2024 EGMO

Part of EGMO

Subcontests

(6)

1

"d"ifferent values

Find all positive integers

d

for which there exists a degree

d

P

with real coefficients such that there are at most

d

different values among

P(0),P(1),P(2),\cdots,P(d^2-d)

1

Ah yes, very interesting

a_1<a_2<\cdots<a_n

of integers, a pair

(a_i,a_j)

1\leq i<j\leq n

is called interesting if there exists a pair

(a_k,a_l)

of integers with

1\leq k<l\leq n

\frac{a_l-a_k}{a_j-a_i}=2.

n\geq 3

, find the largest possible number of interesting pairs in a sequence of length

n

1

An NT Function

Find all functions

f : \mathbb{N} \rightarrow \mathbb{N}

such that the following conditions are true for every pair of positive integers

(x, y)

(i)

x

f(x)

have the same number of positive divisors.

(ii)

x \nmid y

y \nmid x

\gcd(f(x), f(y)) > f(\gcd(x, y))

1

Operations with numbers on a board

Two different integers

u

v

are written on a board. We perform a sequence of steps. At each step we do one of the following two operations:
(i) If

a

b

are different integers on the board, then we can write

a + b

on the board, if it is not already there. (ii) If

a

b

c

are three different integers on the board, and if an integer

x

ax^2 +bx+c = 0

, then we can write

x

on the board, if it is not already there.
Determine all pairs of starting numbers

(u, v)

from which any integer can eventually be written on the board after a finite sequence of steps.

1

Truly peculiar NT

We call a positive integer

n{}

peculiar if, for any positive divisor

d{}

n{}

d(d + 1)

n(n + 1).

Prove that for any four different peculiar positive integers

A, B, C

D{}

the following holds:

\gcd(A, B, C, D) = 1.

1

EGMO 2024 P2

ABC

be a triangle with

AC>AB

, and denote its circumcircle by

\Omega

and incentre by

I

. Let its incircle meet sides

BC,CA,AB

D,E,F

respectively. Let

X

Y

be two points on minor arcs

\widehat{DF}

\widehat{DE}

of the incircle, respectively, such that

\angle BXD = \angle DYC

XY

BC

K

T

be the point on

\Omega

KT

\Omega

T

is on the same side of line

BC

A

. Prove that lines

TD

AI

\Omega

.
[right]Tommy Walker Mackay, United Kingdom[/right]