1

Part of 2008 Cuba MO

Problems(2)

numbers 1-8 in 9x9 board

Source: 2008 Cuba MO 1.1

We place the numbers from

1

81

9\times

board. Prove that exist

k \in \{1,2,...,9\}

so that the product of the numbers in the

k

-th column is diferent to the product of the numbers in the

k

S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2 >=ra^2

Source: 2008 Cuba 2.1

Given a polynomial of degree

2

p(x) = ax^2 +bx+c

define the function

S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.

Determine the real number

r

such that, for any polynomial

p(x)

2

with real roots, holds

S(p) \ge ra^2

algebrainequalitiestrinomial