Olympiad problems

2011 District Olympiad, 3

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function. [b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing. [b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.

1998 VJIMC, Problem 4-M

Tags: riemann sum , calculus , inequalities , integration

Prove the inequality $$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.

2006 Victor Vâlcovici, 2

Tags: derivative , lipschitz continuous , riemann sum , calculus , function , real analysis

Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that [b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $

2011 District Olympiad, 3

1998 VJIMC, Problem 4-M

2006 Victor Vâlcovici, 2

Today's calculation of integrals, 882

2012 Grigore Moisil Intercounty, 3

2001 SNSB Admission, 2

2013 Today's Calculation Of Integral, 882

2016 Korea USCM, 1

2019 Ramnicean Hope, 1