2016 Gulf Math Olympiad

Part of Gulf Math Olympiad

Subcontests

(4)

1

tennis doubles

4. Suppose that four people A, B, C and D decide to play games of tennis doubles. They might first play the team A and B against the team C and D. Next A and C might play B and D. Finally A and D might play B and C. The advantage of this arrangement is that two conditions are satisfied. (a) Each player is on the same team as each other player exactly once. (b) Each player is on the opposing team to each other player exactly twice. Is it possible to arrange a collection of tennis matches satisfying both condition (a) and condition (b) in the following circumstances? (i) There are five players. (ii) There are seven players. (iii) There are nine players.

1

triangle and orthocente

Consider the acute-angled triangle

ABC

X

be a point on the side

BC

Y

be a point on the side

CA

k_1

AX

AC

E'

k_2

BY

BC

B'

M

be the midpoint of

XY

A'M = B'M

. (ii) Suppose that

k_1

k_2

P

Q

. Prove that the orthocentre of

ABC

lies on the line

PQ

1

another number theory problem

x

be a real number that satisfies

x^1 + x^{-1} = 3

x^n + x^{-n}

is an positive integer , then prove that the positive integer

x^{3^{1437}}+x^{3^{-1437}}

is divisible by at least

1439 \times 2^{1437}

positive integers

1

GMO 2016 #1

Consider sequences

a_0

a_1

a_2

\cdots

of non-negative integers defined by selecting any

a_0

a_1

a_2

(not all 0) and for each

n

\geq

a_n

a_n-1

a_n-3

|
1-In the particular case that

a_0

a_1

a_2

= 2, calculate the beginning of the sequence, listing

a_0

a_1

\cdots

a_{19}

a_{20}

.
2-Prove that for each sequence, there is a constant

c

a_i

\leq

c

i

\geq

0. Note that the constant

c

my depend on the numbers

a_0

a_1

a_2

3-Prove that, for each choice of

a_0

a_1

a_2

, the resulting sequence is eventually periodic.
4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values

a_0

a_1

a_2

of the sequence