This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 35

2017 Romania National Olympiad, 1

Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.

2019 Teodor Topan, 3

Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that $$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$ [i]Vlad Mihaly[/i]

2012 District Olympiad, 4

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.

2024 Romania National Olympiad, 4

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.

Kvant 2020, M413

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]

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.

1959 Putnam, B3

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

2013 ELMO Shortlist, 7

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]

1986 Traian Lălescu, 1.3

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

1986 Traian Lălescu, 2.3

Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $