Olympiad problems

2008 Moldova National Olympiad, 12.4

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.

2020 Jozsef Wildt International Math Competition, W2

Tags: limit , real analysis , calculus

Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. [list=1] [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ [/list]

1985 IMO Longlists, 78

Tags: function , recurrence relation , fixed point , calculus , limit

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

1978 Austrian-Polish Competition, 8

Tags: algebra , limit , square root

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side. (a) Show that the sequence converges, for every fixed integer $k\ge 1$. (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

2004 Nicolae Coculescu, 2

Tags: real analysis , sequence , limit

Let bet a sequence $\left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined as $ a_n=\sqrt[n]{1+na_{n-1}} . $ Show that $ \left( a_n \right)_{n\ge 1} $ is convergent and determine its limit. [i]Florian Dumitrel[/i]

2007 Today's Calculation Of Integral, 193

Tags: calculus , integration , parabola , geometry , limit , analytic geometry , conic

For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$. Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin. (1) Find the equation of $l$. (2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.

2020 Jozsef Wildt International Math Competition, W38

Tags: real analysis , sequence , limit

Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

2010 Contests, 522

Tags: calculus , integration , limit , function , geometry , derivative , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2013 Today's Calculation Of Integral, 899

Tags: calculus , integration , limit , calculus computations

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

1952 AMC 12/AHSME, 36

Tags: limit

To be continuous at $ x \equal{} \minus{} 1$, the value of $ \frac {x^3 \plus{} 1}{x^2 \minus{} 1}$ is taken to be: $ \textbf{(A)}\ \minus{} 2 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \infty \qquad\textbf{(E)}\ \minus{} \frac {3}{2}$

2000 Romania National Olympiad, 4

Tags: function , real analysis , darboux , epsilon-delta , second iterate , limit

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions: $ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $ $ \text{(ii)}\quad f $ has Darboux’s property [b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist. [b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?

2021 Belarusian National Olympiad, 10.1

Tags: sequence , algebra , limit

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

1988 Dutch Mathematical Olympiad, 2

Tags: limit , recurrence relation , algebra , sequence , trigonometry

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 $\lim_{n\to \infty}2^{2n+1}(1-c_n)$

1998 Vietnam National Olympiad, 1

Tags: integration , limit , logarithm , real analysis

Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.

2006 Stanford Mathematics Tournament, 15

Tags: limit , geometric series

Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute \[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\] (Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)

2014 JBMO Shortlist, 3

Tags: floor function , modular arithmetic , limit , combinatorics , logarithm

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2008 Harvard-MIT Mathematics Tournament, 6

Tags: limit , ratio , calculus , calculus computations

Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

2008 Teodor Topan, 3

Tags: limit , geometry , geometric transformation , calculus , calculus computations , logarithm

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

1991 Vietnam Team Selection Test, 3

Tags: limit , linear algebra , matrix , vector , invariant , complex number , logarithm

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

2021 Brazil Undergrad MO, Problem 4

Tags: real analysis , mean , limit , exponent

For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$. Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

2006 IMS, 5

Tags: limit , function , algebra

Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.

2019 Jozsef Wildt International Math Competition, W. 30

Tags: zeta function , limit , summation , function

[list=1] [*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$ [*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$ [/list]

2012 Today's Calculation Of Integral, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

1966 Miklós Schweitzer, 3

Tags: limit , advanced fields

Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\] [i]P. Erdos[/i]

2001 District Olympiad, 4

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

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]