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Part of 2010 Polish MO Finals

Problems(2)

Polish MO Final 2010, 1st problem (remainders in the set)

Source:

The integer number

n > 1

is given and a set

S \subset \{0, 1, 2, \ldots, n-1\}

|S| > \frac{3}{4} n

. Prove that there exist integer numbers

a, b, c

such that the remainders after the division by

n

of the numbers:

a, b, c, a+b, b+c, c+a, a+b+c

S

pigeonhole principleceiling functioncombinatorics proposedcombinatorics

Polish MO Final 2010, 4th problem (perimeters' inequality)

Source:

BC

of the triangle

ABC

there are two points

D

E

BD < BE

p_1

p_2

the perimeters of triangles

ABC

ADE

respectively. Prove that

p_1 > p_2 + 2\cdot \min\{BD, EC\}.

inequalitiesgeometryperimeterparallelogramgeometry proposed