Olympiad problems

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: differentiability , primitivableness , antiderivative , integral , indefinite integral , real analysis , function

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having 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. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

2008 Grigore Moisil Intercounty, 1

Tags: differentiability , function , real analysis

Find the differentiable functions $ f:\mathbb{R}\longrightarrow (-\infty ,1) $ with the property $ f(1)=-1 $ and $$ f(x+y)=f(x)+f(y)-f(x)f(y) , $$ for any reals $ x,y. $ [i]Vasile Pop[/i]

2019 Centers of Excellency of Suceava, 3

Tags: differentiability , real analysis , function

For two real intervals $ I,J, $ we say that two functions $ f,g:I\longrightarrow J $ have property $ \mathcal{P} $ if they are differentiable and $ (fg)'=f'g'. $ [b]a)[/b] Provide example of two nonconstant functions $ a,b:\mathbb{R}\longrightarrow\mathbb{R} $ that have property $ \mathcal{P} . $ [b]b)[/b] Find the functions $ \lambda :(2019,\infty )\longrightarrow (0,\infty ) $ having the property that $ \lambda $ along with $ \theta :(2019,\infty )\longrightarrow (0,\infty ), \theta (x)=x^{2019} $ have property $ \mathcal{P} . $ [i]Dan Nedeianu[/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 $|f(1) - f(0)| < |f'(t)|$.

1954 Putnam, B5

Tags: differentiability , limit , function

Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$ Prove that $$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$

2019 Romania National Olympiad, 3

Tags: real analysis , differentiability , calculus

$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$ $\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

2016 Romania National Olympiad, 4

Tags: real analysis , function , find all functions , differentiability , darboux

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

1998 IMC, 6

Tags: differentiability , real analysis

$f: (0,1) \rightarrow [0, \infty)$ is zero except at a countable set of points $a_{1}, a_2, a_3, ... $ . Let $b_n = f(a_n)$. Show that if $\sum b_{n}$ converges, then $f$ is differentiable at at least one point. Show that for any sequence $b_{n}$ of non-negative reals with $\sum b_{n} =\infty$ , we can find a sequence $a_{n}$ such that the function $f$ defined as above is nowhere differentiable.

2007 Mathematics for Its Sake, 1

Tags: real analysis , function , differentiability , extremal

Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\prod_{j=1}^n (x-j)^j, $$ where $ n $ is a natural number.