2017 EGMO

Part of EGMO

Subcontests

(5)

1

Reflections of O, G form SEVEN concurrent circles

ABC

be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid

G

and the circumcentre

O

ABC

BC,CA,AB

G_1,G_2,G_3

O_1,O_2,O_3

, respectively. Show that the circumcircles of triangles

G_1G_2C

G_1G_3B

G_2G_3A

O_1O_2C

O_1O_3B

O_2O_3A

ABC

have a common point.
The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.

1

Expensive n-tuples

n\geq2

be an integer. An

n

(a_1,a_2,\dots,a_n)

of not necessarily different positive integers is expensive if there exists a positive integer

k

(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.

a) Find all integers

n\geq2

for which there exists an expensive

n

-tuple.
b) Prove that for every odd positive integer

m

there exists an integer

n\geq2

m

belongs to an expensive

n

-tuple.
There are exactly

n

factors in the product on the left hand side.

1

Colouring Related Function

Find the smallest positive integer

k

for which there exists a colouring of the positive integers

\mathbb{Z}_{>0}

k

colours and a function

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

with the following two properties:

(i)

For all positive integers

m,n

of the same colour,

f(m+n)=f(m)+f(n).

(ii)

There are positive integers

m,n

f(m+n)\ne f(m)+f(n).

In a colouring of

\mathbb{Z}_{>0}

k

colours, every integer is coloured in exactly one of the

k

colours. In both

(i)

(ii)

the positive integers

m,n

are not necessarily distinct.

2

Lines in Plane

2017

lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

Constructing graphs satisfying conditions on degrees

n\geq1

be an integer and let

t_1<t_2<\dots<t_n

be positive integers. In a group of

t_n+1

people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of

t_1,t_2,\dots,t_n

.
(ii) For every

i

1\leq i\leq n

, there is someone who has played exactly

t_i

games of chess.

1

Cyclic Points

ABCD

be a convex quadrilateral with

\angle DAB=\angle BCD=90^{\circ}

\angle ABC> \angle CDA

Q

R

be points on segments

BC

CD

, respectively, such that line

QR

intersects lines

AB

AD

P

S

, respectively. It is given that

PQ=RS

.Let the midpoint of

BD

M

and the midpoint of

QR

N

.Prove that the points

M,N,A

C

lie on a circle.