MathDB

2

Part of 2010 Czech-Polish-Slovak Match

Problems(2)

Sum of the distances to 60 points on a unit circle ≤ 80

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Given any 6060 points on a circle of radius 11, prove that there is a point on the circle the sum of whose distances to these 6060 points is at most 8080.
geometry unsolvedgeometry
Minimum of cyclic Σ ( x^2 + x/(y^2+z+1) )

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Let xx, yy, zz be positive real numbers satisfying x+y+z≥6x+y+z\ge 6. Find, with proof, the minimum value of x2+y2+z2+xy2+z+1+yz2+x+1+zx2+y+1. x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}.
inequalitiesinequalities unsolved