MathDB

Problem 3

Part of 2012 VJIMC

Problems(2)

inequality, sum 1/(x+1)=1

Source: VJIMC 2012 1.3

5/31/2021
Determine the smallest real number CC such that the inequality xyz⋅1x+1+yzx⋅1y+1+zxy⋅1x+1≤C\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le Cholds for all positive real numbers x,yx,y and zz with 1x+1+1y+1+1z+1=1\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1.
inequalities
ring is commutative, either x^2=1 or x^n=0

Source: VJIMC 2012 2.3

5/31/2021
Let (A,+,⋅)(A,+,\cdot) be a ring with unity, having the following property: for all x∈Ax\in A either x2=1x^2=1 or xn=0x^n=0 for some n∈Nn\in\mathbb N. Show that AA is a commutative ring.
Ring Theoryabstract algebragroup theory