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

1951 Miklós Schweitzer, 1
Tags: real analysis
Choose terms of the harmonic series so that the sum of the chosen terms be finite. Prove that the sequence of these terms is of density zero in the sequence $ 1,\frac12,\frac13,\dots,\frac1n,\dots$

2009 Harvard-MIT Mathematics Tournament, 5
Tags: calculus , limit , trigonometry , derivative , function , real analysis
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]

1993 Putnam, B4
Tags: real analysis
$K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \subseteq [0, 1]$. $\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\int_{0}^{1} g(y) K(x, y) dy = f(x)$ for all $x \subseteq [0, 1]$. Show that $f = g$ on $[0, 1]$.

2015 District Olympiad, 2
Tags: integral , real analysis , calculus , integration
[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

2009 District Olympiad, 1
Tags: function , inequalities , integral , real analysis
Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality: $$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations: [b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $ [b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $

2012 District Olympiad, 1
Tags: limit , real analysis , sequence
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

2020 Jozsef Wildt International Math Competition, W38
Tags: limit , real analysis , sequence
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

2019 Romania National Olympiad, 3
Tags: real analysis , calculus , differentiability
$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$ $\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

2001 Miklós Schweitzer, 7
Tags: function , real analysis
Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.

1995 IMC, 8
Tags: summation , real analysis
Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$

2006 Romania National Olympiad, 4
Tags: integration , function , calculus , derivative , trigonometry , real analysis
Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

2007 Romania National Olympiad, 2
Tags: integration , calculus , real analysis
Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

1971 Miklós Schweitzer, 10
Tags: function , real analysis
Let $ \{\phi_n(x) \}$ be a sequence of functions belonging to $ L^2(0,1)$ and having norm less that $ 1$ such that for any subsequence $ \{\phi_{n_k}(x) \}$ the measure of the set \[ \{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}\] tends to $ 0$ as $ y$ and $ N$ tend to infinity. Prove that $ \phi_n$ tends to $ 0$ weakly in the function space $ L^2(0,1).$ [i]F. Moricz[/i]

1971 Miklós Schweitzer, 9
Tags: function , real analysis
Given a positive, monotone function $ F(x)$ on $ (0, \infty)$ such that $ F(x)/x$ is monotone nondecreasing and $ F(x)/x^{1+d}$ is monotone nonincreasing for some positive $ d$, let $ \lambda_n >0$ and $ a_n \geq 0 , \;n \geq 1$. Prove that if \[ \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,\] or \[ \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,\] then $ \sum_{n=1}^ {\infty} a_n$ is convergent. [i]L. Leindler[/i]

1971 Miklós Schweitzer, 6
Tags: function , limit , integration , real analysis
Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}\] exists and is finite. [i]I. Gyori[/i]

2007 Miklós Schweitzer, 8
Tags: real analysis
For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent? (translated by Miklós Maróti)

2015 District Olympiad, 1
Tags: function , real analysis
Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $ [b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective. [b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.

2008 District Olympiad, 3
Tags: real analysis
Let $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ a sequence of positive real numbers, such that: \[x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*\] a) Prove that the sequences $(x_n+y_n)_{n\ge 1}$ and $(x_ny_n)_{n\ge 1}$ have limit. b) Prove that the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ have limit and that their limits are equal.

2025 SEEMOUS, P4
Tags: real analysis , series , sequence
Let $(a_n)_{n\geq 1}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $\sum_{n=1}^{\infty}\frac{a_n}{n}$ is convergent if and only if the sequence $(a_n\ln n)_{n\geq 1}$ is bounded and $\sum_{n=1}^{\infty} (a_n-a_{n+1})\ln n$ is convergent.

1995 IMC, 11
Tags: function , real analysis
a) Prove that every function of the form $$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$ with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)$. b) Prove that the function $$F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)$$ has at least $40$ zeroes in the interval $(0,1000)$.

2007 Nicolae Coculescu, 2
Tags: limit , function , real analysis , sequence
Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined by $ f(x)=\frac{x}{1+e^x} , $ and let be a sequence $ \left( x_n \right)_{n\ge 0} $ such that $ x_0>0 $ and defined as $ x_n=F\left( x_{n-1} \right) . $ Calculate $ \lim_{n\to\infty } \frac{1}{n}\sum_{k=1}^n \frac{x_k}{\sqrt{x_{k+1}}} $ [i]Florian Dumitrel[/i]

2023 Brazil Undergrad MO, 6
Tags: algebra , real analysis , sequence
Determine all pairs $(c, d) \in \mathbb{R}^2$ of real constants such that there is a sequence $(a_n)_{n\geq1}$ of positive real numbers such that, for all $n \geq 1$, $$a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .$$

2010 Contests, 4
Tags: function , limit , real analysis
A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

2007 Romania National Olympiad, 4
Tags: function , calculus , derivative , symmetry , ratio , real analysis
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

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 $na_n \geq d, \forall n \geq 1$.