ICMC 2

Part of ICMC

Subcontests

(6)

1

ICMC 2018/19 Round 1, Problem 6

A country has four political parties - the Blue Party, the Red Party, the Yellow Party and the Orange Party - and a parliament of 650 seats.
(a) How many ways are there to divide the seats among the four parties so that none of the parties have a majority? (To have a majority that party must hold more than half of the seats.)
The parliament is particularly worried about cyber security. They have decided that all login passwords must be of length exactly 6 and be a combination of a legal set of elements made up of the digits 0-9, the 52 upper and lower case letters (a-z and A-Z), and five special characters: \ICMC 2018/19 Round 1, Problem 1 $, £, *, &, %. For the password to be allowed, it must contain at least one letter or special character and any letter or special character in the password must be followed by a digit (so it must end in a digit).
(b) The Blue members of parliament have decided to choose their password by selecting 6 elements from the legal set without replacement. What is the probability it is allowed?
Note: you may leave your answers as combinatorial or factorial terms.

1

ICMC 2018/19 Round 1, Problem 5

For continuously differentiable function

f : [0, 1] \to\mathbb{R}

f (1/2) = 0

\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x

2

ICMC 2018/19 Round 1, Problem 4

u,v \in\mathbb{R}^4

<u,v>

denote the usual dot product. Define a vector field to be a map

\omega:\mathbb{R}\to\mathbb{R}

<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.

Find a maximal collection of vector fields

\left\{\omega_1,...,\omega_k\right\}

such that the map

\Omega

z

\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)

\lambda_1,\ldots,\lambda_k\in\mathbb{R}

, is nonzero on

\mathbb{R}^4\backslash\{0\}

\lambda_1=\cdots=\lambda_k=0

ICMC 2018/19 Round 2, Problem 4

f:\{0, 1\}^n \to \{0, 1\} \subseteq \mathbb{R}

be a function. Call such a function a Boolean function. Let

\wedge

denote the component-wise multiplication in

\{0,1\}^n

. For example, for

n = 4

(0,0,1,1) \wedge (0,1,0,1) = (0,0,0,1).

S = \left\{i_1,i_2,\ldots ,i_k\right\} \subseteq \left\{1,2,\ldots ,n\right\}

f

is called the oligarchy function over

S

f (x) = x_{i_1},x_{i_2},\ldots,x_{i_k}\ \text{ (with the usual multiplication),}

x_i

i

-th component of

x

. By convention,

f

is called the oligarchy function over

\emptyset

f

is constantly 1.
(i) Suppose

f

is not constantly zero. Show that

f

is an oligarchy function if and only if

f

f(x\wedge y)=f(x)f(y),\ \forall x,y\in\left\{0,1\right\}^n.

Y

be a uniformly distributed random variable over

\left\{0, 1\right\}^n

T

be an operator that maps Boolean functions to functions

\left\{0, 1\right\}^n\to\mathbb{R}

(Tf)(x)=E_Y(f(x\wedge Y)),\ \forall x\in\left\{0,1\right\}^n

E_Y()

denotes the expectation over

Y

f

is called an eigenfunction of

T

\exists\lambda\in\mathbb{R}\backslash\left\{0\right\}

(Tf)(x)=\lambda f(x),\ \forall x\in\left\{0,1\right\}^n

(ii) Prove that

f

is an eigenfunction of

T

f

is an oligarchy function.

2

ICMC 2018/19 Round 1, Problem 3

A ‘magic square’ of size

n

n\times n

array of real numbers such that all the rows, all the columns and the two main diagonals have the same sum. Determine the dimension, over

\mathbb{R}

, of the vector space of

n\times n

magic squares.\\

ICMC 2018/19 Round 2, Problem 3

Show that if the faces of a tetrahedron have the same area, then they are congruent.

2

ICMC 2018/19 Round 1, Problem 2

This question, again, comprises two independent parts.
(i) Show that if

(k+1)

integers are chosen from

\left\{1,2,3,...,2k+1\right\}

, then among the chosen integers there are always two that are coprime.
(ii) Let

A=\left\{1,2,\ldots,n\right\}.

n>11

then there is a bijective map

f: A\to A

with the property that, for every

a\in A

, exactly one of

f(f(f(f(a))))=a

f(f(f(f(f(a)))))=a

ICMC 2018/19 Round 2, Problem 2

In the symmetric group

S_n\ (n \geq 3)

G_{a,b}

be the subgroup generated by the 2-cycle

(a\ b)

and the n-cycle

(1\ 2\ \cdots\ n)

. Find the index

\left|S_n : G_{a,b}\right|

2

ICMC 2018/19 Round 1, Problem 1

This questions comprises two independent parts.
(i) Let

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

be continuous and such that

g(0)=0

g(x)g(-x)>0

x > 0

. Find all solutions

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

to the functional equation

g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}

(ii) Find all continuously differentiable functions

\phi : [a, \infty) \to \mathbb{R}

a > 0

, that satisfies the equation

(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.

ICMC 2018/19 Round 2, Problem 1

Observe that, in the usual chessboard colouring of the two-dimensional grid, each square has 4 of its 8 neighbours black and 4 white. Does there exist a way to colour the three-dimensional grid such that each cube has half of its 26 neighbours black and half white? Is this possible in four dimensions?