ICMC 4

Part of ICMC

Subcontests

(6)

1

game of stones

n+1

squares in a row, labelled from 0 to

n

. Tony starts with

k

stones on square 0. On each move, he may choose a stone and advance the stone up to

m

m

is the number of stones on the same square (including itself) or behind it.
Tony's goal is to get all stones to square

n

. Show that Tony cannot achieve his goal in fewer than

\frac{n}{1} + \frac{n}{2} + \cdots + \frac{n}{k}

moves.
Proposed by Tony Wang

1

sum of divisors is a power of 2

Find all composite positive integers

m

such that, whenever the product of two positive integers

a

b

m

, their sum is a power of

2

.
Proposed by Harun Khan

2

Unique rational sum

Does there exist a set

\mathcal{R}

of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of

\mathcal{R}

?
Proposed by Tony Wang

Almost inversion

\mathbb R^2

denote the Euclidean plane. A continuous function

f : \mathbb R^2 \to \mathbb R^2

maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines.
Proposed by Tony Wang

2

intergal of sin(nx)^n

\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx

.
(a) Prove that

s_n \leq \dfrac 2n

n

.
(b) Find all the limit points of the sequence

s_1, s_2, s_3, \dots

.
Proposed by Cristi Calin

expected mess

f,g,h : \mathbb R \to \mathbb R

be continuous functions and

X

be a random variable such that

E(g(X)h(X))=0

E(g(X)^2) \neq 0 \neq E(h(X)^2)

E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.

You may assume that all expected values exist.
Proposed by Cristi Calin

2

invertible matrix

A

be a square matrix with entries in the field

\mathbb Z / p \mathbb Z

A^n - I

is invertible for every positive integer

n

. Prove that there exists a positive integer

m

A^m = 0

.
(A matrix having entries in the field

\mathbb Z / p \mathbb Z

means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of

p

.)
Proposed by Tony Wang

wonky sequence

p > 3

be a prime number. A sequence of

p-1

a_1,a_2, \dots, a_{p-1}

is called wonky if they are distinct modulo

p

a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p

i \in \{1, 2, \dots, p-1\}

a_p = a_1

a_{p+1} = a_2

. Does there always exist a wonky sequence such that

a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,

are all distinct modulo

p

?
Proposed by Harun Khan

2

Countably many sane points

A set of points in the plane is called sane if no three points are collinear and the angle between any three distinct points is a rational number of degrees.
(a) Does there exist a countably infinite sane set

\mathcal{P}

?
(b) Does there exist an uncountably infinite sane set

\mathcal{Q}

?
Proposed by Tony Wang

union closed subsets

S

be a set with 10 distinct elements. A set

T

S

(possibly containing the empty set) is called union-closed if, for all

A, B \in T

, it is true that

A \cup B \in T

. Show that the number of union-closed sets

T

2^{1023}

.
Proposed by Tony Wang