3

Part of 2008 Brazil National Olympiad

Problems(2)

Find the minimum (try not to use Calculus)

Source: Brazilian Math Olympiad 2008, Problem 3

x,y,z

real numbers such that x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx. Find the minimum value of {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}

calculusinequalitiestrigonometryinequalities unsolved

Count prophetic words!

Source: Brazilian Math Olympiad, Problem 6

The venusian prophet Zabruberson sent to his pupils a

10000

-letter word, each letter being

A

E

: the Zabrubic word. Their pupils consider then that for

1 \leq k \leq 10000

, each word comprised of

k

consecutive letters of the Zabrubic word is a prophetic word of length

k

. It is known that there are at most

7

prophetic words of lenght

3

. Find the maximum number of prophetic words of length

10

combinatorics unsolvedcombinatorics