MathDB

Concyclic points

Source: Romania TST 1 2010, Problem 4

8/25/2012

Two circles in the plane,

\gamma_1

and

\gamma_2

, meet at points

M

and

N

. Let

A

be a point on

\gamma_1

, and let

D

be a point on

\gamma_2

. The lines

AM

and

AN

meet again

\gamma_2

at points

B

and

C

, respectively, and the lines

DM

and

DN

meet again

\gamma_1

at points

E

and

F

, respectively. Assume the order

M

,

N

,

F

,

A

,

E

is circular around

\gamma_1

, and the segments

AB

and

DE

are congruent. Prove that the points

A

,

F

,

C

and

D

lie on a circle whose centre does not depend on the position of the points

A

and

D

on the respective circles, subject to the assumptions above.
***

geometrycircumcircleparallelogramromania

Lattice points inside a convex body

Source: Romania TST 2 2010, Problem 4

8/25/2012

Let

n

be an integer number greater than or equal to

2

, and let

K

be a closed convex set of area greater than or equal to

n

, contained in the open square

(0, n) \times (0, n)

. Prove that

K

contains some point of the integral lattice

\mathbb{Z} \times \mathbb{Z}

.
Marius Cavachi

geometrycalculusintegrationalgebrafunctiondomainsimilar triangles

Additive combinatorics (re Cauchy-Davenport)

Source: Romania TST 3 2010, Problem 4

8/25/2012

Let

X

and

Y

be two finite subsets of the half-open interval

[0, 1)

such that

0 \in X \cap Y

and

x + y = 1

for no

x \in X

and no

y \in Y

. Prove that the set

\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}

has at least

|X| + |Y| - 1

elements.
***

floor functioncombinatorics proposedcombinatoricsCauchy-Davenport theorem

4

Problems(3)