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: 837

1951 Miklós Schweitzer, 3
Tags: limit , real analysis
Consider the iterated sequence (1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$, where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).

1996 Turkey Team Selection Test, 3
Tags: limit , algebra
Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.

2007 District Olympiad, 3
Tags: calculus , integration , function , limit , inequalities , real analysis
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

2004 Harvard-MIT Mathematics Tournament, 3
Tags: calculus , limit
Find \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]

2020 LIMIT Category 1, 13
Tags: geometry , limit
On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.

2020 Jozsef Wildt International Math Competition, W51
Tags: real analysis , summation , limit
Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that $$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$ Show that: $$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

1954 Putnam, B5
Tags: function , limit , differentiability
Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$ Prove that $$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$

2005 Alexandru Myller, 3
Tags: function , limit , topology , real analysis
Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$. [i]Dorin Andrica, Eugen Paltanea[/i]

2020 LIMIT Category 1, 18
Tags: limit , geometry
Let $\triangle ABC$ be a right triangle with $\angle C=90^{\circ}$. Two squares $S_1$ and $S_2$ are inscribed in the triangle $ABC$ such that $S_1$ and $ABC$ share a common vertex $C$ and $S_2$ has one of its sides on $AB$. Suppose that $\text{Area}(S_1)=1+\text{Area}(S_2)=441$, then calculate $AC+BC$ (A)$400$ (B)$420$ (C)$441$ (D)$462$

2005 Today's Calculation Of Integral, 69
Tags: calculus , integration , trigonometry , limit , calculus computations
Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

2001 District Olympiad, 4
Tags: limit , induction , inequalities , real analysis
Prove that: a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic. b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that: \[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\] [i]Radu Gologan[/i]

2012 Pre - Vietnam Mathematical Olympiad, 2
Tags: limit , topology , algebra
Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$

2010 VJIMC, Problem 2
Tags: sequence , limit
Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.

2003 AIME Problems, 13
Tags: probability , counting , distinguishability , modular arithmetic , function , limit , vector
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2014 Bulgaria National Olympiad, 2
Tags: function , induction , limit , geometric series , algebra
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

1996 VJIMC, Problem 3
Tags: real analysis , number theory , limit
Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Put $a_1=1997^{1996^{1997}}$, and $a_{n+1}=\operatorname{cif}(a_n)$ for every $n>0$. Find $\lim_{n\to\infty}a_n$.

2013 Moldova Team Selection Test, 4
Tags: limit , algebra , logarithm
Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

2003 India IMO Training Camp, 7
Tags: algebra , polynomial , limit , number theory
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2003 Bulgaria National Olympiad, 3
Tags: polynomial , limit , induction , algebra
Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

2011 Today's Calculation Of Integral, 708
Tags: calculus , integration , limit , trigonometry , calculus computations
Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

2011 Today's Calculation Of Integral, 730
Tags: calculus , integration , limit , calculus computations , logarithm
Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2009 Today's Calculation Of Integral, 440
Tags: calculus , integration , limit , real analysis , calculus computations
For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$

2011 Today's Calculation Of Integral, 751
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

2008 Vietnam National Olympiad, 1
Tags: calculus , limit , algebra , logarithm
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$

2009 Today's Calculation Of Integral, 450
Tags: calculus , integration , limit , calculus computations , logarithm
Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$