2009 Romanian Master of Mathematics

Part of Romanian Masters of Mathematics Collection

Subcontests

(4)

1

There exists at least one finite set T of positive integers

For a finite set

X

of positive integers, let \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}. Given a finite set

S

of positive integers for which

\Sigma(S) < \frac{\pi}{2},

show that there exists at least one finite set

T

of positive integers for which

S \subset T

and \Sigma(S) \equal{} \frac{\pi}{2}.
Kevin Buzzard, United Kingdom

1

Prove that the four lines AiOi are concurrent or parallel

Given four points

A_1, A_2, A_3, A_4

in the plane, no three collinear, such that A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, denote by

O_i

the circumcenter of

\triangle A_j A_k A_l

with \{i,j,k,l\} \equal{} \{1,2,3,4\}. Assuming

\forall i A_i \neq O_i ,

prove that the four lines

A_iO_i

are concurrent or parallel.
Nikolai Ivanov Beluhov, Bulgaria

1

Number of points in space bounded from above

S

of points in space satisfies the property that all pairwise distances between points in

S

are distinct. Given that all points in

S

have integer coordinates

(x,y,z)

1 \leq x,y, z \leq n,

show that the number of points in

S

is less than \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).
Dan Schwarz, Romania

1

Multiple of multinomial coefficient is an integer

For a_i \in \mathbb{Z}^ \plus{}, i \equal{} 1, \ldots, k, and n \equal{} \sum^k_{i \equal{} 1} a_i, let d \equal{} \gcd(a_1, \ldots, a_k) denote the greatest common divisor of

a_1, \ldots, a_k

. Prove that \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)} is an integer.
Dan Schwarz, Romania