MathDB

2

Part of 2013 Czech-Polish-Slovak Match

Problems(2)

x^n + 1/x^n - 2 > n^2(x + 1/x -2)

Source: Czech - Polish - Slovak Match 2013: P2

5/24/2014
Prove that for every real number x>0x>0 and each integer n>0n>0 we have xn+1xn−2≥n2(x+1x−2)x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)
inequalitiesinequalities proposed
triangular cells of an equilateral triangle are infected

Source: Czech - Polish - Slovak Match 2013: P5

5/29/2014
Triangular grid divides an equilateral triangle with sides of length nn into n2n^2 triangular cells as shown in figure for n=12n=12. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for n=12n=12, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]
geometryperimeterinequalitiescombinatorics unsolvedcombinatorics