MathDB

5

Part of 2008 Hanoi Open Mathematics Competitions

Problems(2)

4x4 (absolute) inequalites imply a 5th inequality (HOMC 2008 J Q5)

Source:

7/25/2019
Suppose x,y,z,tx, y, z, t are real numbers such that {∣x+y+z−t∣≤1∣y+z+t−x∣≤1∣z+t+x−y∣≤1∣t+x+y−z∣≤1\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases} Prove that x2+y2+z2+t2≤1x^2 + y^2 + z^2 + t^2 \le 1.
inequalitiesalgebra
max P(x)- min P(x)= b-a, for all x\in [a,b] (HOMC 2008 Q5)

Source:

7/25/2019
Find all polynomials P(x)P(x) of degree 11 such that maxa≤x≤bP(x)−mina≤x≤bP(x)=b−a\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a , ∀a,b∈R\forall a,b\in R where a<ba < b
polynomialalgebra