MathDB

1

Part of 2019 Centers of Excellency of Suceava

Problems(3)

Four term - inequality

Source: 2019 contest

7/6/2020
For a,b,c,d a,b,c,d positive, prove: 2aa2+bc+2bb2+cd+2cc2+da+2dd2+ab≤1a+1b+1c+1d \frac{2a}{a^2+bc} +\frac{2b}{b^2+cd} +\frac{2c}{c^2+da} +\frac{2d}{d^2+ab}\le \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d}
Dan Popescu
inequalititesinequalities
Combinatoric AM-GM

Source:

7/7/2020
Prove that (m+nmin⁡(m,n))≤(2mm)⋅(2nn), \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , for nonnegative m,n. m,n.
Gheorghe Stoica
inequalitiesAM-GM
sum of four squares prime

Source:

7/7/2020
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two.
Gherghe Stoica
number theoryprime numbers