ENC: [imo-problems] IMO 2001 Problems, First Day

["Marcio" <mcohen@xxxxxxxxxx>]

Acabei de receber isso em outra lista.. Agora vou tentar fazer
(provavelmente sem sucesso :) e depois mandar meus comentarios pra gente
debater na lista!
Abracos,
Marcio

-----Mensagem original-----
De: Mojca Miklavec [mailto:mojca.miklavec@guest.arnes.si]
Enviada em: Domingo, 8 de Julho de 2001 15:41
Para: imo-problems@egroups.fr
Assunto: [imo-problems] IMO 2001 Problems, First Day



1. Let ABC be an acute-angled triangle with circumcentre O. Let P on BC be
the foot of the altitude from A.

Suppose that <)BCA >= <)ABC + 30°.

Prove that <)CAB + <)COP < 90°.


2. Prove that

a/sqrt(a^2+8bc) + b/sqrt(b^2+8ca) + c/sqrt(c^2+8ab) >= 1

for all positive real numbers a, b and c.


3. Twenty-one girls and twenty-one boys took part in a mathematical
contest.

- Each contestant solved at most six problems.
- For each girl and each boy, at leat one problem was solved by both of
them.

Prove that there was a problem that was solved by at least three girls and
at leat three boys.


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