MathDB

1

Part of 2016 VJIMC

Problems(2)

e^(f'(xi))*f(0)^(f(xi)) = f(1)^(f(xi))

Source: 26th annual VJIMC (2016), Category I, Problem 1

4/10/2016
Let f:R→(0,∞)f: \mathbb{R} \to (0, \infty) be a continuously differentiable function. Prove that there exists ξ∈(0,1)\xi \in (0,1) such that ef′(ξ)⋅f(0)f(ξ)=f(1)f(ξ)e^{f'(\xi)} \cdot f(0)^{f(\xi)} = f(1)^{f(\xi)}
functionreal analysis
cyclic product of (1/a + 1/bc) >= 1728

Source: 26th annual VJIMC (2016), Category II, Problem 1

4/10/2016
Let a,b,ca,b,c be positive real numbers such that a+b+c=1a + b + c = 1. Show that
(1a+1bc)(1b+1ca)(1c+1ab)≥1728\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728
Inequalityinequalitiescollege contests