Olympiad problems

2006 Pre-Preparation Course Examination, 6

Tags: limit , algebra

Suppose that $P_c(z)=z^2+c$. You are familiar with the Mandelbrot set: $M=\{c\in \mathbb{C} | \lim_{n\rightarrow \infty}P_c^n(0)\neq \infty\}$. We know that if $c\in M$ then the points of the dynamical system $(\mathbb{C},P_c)$ that don't converge to $\infty$ are connected and otherwise they are completely disconnected. By seeing the properties of periodic points of $P_c$ prove the following ones: a) Prove the existance of the heart like shape in the Mandelbrot set. b) Prove the existance of the large circle next to the heart like shape in the Mandelbrot set. [img]http://astronomy.swin.edu.au/~pbourke/fractals/mandelbrot/mandel1.gif[/img]

2002 Austrian-Polish Competition, 8

Tags: algebra , limit , trigonometry , inequalities

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

2013 BMT Spring, 10

Tags: limit , calculus

Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, $$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$

Today's calculation of integrals, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

1995 VJIMC, Problem 4

Tags: limit , real analysis , trigonometry

Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.

2006 ISI B.Stat Entrance Exam, 8

Tags: algebra , limit , logarithm

Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that \[\frac{\log_2x}{x}=c\] has only one real solution.

1999 Putnam, 1

Tags: trigonometry , limit

Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.

2013 China Team Selection Test, 2

Tags: limit , number theory , modular arithmetic , quadratic

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

1980 All Soviet Union Mathematical Olympiad, 303

Tags: limit , irrational , sequence , digit , algebra

The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

2005 Today's Calculation Of Integral, 56

Tags: limit , integration , calculus computations , calculus

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

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. $

2006 Pre-Preparation Course Examination, 1

Tags: invariant , probability , real analysis , limit , vector , topology , advanced fields

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

2012 ISI Entrance Examination, 2

Tags: limit , derivative , calculus , calculus computations , function

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

1984 All Soviet Union Mathematical Olympiad, 389

Tags: recurrence relation , algebra , sequence , limit

Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.

2020 Jozsef Wildt International Math Competition, W21

Tags: real analysis , limit

Evaluate $$\lim_{n\to\infty}\left(\frac{1+\frac13+\ldots+\frac1{2n+1}}{\ln\sqrt n}\right)^{\ln\sqrt n}$$ [i]Proposed by Ángel Plaza[/i]

2012 Serbia Team Selection Test, 2

Tags: limit , floor function , inequalities , number theory

Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

1966 Miklós Schweitzer, 9

Tags: real analysis , limit

If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

2006 Pre-Preparation Course Examination, 7

Tags: limit , combinatorics , logarithm , function

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

2004 District Olympiad, 1

Tags: real analysis , limit

Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.

2001 IMC, 3

Tags: limit , real analysis , integration

Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

2010 Today's Calculation Of Integral, 568

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

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

1994 Putnam, 4

Tags: matrix , induction , limit , linear algebra

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

2020 LIMIT Category 2, 7

Tags: limit , geometry

A circle $\mathfrak{D}$ is drawn through the vertices $A$ and $B$ of $\triangle ABC$. If $\mathfrak{D}$ intersects $AC$ at a point $M$ and $BC$ at $P$ and $MP$ contains the incenter of $\triangle ABC$, then the length $MP$ is (in standard notation, where $t=\frac{1}{a+b+c}$): (A)$at(b+c)$ (B)$ct(b+a)$ (C)$bct$ (D)$abt$

2011 Today's Calculation Of Integral, 724

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

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2012 Today's Calculation Of Integral, 827

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

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$