MathDB

2

Part of 1990 Polish MO Finals

Problems(2)

cyclic inequality with fractions

Source: Problem 2, Polish NO 1990

9/30/2005
Let x1,x2,...,xnx_1, x_2, . . . , x_n be positive numbers. Prove that ∑i=1nxi2xi2+xi+1xi+2≤n−1 \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 Where xn+1=x1x_{n+1}=x_1 and xn+2=x2x_{n+2}=x_2.
inequalitiesinequalities unsolved
Sequence and perfect squares

Source: Problem 5, Polish NO 1990

9/30/2005
Suppose that (an)(a_n) is a sequence of positive integers such that lim⁡n→∞nan=0\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0 Prove that there exists kk such that there are at least 19901990 perfect squares between a1+a2+...+aka_1 + a_2 + ... + a_k and a1+a2+...+ak+1a_1 + a_2 + ... + a_{k+1}.
limitnumber theory unsolvednumber theory