Olympiad problems

2008 District Olympiad, 4

Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$.

2023 SEEMOUS, P4

Tags: real analysis , convergence , sequence

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$. [list=i] [*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$. [*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\] [/list]

2000 Miklós Schweitzer, 8

Tags: function , real analysis

Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.

1997 Romania National Olympiad, 3

Tags: function , real analysis , differentiable function

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$ a) Prove that there exist nonconstant functions in $\mathcal{F}.$ b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

2018 Putnam, A5

Tags: taylor series , real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.

2008 Romania National Olympiad, 2

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

Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

2017 Romania National Olympiad, 4

Tags: function , asymptote , real analysis

A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $ [b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $ [b]b)[/b] Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.

2020 IMC, 8

Tags: real analysis , logarithm , binomial coefficient

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

2011 VTRMC, Problem 5

Tags: real analysis , limit

Find $\lim_{x\to\infty}\left((2x)^{1+\frac1{2x}}-x^{1+\frac1x}-x\right)$.

2018 Korea USCM, 6

Tags: inequalities , real analysis , derivative , function

Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

2005 Romania National Olympiad, 2

Tags: function , trigonometry , algebra , domain , limit , real analysis

Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

2005 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , function , derivative , real analysis

Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

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 $x \in I.$

2019 Teodor Topan, 2

Tags: arithmetic sequence , ratio , real analysis , seqeunce , newton s binom , cesaro-stolz , squeeze theorem

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression with $ a_1=1 $ and natural ratio. [b]a)[/b] Prove that $$ a_n^{1/a_k} <1+\sqrt{\frac{2\left( a_n-1 \right)}{a_k\left( a_k -1 \right)}} , $$ for any natural numbers $ 2\le k\le n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{1}{a_n}\sum_{k=1}^n a_n^{1/a_k} . $ [i]Nicolae Bourbăcuț[/i]

2017 Brazil Undergrad MO, 4

Tags: real analysis , sequence

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.

2024 SEEMOUS, P1

Tags: real analysis

Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.

2010 Contests, 1

Tags: integration , real analysis

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

1963 Miklós Schweitzer, 8

Tags: trigonometry , function , integration , search , real analysis

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

2013 Bogdan Stan, 1

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

2010 IMC, 3

Tags: limit , function , real analysis

Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.

2007 Today's Calculation Of Integral, 195

Tags: calculus , integration , function , real analysis , algebra , partial fractions , calculus computations

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

2016 Korea USCM, 2

Tags: real analysis , sequence , series

Suppose $\{a_n\}$ is a decreasing sequence of reals and $\lim\limits_{n\to\infty} a_n = 0$. If $S_{2^k} - 2^k a_{2^k} \leq 1$ for any positive integer $k$, show that $$\sum_{n=1}^{\infty} a_n \leq 1$$ (At here, $S_m = \sum_{n=1}^m a_n$ is a partial sum of $\{a_n\}$.)

2018 IMC, 1

Tags: real analysis , sequence

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent: [list=1] [*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*] [*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*] [/list] [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

1950 Miklós Schweitzer, 1

Tags: real analysis

Let $ a>0$, $ d>0$ and put $ f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots$ Give a closed form for $ f(x)$.

1995 IMC, 12

Tags: function , real analysis

Suppose that $(f_{n})_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that $$\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if}\;n=m\\ 0 & \text{if} \;n\ne m \end{cases}$$ and $\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty$. Show that there exists no subsequence $(f_{n_{k}})$ of $(f_{n})$ such that $\lim_{k\to \infty}f_{n_{k}}(x)$ exist for all $x\in [0,1]$.