MathDB

3

Part of 2021 OMpD

Problems(2)

n-variable inequality with absolute values

Source: 2021 2nd OMpD L3 P3 - Brazil - Olimphída Matemáticos por Diversão

7/8/2023
Let aa and bb be positive real numbers, with a<ba < b and let nn be a positive integer. Prove that for all real numbers x1,x2,…,xn∈[a,b]x_1, x_2, \ldots , x_n \in [a, b]: ∣x1−x2∣+∣x2−x3∣+⋯+∣xn−1−xn∣+∣xn−x1∣≤2(b−a)b+a(x1+x2+⋯+xn) |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n) And determine for what values of nn and x1,x2,…,xnx_1, x_2, \ldots , x_n the equality holds.
inequalitiesalgebran-variable inequalityabsolute value
Simple diophantine equation

Source: 2021 2nd OMpD L2 P3 - Brazil - Olimp&iacute;ada Matem&aacute;ticos por Divers&atilde;o

7/8/2023
Determine all pairs of integer numbers (x,y)(x, y) such that: (x−y)2x+y=x−y+6\frac{(x - y)^2}{x + y} = x - y + 6
Diophantine equationnumber theory