MathDB

2

Part of 2011 Bosnia Herzegovina Team Selection Test

Problems(2)

semicircle with diameter d

Source: Bosnia and Herzegovina 2011

5/16/2011
On semicircle, with diameter ∣AB∣=d|AB|=d, are given points CC and DD such that: ∣BC∣=∣CD∣=a|BC|=|CD|=a and ∣DA∣=b|DA|=b where a,b,da, b, d are different positive integers. Find minimum possible value of dd
geometrytrigonometrynumber theorynumber theory proposed
a+b+c=1, trivial

Source: Bosnia and Herzegovina 2011

5/16/2011
Let a,b,ca, b, c be positive reals such that a+b+c=1a+b+c=1. Prove that the inequality a1+b−c3+b1+c−a3+c1+a−b3≤1a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1 holds.
inequalitiesinequalities proposed