MathDB

5

Part of 1996 IMO Shortlist

Problems(2)

Cubic polynomial with absolute value inequality

Source: IMO Shortlist 1996, A5

8/9/2008
Let P(x) P(x) be the real polynomial function, P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d. Prove that if ∣P(x)∣≤1 |P(x)| \leq 1 for all x x such that ∣x∣≤1, |x| \leq 1, then |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.
algebrapolynomialinequalitiesfunctionmaximizationIMO Shortlist
f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n)

Source: IMO Shortlist 1996, N5

8/9/2008
Show that there exists a bijective function f:N0→N0 f: \mathbb{N}_{0}\to \mathbb{N}_{0} such that for all m,n∈N0 m,n\in \mathbb{N}_{0}: f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n).
functionnumber theoryFunctional EquationsIMO Shortlist