▪ Pre – Vietnam Mathematical Olympiad 2012

1η Ημέρα – 26/11/2011

1: Let a sequence defined by:

Denote and . Prove that , have finite limit.

2: Let be a set of finite distinct positive real numbers. Two other sets , are defined by:

. Prove that .

3: Two circles and intersect at and . Take two points on and , respectively, such that . The line intersects and respectively at . Let respectively be the centers of the two arcs and (which don’t contains ). Prove that is a cyclic quadrilateral.

4: For a table ( rows and columns), determine the maximum of that we can write one number in the set in each cell such that these conditions are satisfied:

a. Each row contains enough numbers of the set .

b. Any two rows are distinct.

c. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

2η Ημέρα – 26/11/2011

5: Determine all values of satisfied the following condition: there’s exist a cyclic of such that is a complete residue systems modulo .

6: Find all function such that

for all .

7: There are students. Denoted the number of the selections to select two students (with their weights are and ) such that (kg) and (kg) by and , respectively. Prove that .