Olympiad problems

1975 Miklós Schweitzer, 5

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

2022 Thailand TST, 2

Tags: real analysis , inequalities , algebra

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

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 $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

2017 VJIMC, 4

Tags: integration , real analysis , calculus

Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that \[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]

1973 Miklós Schweitzer, 6

Tags: integration , function , real analysis , search

If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.\] [i]Z. Daroczy[/i]

1994 IMC, 3

Tags: real analysis , derivative , calculus , function

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$

2010 Contests, 3

Tags: real analysis , trigonometry , limit

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

1983 Miklós Schweitzer, 3

Tags: real analysis

Let $ f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable, $ 2 \pi$-periodic even function. Prove that if \[ f''(x)\plus{}f(x)\equal{}\frac{1}{f(x\plus{} 3 \pi /2 )}\] holds for every $ x$, then $ f$ is $ \pi /2$-periodic. [i]Z. Szabo, J. Terjeki[/i]

2014 Romania National Olympiad, 3

Tags: function , real analysis

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ [b]a)[/b] Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ [b]b)[/b] Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

2006 District Olympiad, 1

Tags: function , real analysis , integration , calculus , inequalities

Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]

2021 SEEMOUS, Problem 4

Tags: convergence , real analysis

For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by \[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \] Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.

2000 IMC, 5

Tags: real analysis , function , domain , symmetry , algebra

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

2013 BMT Spring, 3

Tags: limit , real analysis

Evaluate $$\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}$$

2005 Unirea, 4

Tags: function , parameterization , inequalities , integration , real analysis

$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable suppose $f(a)=f(-a)$ $ f'(-a)=f'(a)=a^2$ Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$ Study equality case ? Radu Miculescu

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} . $

2020 Miklós Schweitzer, 7

Tags: analysis , real analysis

Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\] Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.

2010 IMC, 2

Tags: integration , infinite series , logarithm , real analysis

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

2007 Gheorghe Vranceanu, 1

Tags: function , group theory , real analysis , primitive

Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group. [b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.

2014 Putnam, 2

Tags: function , absolute value , real analysis , integration

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2009 District Olympiad, 3

Tags: real analysis , limit , induction

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

2023 Miklós Schweitzer, 9

Tags: polynomial , analysis , functional analysis , real analysis

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

1993 Putnam, A5

Tags: real analysis

Let U be the set formed as the union of three open intervals, $U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)$. Show that $\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx$ is rational.

2000 District Olympiad (Hunedoara), 3

Tags: real analysis , function , wilkosz , primitive

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that: $ \text{(i)}\quad f(0)=0 $ $ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $ $ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $ Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$ is primitivable.

2024 Romania National Olympiad, 1

Tags: function , real analysis

Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$

1999 Putnam, 5

Tags: integration , inequalities , polynomial , real analysis , calculus

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]