MathDB

4

Part of 2000 Tuymaada Olympiad

Problems(3)

Like vasc's one or cmo i don't remember

Source: Tuymaada 2000

1/17/2006
Prove for real x1x_1, x2x_2, ....., xnx_n, 0<xk≤120 < x_k \leq {1\over 2}, the inequality (nx1+⋯+xn−1)n≤(1x1−1)…(1xn−1). \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right).
inequalitieslogarithmsinequalities proposed
10^{-n} \neq \sum 1/a_i!

Source: Tuymaada 2000, day 2, problem 4.

4/30/2007
Prove that no number of the form 10−n10^{-n}, n≥1,n\geq 1, can be represented as the sum of reciprocals of factorials of different positive integers.
factorialnumber theory proposednumber theory
1/a(a+1)+ 1/b(b+1)+1/c(c+1) &gt;= 3/2 if abc=1 and a,b,c&gt;0

Source: Tuymaada 2000 Juniors 4

4/28/2019
Prove that if the product of positive numbers a,ba,b and cc equals one, then 1a(a+1)+1b(b+1)+1c(c+1)≥32\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}
algebrainequalitiesthree variable inequality