Olympiad problems

1998 VJIMC, Problem 3

Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.

1986 Traian Lălescu, 1.3

Tags: continuity , real analysis , fractional part

Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.

Kvant 2020, M413

Tags: real analysis , function , continuity

Determine the positive numbers $a{}$ for which the following statement true: for any function $f:[0,1]\to\mathbb{R}$ which is continuous at each point of this interval and for which $f(0)=f(1)=0$, the equation $f(x+a)-f(x)=0$ has at least one solution. [i]Proposed by I. Yaglom[/i]

2018 District Olympiad, 1

Tags: function , continuity , supremum

Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$, \[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\] a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$. b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.

2021 Alibaba Global Math Competition, 7

Tags: convergence , sobolev space , continuity

A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that \[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\] Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.

2013 ELMO Shortlist, 7

Tags: function , induction , algebra , 3n-3 theorem , continuity

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2024 Romania National Olympiad, 4

Tags: function , real analysis , continuity

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$ a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin. b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.

2023 District Olympiad, P3

Tags: real analysis , continuity , function

Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.

2011 N.N. Mihăileanu Individual, 3

Tags: function , real analysis , continuity

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function having the property that $$ f(f(x))=f(x)-\frac{1}{4}x +1, $$ for all real numbers $ x. $ [b]a)[/b] Prove that $ f $ is increasing. [b]b)[/b] Show that the equation $ f(x)=ax $ has at least a real solution in $ x, $ for any real number $ a\ge 1. $ [b]c)[/b] Calculate $ \lim_{x\to\infty } \frac{f(x)}{x} $ supposing that it exists, it's finite, and that $ \lim_{x\to\infty } f(f(x))=\infty . $

2012 District Olympiad, 4

Tags: function , real analysis , continuity , monotone

A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $ [b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $ [b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.