2019 EGMO

Part of EGMO

Subcontests

(6)

1

two sequences of positive integers and inequalities

n\ge 2

be an integer, and let

a_1, a_2, \cdots , a_n

be positive integers. Show that there exist positive integers

b_1, b_2, \cdots, b_n

satisfying the following three conditions:

\text{(A)} \ a_i\le b_i

i=1, 2, \cdots , n;

\text{(B)} \

the remainders of

b_1, b_2, \cdots, b_n

n

are pairwise different; and

\text{(C)} \

b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)

\lfloor x \rfloor

denotes the integer part of real number

x

, that is, the largest integer that does not exceed

x

1

Alina draws chords

On a circle, Alina draws

2019

chords, the endpoints of which are all different. A point is considered marked if it is either

\text{(i)}

4038

endpoints of a chord; or

\text{(ii)}

an intersection point of at least two chords.
Alina labels each marked point. Of the

4038

points meeting criterion

\text{(i)}

2019

0

2019

1

. She labels each point meeting criterion

\text{(ii)}

with an arbitrary integer (not necessarily positive). Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with

k

marked points has

k-1

such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. Alina finds that the

N + 1

yellow labels take each value

0, 1, . . . , N

exactly once. Show that at least one blue label is a multiple of

3

. (A chord is a line segment joining two different points on a circle.)

1

more incircles and tangents

ABC

be a triangle with incentre

I

. The circle through

B

AI

I

AB

P

. The circle through

C

AI

I

AC

Q

PQ

is tangent to the incircle of

ABC.

1

Concurrent angle bisectors

ABC

be a triangle such that

\angle CAB > \angle ABC

I

be its incentre. Let

D

be the point on segment

BC

\angle CAD = \angle ABC

\omega

be the circle tangent to

AC

A

and passing through

I

X

be the second point of intersection of

\omega

and the circumcircle of

ABC

. Prove that the angle bisectors of

\angle DAB

\angle CXB

intersect at a point on line

BC

1

Yet another domino problem

n

be a positive integer. Dominoes are placed on a

2n \times 2n

board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each

n

, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size

2 \times 1

1 \times 2

. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

1

Real triples

Find all triples

(a, b, c)

of real numbers such that

ab + bc + ca = 1

a^2b + c = b^2c + a = c^2a + b.