MathDB

1

Part of 1994 China Team Selection Test

Problems(2)

5n numbers inequality

Source: China TST 1994, problem 4

5/17/2005
Given 5n5n real numbers ri,si,ti,ui,vi≥1(1≤i≤n)r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n), let R=1n∑i=1nriR = \frac {1}{n} \sum_{i=1}^{n} r_i, S=1n∑i=1nsiS = \frac {1}{n} \sum_{i=1}^{n} s_i, T=1n∑i=1ntiT = \frac {1}{n} \sum_{i=1}^{n} t_i, U=1n∑i=1nuiU = \frac {1}{n} \sum_{i=1}^{n} u_i, V=1n∑i=1nviV = \frac {1}{n} \sum_{i=1}^{n} v_i. Prove that ∏i=1nrisitiuivi+1risitiuivi−1≥(RSTUV+1RSTUV−1)n\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n.
inequalitieslogarithmsfunctionalgebradomainlinear algebramatrix
all sets comprising of 4 natural numbers

Source: China TST 1994, problem 1

5/17/2005
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
number theory unsolvednumber theory