Olympiad problems

1995 Turkey Team Selection Test, 3

Tags: limit , algebra

The sequence $\{x_n\}$ of real numbers is defined by \[x_1=1 \quad\text{and}\quad x_{n+1}=x_n+\sqrt[3]{x_n} \quad\text{for}\quad n\geq 1.\] Show that there exist real numbers $a, b$ such that $\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1$.

2000 District Olympiad (Hunedoara), 4

Tags: limit , function , real analysis , calculus , integration

Let $ f:[0,1]\longrightarrow\mathbb{R}_+^* $ be a Riemann-integrable function. Calculate $ \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) . $

2014 Contests, 4

Tags: limit , real analysis , number theory , analytic number theory

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

1996 China Team Selection Test, 2

Tags: function , limit , induction , strong induction , algebra

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2007 Today's Calculation Of Integral, 186

Tags: calculus , integration , limit , calculus computations

For $a>0,$ find $\lim_{a\to\infty}a^{-\left(\frac{3}{2}+n\right) }\int_{0}^{a}x^{n}\sqrt{1+x}\ dx\ (n=1,\ 2,\ \cdots).$

2005 Today's Calculation Of Integral, 62

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

For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$. Evaluate \[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]

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 LIMIT Category 2, 11

Tags: limit , geometry

$\triangle PQR$ is isosceles and right angled at $R$. Point $A$ is inside $\triangle PQR$, such that $PA=11, QA=7$, and $RA=6$. Legs $\overline{PR}$ and $\overline{QR}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

2025 District Olympiad, P1

Tags: real analysis , limit

Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$ [i]Mathematical Gazette[/i]

2007 Princeton University Math Competition, 7

Tags: limit , linear algebra , matrix , algebra , polynomial , vector

Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$, \[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\] find $\lim_{n \to \infty} \frac{x_n}{y_n}$.

2007 Vietnam National Olympiad, 2

Tags: function , limit , algebra

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

2005 Vietnam National Olympiad, 3

Tags: limit , induction , algebra

Let $\{x_n\}$ be a real sequence defined by: \[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\] For all $n=1,2,3...$ and a is a real number. Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.

1985 IMO Longlists, 17

Tags: limit , algebra

Set \[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\] Find $\lim_{n\to\infty} A_n.$

1990 IMO Longlists, 66

Tags: function , limit , trigonometry , algebra

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that \[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]

1963 Miklós Schweitzer, 6

Tags: function , integration , limit , real analysis , inequalities

Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx <\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]

2019 Jozsef Wildt International Math Competition, W. 53

Tags: limit , summation , calculus

Compute $$\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}$$

1972 Miklós Schweitzer, 3

Tags: advanced fields , limit , calculus , integration

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

2012 Today's Calculation Of Integral, 832

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

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

1976 IMO Longlists, 32

Tags: function , limit , random walk , martingale , combinatorics

We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.

2000 VJIMC, Problem 3

Tags: limit , real analysis

Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact $$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?

2002 Moldova National Olympiad, 1

Tags: limit

The sequence $ (a_n)$ is defined by $ a_1\in (0,1)$ and $ a_{n\plus{}1}\equal{}a_n(1\minus{}a_n)$ for $ n\ge 1$. Prove that $ \lim_{n\rightarrow \infty} na_n\equal{}1$

2006 Grigore Moisil Urziceni, 1

Tags: limit , sequence

[b]a)[/b] $ \lim_{n\to\infty } \sum_{j=1}^n\frac{n}{n^2+n+j} =1 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n- \sum_{j=1}^n\frac{n^2}{n^2+n+j} \right) =3/2 $ [i]Cristinel Mortici[/i]

2005 Gheorghe Vranceanu, 4

Tags: real analysis , convergence , limit

Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $ [b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent. [b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $

1980 IMO, 4

Tags: limit , real analysis

Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]