Olympiad problems

1983 Putnam, A6

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

2019 Centers of Excellency of Suceava, 3

Tags: function , real analysis , differentiability

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]

2024 District Olympiad, P2

Tags: integral , derivative , real analysis

Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

2017 CIIM, Problem 2

Tags: college math , real analysis

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.

1998 VJIMC, Problem 2

Tags: limit , real analysis

Find the limit $$\lim_{n\to\infty}\left(\frac{\left(1+\frac1n\right)^n}e\right)^n.$$

2020 Miklós Schweitzer, 2

Tags: analysis , function , real analysis

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$.

2000 IMC, 5

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

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

2010 Today's Calculation Of Integral, 547

Tags: calculus , integration , function , derivative , real analysis , calculus computations , logarithm

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2006 Moldova MO 11-12, 2

Tags: function , limit , real analysis

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

2021 Miklós Schweitzer, 4

Tags: real analysis

Let $I$ be a nonempty open subinterval of the set of positive real numbers. For which even $n \in \mathbb{N}$ are there injective function $f: I \to \mathbb{R}$ and positive function $p: I \to \mathbb{R}$, such that for all $x_1 , \ldots , x_n \in I$, \[ f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)} \] holds?

2004 Gheorghe Vranceanu, 2

Tags: function , divt , real analysis , primitivability

Let be two real numbers $ a<b, $ a nonempty and non-maximal subset $ K $ of the interval $ (a,b) $ and three functions $$ f:(a,b)\longrightarrow\mathbb{R}, g,h:\mathbb{R}\longrightarrow\mathbb{R} $$ satisfying the following relations. $ \text{(i)} g $ and $ h $ are primitivable. $ \text{(ii)} g-h $ hasn't any root in $ (a,b). $ $ \text{(iii)} $ The restrictions of $ f $ at $ K $ and $ (a,b)\setminus K $ are equal to $ g,h, $ respectively. Prove that $ f $ is not primitivable.

2009 Today's Calculation Of Integral, 456

Tags: calculus , integration , limit , trigonometry , real analysis , calculus computations , logarithm

Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$

2016 VJIMC, 4

Tags: real analysis , infinite sum

Find the value of sum $\sum_{n=1}^\infty A_n$, where $$A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.$$

2007 IMS, 8

Tags: function , topology , real analysis

Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.

2011 Romania National Olympiad, 3

Tags: function , real analysis

[color=darkred]Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ . [/color]

1999 Romania National Olympiad, 3

Tags: real analysis , mean value theorem

Let $a,b \in \mathbb{R},$ $a<b$ and $f,g:[a,b] \to \mathbb{R}$ two differentiable functions with increasing derivatives and $f'(a)>0,$ $g'(a)>0.$ Prove that there exists $c \in [a,b]$ such that $$\frac{f(b)-f(a)}{b-a} \cdot \frac{g(b)-g(a)}{b-a}=f'(c)g'(c).$$

2006 Miklós Schweitzer, 6

Tags: geometric series , limit , real analysis

Let G (n) = max | A(n) |, where A(n) ranges over all subsets of {1,2,...,n} and contains no three-member geometric series, ie, there is no $x, y, z \in A$ such that x < y < z and xz = y^2. Prove that $\lim_{n \to \infty} \frac{G (n)}{n}$ exists.

2009 National Olympiad First Round, 14

Tags: real analysis , function , number theory , relatively prime

For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ? $\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$

2014 Paenza, 6

Tags: function , integration , real analysis

(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then: \[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\] (b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then: \[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]

2014 Cezar Ivănescu, 3

Tags: function , real analysis

Find the real numbers $ \lambda $ that have the property that there is a nonconstant, continuous function $ u: [0,1]\longrightarrow\mathbb{R} $ satisfying $$ u(x)=\lambda\int_0^1 (x-3y)u(y)dy , $$ for any $ x $ in the interval $ [0,1]. $

2003 District Olympiad, 4

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

Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $ f $ has a finite limit at $ \infty . $ Show that: $$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$

Kvant 2020, M365

Tags: real analysis , convergence , divergence

[list=a] [*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one? [*]The same question as before, for numbers not exceeding one. [*]Can it happen that the series $a_1+a_2+\cdots$ converges, but the series $a_1^3+a_2^3+\cdots$ diverges? [/list]

2019 VJIMC, 4

Tags: algebra , inequalities , real analysis

Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]

2003 IMC, 6

Tags: algebra , polynomial , real analysis

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

2006 Victor Vâlcovici, 1

Tags: monotone , function , real analysis , ftc , definite integral , calculus , integration

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]