Olympiad problems

1982 IMO Longlists, 53

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

2019 Centers of Excellency of Suceava, 2

Tags: limit , convergence , real analyssi , real analysis , sequence

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]

2012 Today's Calculation Of Integral, 818

Tags: calculus , integration , function , limit , absolute value , logarithm

For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

1966 Swedish Mathematical Competition, 1

Tags: limit , algebra , sequence

Let $\{x\}$ denote the fractional part of $x$, $x - [x]$. The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$. Is it true that $\lim \{x_n + y_n\} = 0$? $\lim \{x_n - y_n\} = 0$?

1972 Putnam, A3

Tags: limit , sequence

A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists. Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.

2002 Tuymaada Olympiad, 1

Tags: limit , number theory

A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite. [i]Proposed by A. Golovanov[/i]

2024 Olimphíada, 3

Tags: algebra , limit , bounded , unbounded

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$ for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?

1998 Turkey MO (2nd round), 3

Tags: limit , combinatorics

Some of the vertices of unit squares of an $n\times n$ chessboard are colored so that any $k\times k$ ( $1\le k\le n$) square consisting of these unit squares has a colored point on at least one of its sides. Let $l(n)$ denote the minimum number of colored points required to satisfy this condition. Prove that $\underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}$.

2006 Switzerland Team Selection Test, 3

Tags: function , domain , limit , algebra

Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.

2020 LIMIT Category 2, 1

Tags: function , limit , involution , counting

Find the number of $f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}$ such that $f(f(x))=x$ (A)$26$ (B)$41$ (C)$120$ (D)$60$

2011 Laurențiu Duican, 2

Tags: limit , integration , real analysis , squeeze theorem , definite integral , calculus

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

2023 CIIM, 6

Tags: limit , real analysis , inequalities , algebra

Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$

2010 Today's Calculation Of Integral, 625

Tags: calculus , integration , limit , trigonometry , calculus computations

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2005 Today's Calculation Of Integral, 42

Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

2007 Brazil National Olympiad, 1

Tags: quadratic , induction , algebra , polynomial , limit , quadratic formula

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

2018 VJIMC, 4

Tags: function , functional equation , decreasing , limit

Determine all possible (finite or infinite) values of \[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\] if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying \[f(f(x))^4-f(f(x))+f(x)=1\] for all $x \in \mathbb{R}$.

2009 Today's Calculation Of Integral, 430

Tags: calculus , integration , trigonometry , limit , calculus computations

For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

2020 LIMIT Category 2, 17

Tags: limit , triangle , angle , calculus

Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ (A) $\pi/2$ (B) $\pi/4$ (C) $\pi $ (D) $\pi/3$

1986 IMO Longlists, 73

Tags: limit , algebra

Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$

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]