Olympiad problems

2019 LIMIT Category B, Problem 1

Tags: real analysis

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n\ge2$. Define $$p_n=\prod_{i=1}^n\left(1+\frac1{a_i}\right)$$Then $\lim_{n\to\infty}p_n$ is $\textbf{(A)}~1+e$ $\textbf{(B)}~e$ $\textbf{(C)}~1$ $\textbf{(D)}~\infty$

2016 Korea USCM, 7

Tags: holder inequality , integral , inequalities , real analysis , function

$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

2001 IMC, 6

Tags: function , real analysis

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R} $ satisfy \[ f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, \] \[\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,\] and \[\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).\] Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$.

2001 SNSB Admission, 3

Tags: positive definite , matrix , lebesgue integral , real analysis

Let be an $ n\times n $ positive-definite symmetric real matrix $ A. $ Prove the following equality. $$ \tiny\int_{\mathbb{R}^n} \exp\left( -\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}^\intercal A\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\right) dx_1dx_2\cdots dx_n=\normalsize\frac{\pi^{n/2}}{\sqrt{\det A} } $$

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]

2007 Grigore Moisil Intercounty, 4

Tags: inequalities , convergence , sequence , real analysis

Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.

2003 District Olympiad, 4

Tags: real analysis , function

Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that: a)$f$ has at least one point of discontinuity; b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity; c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity. [i]Radu Mortici [/i]

2013 Bogdan Stan, 4

Tags: seqeunce , cesaro-stolz , sequence , real analysis

Let be a sequence $ \left( x_n \right)_{n\ge 1} $ having the property that $$ \lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13. $$ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent and calculate its limit. [i]Cosmin Nițu[/i]

2011 District Olympiad, 4

Tags: inequalities , analytic geometry , slope , triangle inequality , real analysis , function , graphing lines

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

2019 SEEMOUS, 1

Tags: real analysis

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

2012 IFYM, Sozopol, 5

Tags: limit , real analysis

Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies \[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \] Prove that $\lim_{n\to \infty}c_n$ exists and find its value. [i]Proposed by Sadovnichy-Grigorian-Konyagin[/i]

2008 Moldova MO 11-12, 8

Tags: trigonometry , real analysis , logarithm , integration

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

1986 Traian Lălescu, 1.1

Tags: real analysis , limit

Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $

2020 CIIM, 6

Tags: combinatorics , algebra , real analysis

For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]

2008 Alexandru Myller, 3

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

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

2005 Grigore Moisil Urziceni, 2

Tags: real analysis , limit

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

2025 Romania National Olympiad, 3

Tags: derivative , function , calculus , real analysis

Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent: a) $f$ is differentiable, with continuous first derivative. b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.

1987 Traian Lălescu, 1.4

Tags: limit , sequence , real analysis

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

1950 Miklós Schweitzer, 9

Tags: real analysis , conic

Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.

2006 Miklós Schweitzer, 7

Tags: periodic , real analysis , algebra

Suppose that the function $f: Z \to Z$ can be written in the form $f = g_1+...+g_k$ , where $g_1,. . . , g_k: Z \to R$ are real-valued periodic functions, with period $a_1,...,a_k$. Does it follow that f can be written in the form $f = h_1 +. . + h_k$ , where $h_1,. . . , h_k: Z \to Z$ are periodic functions with integer values, also with period $a_1,...,a_k$?

1983 Miklós Schweitzer, 5

Tags: real analysis , limit , function , integration

Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists. [i]T. Krisztin[/i]

2019 Centers of Excellency of Suceava, 2

Tags: real analysis , limit , convergence , real analyssi , sequence

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]

1997 Miklós Schweitzer, 5

Tags: geometry , real analysis

Let $a_1>a_2>a_3>\cdots$ be a sequence of real numbers which converges to 0. We put circles of radius $a_1$ into a unit square until no more can fit. (A previously laid circle must not be moved.) Then we put circles of radius $a_2$ in the remaining space until no more can fit, continuing the process for $a_3$,... What can the area covered by the circles be? a similar problem involving circles in a square: [url]https://artofproblemsolving.com/community/c7h1979044[/url]

2003 IMC, 5

Tags: function , real analysis

a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$. b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.

2014 IMS, 2

Tags: real analysis , topology

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.