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

Tags were heavily modified to better represent problems.

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Found problems: 884

2007 IMS, 3
Tags: search , real analysis
Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.

1957 Miklós Schweitzer, 3
Tags: real analysis , topology
[b]3.[/b] Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$, the subset $A$ contains the mid point of the line segment beteween $x$ and $y$. Show that $A$ consists of a convex open set and of some of its boundary points. [b](St. 1)[/b]

1999 VJIMC, Problem 2
Tags: limit , real analysis , function
Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

2008 Romania National Olympiad, 2
Tags: calculus , geometry , real analysis , inequalities , derivative , function , integration
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 \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

2007 Today's Calculation Of Integral, 195
Tags: calculus computations , integration , algebra , real analysis , partial fractions , calculus , function
Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

2025 District Olympiad, P3
Tags: real analysis , function
[list=a] [*] Let $a<b$ and $f:[a,b]\rightarrow\mathbb{R}$ be a strictly monotonous function such that $\int_a^b f(x) dx=0$. Show that $f(a)\cdot f(b)<0$. [*] Find all convergent sequences $(a_n)_{n\geq 1}$ for which there exists a scrictly monotonous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{a_{n-1}}^{a_n} f(x)dx = \int_{a_n}^{a_{n+1}} f(x)dx,\text{ for all }n\geq 2.$$

2007 Miklós Schweitzer, 1
Tags: real analysis
Prove that there exist subfields of $\mathbb R$ that are a) non-measurable and b) of measure zero and continuum cardinality. (translated by Miklós Maróti)

2020 Jozsef Wildt International Math Competition, W10
Tags: real analysis , sequence , limit
Let there be $(a_n)_{n\ge1},(b_n)_{n\ge1},a_n,b_n\in\mathbb R^*_+=(0,\infty)$ such that $\lim_{n\to\infty}a_n=a\in\mathbb R^*_+$ and $(b_n)_{n\ge1}$ is a bounded sequence. If $(x_n)_{n\ge1}$, $x_n=\prod_{k=1}^n(ka_h+b_h)$ find: $$\lim_{n\to\infty}\left(\sqrt[n+1]{x_{n+1}}-\sqrt[n]{x_n}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

2010 Laurențiu Panaitopol, Tulcea, 3
Tags: calculus , function , real analysis , derivative , monotone
Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that: $ \text{(i) supp} f''=f\left(\mathbb{R}\right) $ $ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $ Prove that: [b]a)[/b] any such $ g $ is injective. [b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $ $$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$ [i]Laurențiu Panaitopol[/i]

1998 IMC, 4
Tags: real analysis , calculus , derivative , function
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$. Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.

2005 Brazil Undergrad MO, 2
Tags: function , integration , real analysis , induction , inequalities
Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

2007 Today's Calculation Of Integral, 184
Tags: function , calculus , real analysis , integration , calculus computations , logarithm , inequalities
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

2011 N.N. Mihăileanu Individual, 2
Tags: matrix , linear algebra , real analysis
Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that $$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$ for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $ [i]Nelu Chichirim[/i]

ICMC 6, 6
Tags: real analysis , convergence , floor function , sequence
Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

KoMaL A Problems 2017/2018, A. 723
Tags: real analysis
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that the limit $$g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}$$ exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.

1998 Miklós Schweitzer, 9
Tags: real analysis
Let G be a domain (connected open set) in the $R^2$ plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc $v_q$ in which $q\in v_q$ and $v_q\setminus\{q\}\subset G$. other questions: (i) Show that local connectedness cannot be replaced by connectedness. (ii) Show that if we replace the $R^2$ plane with $R^3$ space, the statement does not hold.

2002 District Olympiad, 4
Tags: function , real analysis
Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: 1. $f$ has one-side limits in any $a\in \mathbb{R}$ and $f(a-0)\le f(a)\le f(a+0)$. 2. for any $a,b\in \mathbb{R},\ a<b$, we have $f(a-0)<f(b-0)$. Prove that $f$ is strictly increasing. [i]Mihai Piticari & Sorin Radulescu[/i]

2012 District Olympiad, 4
Tags: function , real analysis , continuity , monotone
A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $ [b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $ [b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.

2007 Mathematics for Its Sake, 1
Tags: number theory , floor function , real analysis , sequence , parity
Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

2011 Miklós Schweitzer, 3
Tags: real analysis , algebraic combinatorics , euclidean space
In $R^d$ , all $n^d$ points of an n × n × ··· × n cube grid are contained in 2n - 3 hyperplanes. Prove that n ($n\geq3$) hyperplanes can be chosen from these so that they contain all points of the grid.

2012 Pre-Preparation Course Examination, 3
Tags: integration , real analysis , function
Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$. Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?

2002 IMC, 12
Tags: inequalities , gradient , real analysis
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$ exists at every point of $\mathbb{R}^{n}$ and satisfies the condition $$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$ Prove that $$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$

1979 Spain Mathematical Olympiad, 8
Tags: calculus , derivative , real analysis
Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$ Express the numerical value of its derivative of order $325$ for $x = 0$.

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.

1995 VJIMC, Problem 4
Tags: limit , real analysis , trigonometry
Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.