2015 EGMO

Part of EGMO

Subcontests

(6)

1

interesting combinatorics EGMO P5

m, n

be positive integers with

m > 1

. Anastasia partitions the integers

1, 2, \dots , 2m

m

pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to

n

1

reflection with circumcentre EGMO P6

H

be the orthocentre and

G

be the centroid of acute-angled triangle

ABC

AB\ne AC

AG

intersects the circumcircle of

ABC

A

P

P'

be the reflection of

P

BC

\angle CAB = 60

HG = GP'

1

EGMO P4 infinite sequence

Determine whether there exists an infinite sequence

a_1, a_2, a_3, \dots

of positive integers which satisfies the equality

a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}}

for every positive integer

n

1

[EGMO3] GCD Bounds

n, m

be integers greater than

1

a_1, a_2, \dots, a_m

be positive integers not greater than

n^m

. Prove that there exist positive integers

b_1, b_2, \dots, b_m

not greater than

n

\gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n,

\gcd(x_1, x_2, \dots, x_m)

denotes the greatest common divisor of

x_1, x_2, \dots, x_m

1

[EGMO2] Domino tilings

2 \times 1

1 \times 2

tile. Determine in how many ways exactly

n^2

dominoes can be placed without overlapping on a

2n \times 2n

chessboard so that every

2 \times 2

square contains at least two uncovered unit squares which lie in the same row or column.

1

EGMO 2015, Problem 1

\triangle ABC

be an acute-angled triangle, and let

D

be the foot of the altitude from

C.

The angle bisector of

\angle ABC

CD

E

and meets the circumcircle

\omega

\triangle ADE

F.

\angle ADF = 45^{\circ}

CF

\omega .