2018 EGMO

Part of EGMO

Subcontests

(6)

1

egmo 2018 p6

[*]Prove that for every real number

t

0 < t < \tfrac{1}{2}

there exists a positive integer

n

with the following property: for every set

S

n

positive integers there exist two different elements

x

y

S

, and a non-negative integer

m

m \ge 0

|x-my|\leq ty.

[*]Determine whether for every real number

t

0 < t < \tfrac{1}{2}

there exists an infinite set

S

of positive integers such that

|x-my| > ty

for every pair of different elements

x

y

S

and every positive integer

m

m > 0

1

egmo 2018 p5

\Gamma

be the circumcircle of triangle

ABC

\Omega

is tangent to the line segment

AB

and is tangent to

\Gamma

at a point lying on the same side of the line

AB

C

. The angle bisector of

\angle BCA

\Omega

at two different points

P

Q

\angle ABP = \angle QBC

1

egmo 2018 p4

1 \times 2

2 \times 1

n \ge 3

be an integer. Dominoes are placed on an

n \times n

board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some

k \ge 1

such that each row and each column has a value of

k

. Prove that a balanced configuration exists for every

n \ge 3

, and find the minimum number of dominoes needed in such a configuration.

1

NT from EGMO 2018

Consider the set

A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.

[*]Prove that every integer

x \geq 2

can be written as the product of one or more elements of

A

, which are not necessarily different.
[*]For every integer

x \geq 2

f(x)

denote the minimum integer such that

x

can be written as the product of

f(x)

A

, which are not necessarily different. Prove that there exist infinitely many pairs

(x,y)

of integers with

x\geq 2

y \geq 2

f(xy)<f(x)+f(y).

(x_1,y_1)

(x_2,y_2)

are different if

x_1 \neq x_2

y_1 \neq y_2

1

Geometry from EGMO 2018

ABC

be a triangle with

CA=CB

\angle{ACB}=120^\circ

M

be the midpoint of

AB

P

be a variable point of the circumcircle of

ABC

Q

be the point on the segment

CP

QP = 2QC

. It is given that the line through

P

and perpendicular to

AB

intersects the line

MQ

at a unique point

N

. Prove that there exists a fixed circle such that

N

lies on this circle for all possible positions of

P

1

Combinatorics from EGMO 2018

n

contestant of EGMO are named

C_1, C_2, \cdots C_n

. After the competition, they queue in front of the restaurant according to the following rules.
[*]The Jury chooses the initial order of the contestants in the queue. [*]Every minute, the Jury chooses an integer

i

1 \leq i \leq n

.
[*]If contestant

C_i

i

other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly

i

positions. [*]If contestant

C_i

i

other contestants in front of her, the restaurant opens and process ends.

[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices. [*]Determine for every

n

the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.