Olympiad problems

1996 IMC, 7

Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if $$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$

2015 BMT Spring, 9

Tags: real analysis , limit

Find $$\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).$$

2005 Miklós Schweitzer, 11

Tags: real analysis , homogeneous function , positive definite , linear algebra

Let $E: R^n \backslash \{0\} \to R^+$ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and $p \in R^n \backslash \{0\}$ , $E (\lambda p) = \lambda^2 E (p)$). Prove that if the second derivative of $E''(p): R^n \times R^n \to R$ is a non-degenerate bilinear form at any point $p \in R^n \backslash \{0\}$, then $E''(p)$ ($p \in R^n \backslash \{0\}$) is positive definite.

2013 ISI Entrance Examination, 3

Tags: real analysis , inequalities , function

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

2010 Romania National Olympiad, 4

Tags: real analysis , logarithm , limit

Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

2012 Grigore Moisil Intercounty, 3

Tags: real analysis , limit , riemann sum , definite integral , claculus

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

2010 Romania National Olympiad, 3

Tags: function , real analysis

Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.

1969 Miklós Schweitzer, 3

Tags: real analysis , abstract algebra , group theory , function

Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\] [i]A. Mate[/i]

2010 Romania National Olympiad, 1

Tags: limit , calculus , derivative , real analysis , function , integration

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

2016 Miklós Schweitzer, 5

Tags: real analysis , algebra , density , combinatorics

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

2012 Romania National Olympiad, 1

Tags: real analysis , function , integration

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

2017 CIIM, Problem 2

Tags: college math , real analysis

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.

1949 Miklós Schweitzer, 2

Tags: trigonometry , limit , real analysis , integration

Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .

2010 Today's Calculation Of Integral, 621

Tags: calculus , logarithm , limit , real analysis , calculus computations , integration

Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

1977 Miklós Schweitzer, 9

Tags: function , real analysis , vector , integration

Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase. [i]E. Gesztelyi[/i]

1995 Miklós Schweitzer, 3

Tags: real analysis

Denote $\langle x\rangle$ the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers $a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n$ such that $$f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle$$ for every x iff there is a k such that $$\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)$$ is constant.

2012 Romania National Olympiad, 4

Tags: inequalities , calculus , ratio , real analysis , integration , function

[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]

2011 Miklós Schweitzer, 8

Tags: real analysis

Given a nonzero real number $a\leq 1/e$, let $z_1, ..., z_n \in C$ be non-real numbers for which $ze^z + a = 0$ holds, and let $c_1, ..., c_n \in C$ be arbitrary. Show that the function $f(x)=Re(\sum_{j=1}^n c_j e^{z_j x})$ ($x \in R$) has a zero in every closed interval of length 1.

2006 Cezar Ivănescu, 3

Tags: function , real analysis , ftc , newton-leibniz

[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $ [b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $

1968 Miklós Schweitzer, 5

Tags: real analysis , inequalities

Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\] [i]P. Turan[/i]

1969 Miklós Schweitzer, 8

Tags: function , real analysis , integration

Let $ f$ and $ g$ be continuous positive functions defined on the interval $ [0, +\infty)$, and let $ E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $ E \times E$ by the relation \[ F(x,y)= %Error. "dispalymath" is a bad command. \int_0^xf(t)dt+ %Error. "dispalymath" is a bad command. \int_0^y g(t)dt\] has a nonvoid interior. [i]L. Losonczi[/i]

2020 Miklós Schweitzer, 2

Tags: real analysis , analysis , function

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$.

1951 Miklós Schweitzer, 1

Tags: real analysis

Choose terms of the harmonic series so that the sum of the chosen terms be finite. Prove that the sequence of these terms is of density zero in the sequence $ 1,\frac12,\frac13,\dots,\frac1n,\dots$

2004 Alexandru Myller, 4

Tags: limit , analysis , real analysis , definite integral , factorial , function

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

1994 IMC, 2

Tags: real analysis

Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$. a) Prove that $f$ attains its minimum and its maximum. b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or local minimum or maximum.