Olympiad problems

2016 District Olympiad, 3

Tags: function , continuity , real analysis , algebra , functional equation

Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$

1958 February Putnam, B7

Tags: continuity , integral

Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and $$\int_{a}^{b} x^n f(x) \, dx =0$$ for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$

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 . $

2013 ELMO Shortlist, 7

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

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]

2021 Alibaba Global Math Competition, 7

Tags: convergence , continuity , sobolev space

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 $H^s(\mathbb{R})$.

2022 SEEMOUS, 2

Tags: analysis , functional equation , continuity , palic , differentiation , integration

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

2000 Romania National Olympiad, 1

Tags: calculus , continuity , integral , limit , real analysis , integration

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

1987 Traian Lălescu, 2.2

Tags: real analysis , darboux s property , darboux , continuity , pathological , function

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $ [b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property. [b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $

2016 Mathematical Talent Reward Programme, MCQ: P 12

Tags: continuity , function

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]

2003 Romania National Olympiad, 3

Tags: limit , function , continuity , real analysis

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

2007 Gheorghe Vranceanu, 3

Tags: real analysis , find all functions , continuity , ftc , function

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $

2023 District Olympiad, P3

Tags: function , continuity , real analysis

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.

1985 Traian Lălescu, 2.1

Tags: real analysis , continuity , function

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. $

1970 Putnam, B5

Tags: function , continuity

Let $u_n$ denote the ramp function $$ u_n (x) =\begin{cases} -n \;\; \text{for} \;\; x \leq -n, \\ \; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\ \;n \;\; \; \text{for} \;\; n \leq x, \end{cases}$$ and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$

2004 Alexandru Myller, 4

Tags: real analysis , darboux s property , darboux , monotone , continuity , function

Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous. [i]Mihai Piticari[/i]

2023 ISI Entrance UGB, 8

Tags: calculus , continuity , differentiability , function

Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.

2018 District Olympiad, 4

Tags: real analysis , continuity , function

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 $(a, b)$.

2018 District Olympiad, 1

Tags: continuity , supremum , function

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 $\sup\{I(f) \mid f \in \mathcal{F}\}$.

2015 District Olympiad, 3

Tags: function , real analysis , continuity , monotone functions , find all functions

Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality: $$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$

2013 Bogdan Stan, 1

Tags: real analysis , side-continuity , continuity , limit function , local analysis

Let be a real function that admits finite right-limits everywhere. Prove that the function that maps every real number to its right-limit is right-continuous everywhere. [i]Tolosi Marin[/i]

2023 District Olympiad, P1

Tags: algebra , functional equation , continuity

Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

2013 ELMO Problems, 6

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

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, 2.4

Tags: function , real analysis , continuity

Prove that, if a continuous function has limits at $ \pm\infty , $ and these are equal, then it touches its maximum or minimum at one point.

2007 Gheorghe Vranceanu, 2

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

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} . $

2011 Laurențiu Duican, 3

Tags: real analysis , function , 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]