MathDB

1

Expression is a perfect square implies the polynomial is constant

a\equiv 1\pmod{4}

be a positive integer. Show that any polynomial

Q\in\mathbb{Z}[X]

with all positive coefficients such that

Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})

is a perfect square for any

n\in\mathbb{N}^{\ast}

must be a constant polynomial.
Proposed by Vlad Matei, Romania

5

1

Strictly monotone polynomial with an extra condition

Let

\mathbb{R}_{>0}

be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions

f:\mathbb{R}_{>0} \to \mathbb{R}

such that there exists a two-variable polynomial

P(x, y)

with real coefficients satisfying

f(xy)=P(f(x), f(y))

for all

x, y\in\mathbb{R}_{>0}

.\\
Proposed by Navid Safaei, Iran

4

1

Ana plays with pawns

Ana plays a game on a

100\times 100

chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else.\\ Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways:
[*] it moves to the square directly in front of it if there is no other pawn on it; [*] it captures a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it.
(We say a pawn

P

captures a pawn

Q

of the opposite color if we remove

Q

from the board and move

P

to the square that

Q

was previously on.)\\ \\ Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made?
Proposed by José Alejandro Reyes González, Mexico

3

1

Alice an Bob play a game on a square grid

Alice and Bob play the following game on a square grid with

2024 \times 2024

unit squares. They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. \\ On the

k

-th turn, let

n_k

be the least integer such that

n_k\geqslant\tfrac{k}{2024}

. If there is at least one square without a sticker, then the player taking the turn: [list = i] [*] selects at most

n_k

unit squares on the grid such that at least one of the chosen unit squares does not have a sticker. [*] covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first.
At the end of their turn, a player wins if there exist

123

unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. \\ Does Alice have a winning strategy?
Proposed by Erik Paemurru, Estonia

2

1

Equal angles imply equal lengths

Let

ABC

be an acute angled triangle and let

P, Q

be points on

AB, AC

respectively, such that

PQ

is parallel to

BC

. Points

X, Y

are given on line segments

BQ, CP

respectively, such that

\angle AXP = \angle XCB

and

\angle AYQ = \angle YBC

. Prove that

AX = AY

.
Proposed by Ervin Maci

\acute{c},

Bosnia and Herzegovina

1

Consecutive string with a nice property

For a positive integer

n

denote by

P_0(n)

the product of all non-zero digits of

n

. Let

N_0

be the set of all positive integers

n

such that

P_0(n)|n

. Find the largest possible value of

\ell

such that

N_0

contains infinitely many strings of

\ell

consecutive integers.
Proposed by Navid Safaei, Iran

IMSC 2024

Subcontests

Expression is a perfect square implies the polynomial is constant

Strictly monotone polynomial with an extra condition

Ana plays with pawns

Alice an Bob play a game on a square grid

Equal angles imply equal lengths

Consecutive string with a nice property