MathDB

1

At least one of two limits is equal to infinity

f:\mathbb{N}\to\mathbb{N}

be a bijection of the positive integers. Prove that at least one of the following limits is true:

\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.

Proposed by Dylan Toh

5

2

Representing integers using integer polynomials

[*]Is there a non-linear integer-coefficient polynomial

P(x)

and an integer

N{}

such that all integers greater than

N{}

may be written as the greatest common divisor of

P(a){}

and

P(b){}

for positive integers

a>b

? [*]Is there a non-linear integer-coefficient polynomial

Q(x)

and an integer

M{}

such that all integers greater than

M{}

may be written as

Q(a) - Q(b)

for positive integers

a>b

? Proposed by Dylan Toh

Equilateral triangle dissection

Is it possible to dissect an equilateral triangle into three congruent polygonal pieces (not necessarily convex), one of which contains the triangle’s centre in its interior?
Note: The interior of a polygon is the polygon without its boundary.
Proposed by Dylan Toh

4

2

Classical tetrahedron maximisation problem

Points

A, B, C,

and

D{}

lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron

ABCD.

Proposed by Fredy Yip

Convergence of infinite series

Let

(t_n)_{n\geqslant 1}

be the sequence defined by

t_1=1, t_{2k}=-t_k

and

t_{2k+1}=t_{k+1}

for all

k\geqslant 1.

Consider the series

\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.

Prove that this series converges to a positive real number.
Proposed by Dylan Toh

3

2

Encryprion keys on Mathsenger

There are 105 users on the social media platform Mathsenger, every pair of which has a direct messaging channel. Prove that each messaging channel may be assigned one of 100 encryption keys, such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key.
Proposed by Fredy Yip

Expected value with polynomials

Let

N{}

be a fixed positive integer,

S{}

be the set

\{1, 2,\ldots , N\}

and

\mathcal{F}

be the set of functions

f:S\to S

such that

f(i)\geqslant i

for all

i\in S.

For each

f\in\mathcal{F}

let

P_f

be the unique polynomial of degree less than

N{}

satisfying

P_f(i) = f(i)

for all

i\in S.

If

f{}

is chosen uniformly at random from

\mathcal{F}

determine the expected value of

P_f'(0)

where

P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.

Proposed by Ishan Nath

2

Unusual movement on the lattice grid

Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less?
Proposed by Tony Wang

Operations on circular necklaces

Let

n\geqslant 3

be a positive integer. A circular necklace is called fun if it has

n{}

black beads and

n{}

white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. Prove that it is possible to change any fun necklace into any other fun necklace using at most

(n-1)

moves.
Note: Necklaces related by rotations or reflections are considered to be the same.
Proposed by Dylan Toh

1

2

Divisibility of Fibonacci sum

Let

F_n{}

denote the

n{}

-th Fibonacci number. Prove that

3^{2023}

divides

3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.

Proposed by Dylan Toh

Divisibility with 2024 integers

Prove that there exist distinct positive integers

a_1, a_2,\ldots , a_{2024}

such that for each

i\in\{1,2,\ldots,2024\}

a_i\mid a_1a_2\cdots a_{i-1}a_{i+1}\cdots a_{2024}+k,

where a)

k=1

and b)

k=2024.

Proposed by Ishan Nath

ICMC 7

Subcontests

At least one of two limits is equal to infinity

Representing integers using integer polynomials

Equilateral triangle dissection

Classical tetrahedron maximisation problem

Convergence of infinite series

Encryprion keys on Mathsenger

Expected value with polynomials

Unusual movement on the lattice grid

Operations on circular necklaces

Divisibility of Fibonacci sum

Divisibility with 2024 integers