IMSC 2023

Part of IMSC

Subcontests

(6)

1

Divisibility with iteration of a polynomial

Find all polynomials

P(x)

with integer coefficients, such that for all positive integers

m, n

m+n \mid P^{(m)}(n)-P^{(n)}(m).

Proposed by Navid Safaei, Iran

1

Colorful combo geo

2022

points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points?
Proposed by Art Waeterschoot, Belgium

1

Incircle geo with equal segments

ABC

be a triangle with incenter

I

AI

BC

D

E

be a point on the segment

AC

CD=CE

F

be on the segment

AB

BF=BD

(CEI) \cap (DFI)=P \neq I

(BFI) \cap (DEI)=Q \neq I

PQ \perp BC

.
Proposed by Leonardo Franchi, Italy

1

Grid 9x9 with 0 and 1

9 \times 9

grid that is divided into nine

3 \times 3

subgrids with the following properties: - each cell contains either a

0

1

, - each row contains at least one

0

and at least one

1

, - each column contains at least one

0

and at least one

1

, and - each of the nine subgrids contains at least one

0

and at least one

1

.
An incomplete binoku is obtained from a binoku by removing the numbers from some of the cells. What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way?
Proposed by Stijn Cambie, South Korea

1

Classical NT FE

Find all functions

f:\mathbb{Z} \rightarrow \mathbb{Z}

f(1) \neq f(-1)

f(m+n)^2 \mid f(m)-f(n)

for all integers

m, n

.
Proposed by Liam Baker, South Africa

1

Number theory

n!

empty baskets in a row, labelled

1, 2, . . . , n!

. Caesar first puts a stone in every basket. Caesar then puts 2 stones in every second basket. Caesar continues similarly until he has put

n

stones into every nth basket. In other words, for each

i = 1, 2, . . . , n,

i

stones into the baskets labelled

i, 2i, 3i, . . . , n!.

x_i

be the number of stones in basket

i

after all these steps. Show that

n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i}