MathDB

3

Part of 1987 Greece National Olympiad

Problems(3)

\sqrt{a+x}+ \sqrt{a-x}>a

Source: 1987 Greece MO Grade XI p3

9/6/2024
Solve for real values of parameter aa, the inequality : a+x+a−x>a,  x∈R\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}
parameterizationinequalitiesalgebra
min (a-1)(a-3)(a-4)(a-6)+10 1987 Greece MO Grade X p3

Source:

9/6/2024
Prova that for any real aa, expresssion A=(a−1)(a−3)(a−4)(a−6)+10A=(a-1)(a-3)(a-4)(a-6)+10 is always positive. What is the minimum value that expression AA can take and for which values of aa?
algebrainequalities
x_n+x_{k}=x_{nk}, strictly increasing with terms naturals

Source: 1987 Greece MO Grade XII p3

9/6/2024
There is no sequence xnx_n strictly increasing with terms natural numbers such that : xn+xk=xnk,  for any   n,k∈N∗ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*
algebraSequencerecurrence relation