Prove that for a,b,c>0, the following inequality is valid: (a+b+c3)2≥ab+bc+ca3.
Prove that amongst any 7 different numbers it is always possible to choose two of them, x and y, so that the following inequality was true: 0<x−y1+xy<13.
Is it true that if a is a positive number, then a2≥a? What about a2+1≥a?
Show for positive a and b that a2+b2≥2ab.
Is it true that if b is a positive number, then b3+b2≥b? What about b3+1≥b?
Show that if a is positive, then 1+a≥2a.
Let k be a natural number, prove the following inequality. 1k2>1k−1k+1.
Show that if a is a positive number, then a3+2≥2aa.
The numbers a, b and c are positive. By completing the square, show that a24+b2+c2≥ab−ac+2bc.
Let m and n be natural numbers such that m>n. Show that: 1n2+1(n+1)2+1(n+2)2+⋯+1m2>1n−1m.