MathDB

4.3

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

x_n <\1/\sqrt{n! H_n}

Source: 2011 Saudi Arabia Pre-TST November 4.3

1/1/2022
Let n≥2n \ge 2 be a positive integer and let xnx_n be a positive real root to the equation x(x+1)...(x+n)=1x(x+1)...(x + n) = 1. Prove that xn<1n!Hnx_n <\frac{1}{\sqrt{n! H_n}} where Hn=1+12+...+1nH_n = 1+\frac12+...+\frac{1}{n}.
algebrainequalities
x_1x_2...x_n &gt;= (n -1)^n if sum 1/(1+x_k)=1

Source: 2011 Saudi Arabia Pre-TST February 4.3

1/1/2022
Let x1,x2,...,xnx_1,x_2,...,x_n be positive real numbers for which 11+x1+11+x2+...+11+xn=1\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1 Prove that x1x2...xn≥(n−1)nx_1x_2...x_n \ge (n -1)^n.
algebrainequalities