MathDB

Problem 4-M

Part of 1998 VJIMC

Problems(2)

sum of sqrt(1-k^2/n^2) inequality

Source: VJIMC 1998 1.4-M

8/2/2021
Prove the inequality nπ4−18n≤12+∑k=1n−11−k2n2≤nπ4\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4for every integer n≥2n\ge2.
inequalitiesintegrationcalculusRiemann sum
Jensen with weaker condition?

Source: VJIMC 1998 2.4-M

8/24/2021
A function f:R→Rf:\mathbb R\to\mathbb R has the property that for every x,y∈Rx,y\in\mathbb R there exists a real number tt (depending on xx and yy) such that 0<t<10<t<1 and f(tx+(1−t)y)=tf(x)+(1−t)f(y).f(tx+(1-t)y)=tf(x)+(1-t)f(y). Does it imply that f(x+y2)=f(x)+f(y)2f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2 for every x,y∈Rx,y\in\mathbb R?
inequalitiesFunctional Analysisfunctionreal analysis