MathDB

1

Part of 1988 Polish MO Finals

Problems(2)

inequality with reals in (0,1)

Source: Problem 1, Polish NO 1988

10/16/2005
The real numbers x1,x2,...,xnx_1, x_2, ... , x_n belong to the interval (0,1)(0,1) and satisfy x1+x2+...+xn=m+rx_1 + x_2 + ... + x_n = m + r, where mm is an integer and r∈[0,1)r \in [0,1). Show that x12+x22+...+xn2≤m+r2x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2.
inequalitiesfunctionalgebradomainanalytic geometryinequalities unsolved
continous function f: [0,d] to R

Source: Problem 4, Polish NO 1988

10/16/2005
dd is a positive integer and f:[0,d]→Rf : [0,d] \rightarrow \mathbb{R} is a continuous function with f(0)=f(d)f(0) = f(d). Show that there exists x∈[0,d−1]x \in [0,d-1] such that f(x)=f(x+1)f(x) = f(x+1).
functionalgebra unsolvedalgebra