2

Part of 2009 Bundeswettbewerb Mathematik

Problems(2)

Maximum of m(a,b)

Source: Bundeswettbewerb Mathematik 2009 Round 1

a,b

be positive real numbers. Define

m(a,b)

as the minimum of $ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.

Find the maximum of

algebraalgebra unsolved

trinomial has 2 real roots such |x_2-x_1|><=1/n <=> n has 2 prime divisors

Source: Germany Federal - Bundeswettbewerb Mathematik 2009, round 2, p2

n

be an integer that is greater than

1

. Prove that the following two statements are equivalent: (A) There are positive integers

a, b

c

that are not greater than

n

and for which that polynomial

ax^2 + bx + c

has two different real roots

x_1

x_2

| x_2- x_1 | \le \frac{1}{n}

n

has at least two different prime divisors.

number theoryquadratic trinomialtrinomialprime divisorsprimeDivisors