4

Part of 2016 European Mathematical Cup

Problems(2)

European Mathematical Cup 2016 problem 4 senior division

Source:

C_{1}

C_{2}

be circles intersecting in

X

Y

A

D

C_{1}

B

C

C_2

A

X

C

are collinear and

D

X

B

are collinear. The tangent to circle

C_{1}

D

BC

and the tangent to

C_{2}

B

P

R

respectively. The tangent to

C_2

C

AD

C_1

A

Q

S

respectively. Let

W

be the intersection of

AD

with the tangent to

C_{2}

B

Z

the intersection of

BC

with the tangent to

C_1

A

. Prove that the circumcircles of triangles

YWZ

RSY

PQY

have two points in common, or are tangent in the same point.
Proposed by Misiakos Panagiotis

European Mathematical Cup 2016 junior division problem 4

Source:

We will call a pair of positive integers

(n, k)

k > 1

lovely

couple

if there exists a table

nxn

consisting of ones and zeros with following properties: • In every row there are exactly

k

ones. • For each two rows there is exactly one column such that on both intersections of that column with the mentioned rows, number one is written. Solve the following subproblems: a) Let

d \neq 1

be a divisor of

n

. Determine all remainders that

d

can give when divided by

6

. b) Prove that there exist infinitely many lovely couples.
Proposed by Miroslav Marinov, Daniel Atanasov