3

Part of ICMC 3

Problems(2)

ICMC 2019/20 Round 1, Problem 3

Source: Imperial College Mathematics Competition 2019/20 - Round 1

Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a table denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called balanced if one diagonal pair of points are coloured white and the other diagonal pair black.
Determine all possible values of

k \geq 2

for which there exists a colouring of a

k\times 2019

grid with no balanced tables.
proposed by the ICMC Problem Committee

college contests

ICMC 2019/20 Round 2, Problem 3

Source: Imperial College Mathematics Competition 2019/20 - Round 2

\mathbb{R}

denote the set of real numbers. A subset

S\subseteq\mathbb{R}

is called dense if any non-empty open interval of

\mathbb{R}

contains at least one element in

S

. For a function

f:\mathbb{R}\to\mathbb{R}

\mathcal{O}_f(x)

\left\{x,f(x),f(f(x)),\ldots\right\}

.
(a) Is there a function

g:\mathbb{R}\to\mathbb{R}

, continuous everywhere in

\mathbb{R}

\mathcal{O}_g(x)

is dense for all

x\in\mathbb{R}

x\in\mathbb{R}

?
(b) Is there a function

h:\mathbb{R}\to\mathbb{R}

, continuous at all but a single

x_0\in\mathbb{R}

\mathcal{O}_h(x)

is dense for all

x\in\mathbb{R}

?
Proposed by the ICMC Problem Committee

college contests