This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 884

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 $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

2008 Alexandru Myller, 3
Tags: integral , function , real analysis , inequalities , calculus , integration
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]

1996 IMC, 12
Tags: inequalities , real analysis
i) Prove that for every sequence $(a_{n})_{n\in \mathbb{N}}$, such that $a_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}a_{n}<\infty$, we have $$\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.$$ ii) Prove that for every $\epsilon>0$ there exists a sequence $(b_{n})_{n\in \mathbb{N}}$ such that $b_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}b_{n}<\infty$ and $$\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.$$

2023 USEMO, 2
Tags: real analysis , algebra
Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

2006 IMC, 6
Tags: function , algebra , polynomial , trigonometry , real analysis
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

2010 Laurențiu Panaitopol, Tulcea, 1
Tags: function , primitivableness , antiderivative , differentiability , integral , indefinite integral , real analysis
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]

2009 IMS, 5
Tags: calculus , integration , function , real analysis
Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.

2019 VJIMC, 4
Tags: complex analysis , real analysis , complex number
Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/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 $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.

2019 Teodor Topan, 3
Tags: limit , density , mod 1 , real analysis , convergence , function , sequence
Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

1999 Putnam, 5
Tags: polynomial , integration , calculus , inequalities , real analysis
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

2013 Romania National Olympiad, 4
Tags: function , inequalities , real analysis
a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$, for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$ b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$

2000 IMC, 2
Tags: function , real analysis
Let $f$ be continuous and nowhere monotone on $[0,1]$. Show that the set of points on which $f$ obtains a local minimum is dense.

2024 OMpD, 4
Tags: function , real analysis
Let \( n \) be a positive integer. Determine the largest possible value of \( k \) with the following property: there exists a bijective function \( \phi: [0, 1] \to [0, 1]^k \) and a constant \( C > 0 \) such that, for all \( x, y \in [0, 1] \), \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: \( \| \cdot \| \) denotes the Euclidean norm, that is, \( \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} \).

2007 District Olympiad, 1
Tags: limit , induction , real analysis
Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.

2016 Miklós Schweitzer, 5
Tags: real analysis , algebra , density , combinatorics
Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

2004 Gheorghe Vranceanu, 2
Tags: function , real analysis
Prove that there is exactly a function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{\ge 0} $ satisfying the following two properties: $ \text{(i)} x\in\mathbb{R}_{> 0}\implies \left( f(x)+f(f(x)) =4018020x \wedge f(x)>0 \right) $ $ \text{(ii)} 0=f(0)+f(f(0)) $

2021 Brazil Undergrad MO, Problem 2
Tags: real analysis , function
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.

1949 Miklós Schweitzer, 2
Tags: limit , integration , trigonometry , real analysis
Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .

2007 Romania National Olympiad, 2
Tags: function , algebra , polynomial , real analysis
Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$). Show that there is an interval $I$ such that $f(I)=[a,b]$.

1995 IMC, 6
Tags: inequalities , real analysis
Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$, we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

2006 Pre-Preparation Course Examination, 3
Tags: function , real analysis
Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$. Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.

1968 Miklós Schweitzer, 2
Tags: search , inequalities , real analysis
Let $ a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that \[ ( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).\] [i]J. Suranyi[/i]

2013 Romania National Olympiad, 1
Tags: function , integration , calculus , real analysis
Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .

2006 IberoAmerican Olympiad For University Students, 6
Tags: real analysis , algebra
Let $x_0(t)=1$, $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$; $y_{n,0}(t)=1$, $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$, $1\leq k \leq n$. Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$.