2

Part of 2013 Turkey MO (2nd round)

Problems(2)

Source: Turkey National Olympiad Second Round 2013 P2

m

be a positive integer. a. Show that there exist infinitely many positive integers

k

1+km^3

is a perfect cube and

1+kn^3

is not a perfect cube for all positive integers

n<m

m=p^r

p \equiv 2 \pmod 3

is a prime number and

r

is a positive integer. Find all numbers

k

satisfying the condition in part a.

Find the maximum value of M

Source: Turkey National Olympiad Second Round 2013 P5

Find the maximum value of

M

for which for all positive real numbers

a, b, c

a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc)

inequalities proposedinequalities