; 0)$ for sufficiently large $m$.

Let $\alpha:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\alpha(0)=\alpha(1)$ and $\left|\alpha(t)-e^{2 \pi i t}\right| \leqslant 1$ for each $t \in[0,1]$