Olympiad problems

1988 Dutch Mathematical Olympiad, 2

Given is a number $a$ with 0 $\le \alpha \le \pi$. A sequence $c_0,c_1, c_2,...$ is defined as $$c_0=\cos \alpha$$ $$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...$$ Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$

1997 VJIMC, Problem 4-M

Tags: real analysis , limit , digit , number theory

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2002 India Regional Mathematical Olympiad, 6

Tags: inequalities , limit

Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]

2009 Romania National Olympiad, 1

Tags: real analysis , limit

Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by \[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\] and $x_0,y_0$ are given real numbers. a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit. b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.

1997 Canada National Olympiad, 3

Tags: induction , limit , floor function , inequalities

Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.

1970 Miklós Schweitzer, 10

Tags: real analysis , limit

Prove that for every $ \vartheta$, $ 0<\vartheta<1$, there exist a sequence $ \lambda_n$ of positive integers and a series $ \sum_{n=1}^{\infty} a_n$ such that (i)$ \lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}$, (ii) $ \lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}$ exists, (iii) $ \sum _{n=1}^{\infty} a_n$ is divergent. [i]P. Turan[/i]

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2001 VJIMC, Problem 3

Tags: function , limit , calculus , integration

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

2011 Today's Calculation Of Integral, 730

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

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 $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2020 LIMIT Category 1, 14

Tags: number theory , limit , algebra

Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.

1973 Poland - Second Round, 4

Tags: limit , algebra

Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $. Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.

2012 Waseda University Entrance Examination, 3

Tags: probability , limit , probability and stats

An unfair coin, which has the probability of $a\ \left(0<a<\frac 12\right)$ for showing $Heads$ and $1-a$ for showing $Tails$, is flipped $n\geq 2$ times. After $n$-th trial, denote by $A_n$ the event that heads are showing on at least two times and by$B_n$ the event that are not showing in the order of $tails\rightarrow heads$, until the trials $T_1,\ T_2,\ \cdots ,\ T_n$ will be finished . Answer the following questions: (1) Find the probabilities $P(A_n),\ P(B_n)$. (2) Find the probability $P(A_n\cap B_n )$. (3) Find the limit $\lim_{n\to\infty} \frac{P(A_n) P(B_n)}{P(A_n\cap B_n )}.$ You may use $\lim_{n\to\infty} nr^n=0\ (0<r<1).$

1974 Miklós Schweitzer, 6

Tags: logarithm , function , induction , limit , real analysis , integration

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

1990 IMO Longlists, 66

Tags: algebra , limit , trigonometry , function

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

2012 Bogdan Stan, 3

Tags: inequalities , floor , bernoulli s inequality , limit , limit of sequence , logarithm

$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $ [i]Taclit Daniela Nadia[/i]

2014 Contests, 2

Tags: algebra , function , limit , functional equation

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

2016 Mathematical Talent Reward Programme, MCQ: P 2

Tags: function , limit , summation , infinite sum

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]

2003 Brazil National Olympiad, 2

Tags: limit , algebra , function

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

1985 Iran MO (2nd round), 5

Tags: algebra , derivative , limit , calculus , function

Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying \[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\] Find the derivative of $f$ in an arbitrary point $x.$

2005 Vietnam National Olympiad, 3

Tags: algebra , induction , limit

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.

2020 LIMIT Category 1, 15

Tags: limit , chessboard , counting

In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?

2009 Today's Calculation Of Integral, 461

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

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

2005 Alexandru Myller, 3

Tags: limit , real analysis , topology , function

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]

1986 Federal Competition For Advanced Students, P2, 3

Tags: limit , algebra , polynomial

Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by: $ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$ contains infinitely many natural numbers.

1994 Polish MO Finals, 3

Tags: limit , algebra , function

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.