2

Part of 2003 Federal Competition For Advanced Students, Part 2

Problems(2)

the minimum value of [(a^3+b^3+c^3)/abc]^2

Source: Austrian Mathematical Olympiad 2003, Part 2, D1, P2

a, b, c

be nonzero real numbers for which there exist

\alpha, \beta, \gamma \in\{-1, 1\}

with \alpha a + \beta b + \gamma c = 0. What is the smallest possible value of

\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?

geometrygeometric transformationrotationinequalities unsolvedinequalities

In how many ways can one tile the rectangle?

Source: Austrian Mathematical Olympiad 2003, Part 2, D2, P2

We are given sufficiently many stones of the forms of a rectangle

2\times 1

1\times 1

n > 3

be a natural number. In how many ways can one tile a rectangle

3 \times n

using these stones, so that no two

2 \times 1

rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?

geometryrectanglecombinatorics unsolvedcombinatorics