MathDB

1

Part of 2007 South East Mathematical Olympiad

Problems(2)

cubic equation

Source: China south east mathematical Olympiad 2007 problem1

7/12/2013
Determine the number of real number aa, such that for every aa, equation x3=ax+a+1x^3=ax+a+1 has a root x0x_0 satisfying following conditions: (a) x0x_0 is an even integer; (b) ∣x0∣<1000|x_0|<1000.
algebra unsolvedalgebra
function f(x)

Source: China south east mathematical Olympiad 2007 problem 5

7/12/2013
Let f(x)f(x) be a function satisfying f(x+1)−f(x)=2x+1(x∈R)f(x+1)-f(x)=2x+1 (x \in \mathbb{R}).In addition, ∣f(x)∣≤1|f(x)|\le 1 holds for x∈[0,1]x\in [0,1]. Prove that ∣f(x)∣≤2+x2|f(x)|\le 2+x^2 holds for x∈Rx \in \mathbb{R}.
functioninequalitiesalgebra unsolvedalgebra