Olympiad problems

2018 District Olympiad, 4

Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.

2011 Laurențiu Duican, 3

Tags: function , real analysis , continuity , fractional part

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

1985 Traian Lălescu, 2.1

Tags: function , real analysis , continuity

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a bounded function in some neighbourhood of $ 0, $ such that there are three real numbers $ a>0, b>1, c $ with the property that $$ f(ax)=bf(x)+c,\quad\forall x\in\mathbb{R} . $$ Show that $ f $ is continuous at $ 0 $ if and only if $ c=0. $

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.

2013 ELMO Shortlist, 7

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

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]

2016 Mathematical Talent Reward Programme, MCQ: P 12

Tags: function , continuity

Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are [list=1] [*] 0 [*] 1 [*] 2 [*] More than 2 [/list]

1986 Traian Lălescu, 2.3

Tags: limit , calculus , real analysis , continuity

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

2007 Gheorghe Vranceanu, 2

Tags: function , darboux , real analysis , continuity , primitivability

Let be areal number $ r, $ a nonconstant and continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with period $ T $ and $ F $ be its primitive having $ F(0)=0. $ Define the funtion $ g:\mathbb{R}\longrightarrow\mathbb{R} $ as $$ g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right. $$ Prove that: [b]a)[/b] the image of $ f $ is closed. [b]b)[/b] $ g $ has the intermediate value property if and only if $ r\in f\left(\mathbb{R}\right) . $ [b]c)[/b] $ g $ is primitivable if and only if $ r=\frac{F(T)}{T} . $

2013 ELMO Problems, 6

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]

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]