Olympiad problems

2000 IMC, 6

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

1970 IMO, 3

Tags: limit , inequality , calculus , integration , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

1998 AIME Problems, 8

Tags: limit

Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?

2009 Romania National Olympiad, 1

Tags: limit , real analysis

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.

2019 Jozsef Wildt International Math Competition, W. 57

Tags: limit , integration , calculus , sequence

Let be $x_1=\frac{1}{\sqrt[n+1]{n!}}$ and $x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}$ for all $n\in \mathbb{N}^*$ and $f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}$ where $$f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}$$Prove that the sequence $(a_n)_{n\geq1}$ when $a_n=\int \limits_{x_1}^{x_2}f(x)dx$ is convergent and compute $$\lim \limits_{n \to \infty}a_n$$

1982 IMO Shortlist, 3

Tags: limit , geometric sequence , geometric series , inequality , minimization

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 $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

2012 Today's Calculation Of Integral, 786

Tags: calculus , integration , limit , induction , algebra , polynomial , probability

For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

2020 LIMIT Category 1, 15

Tags: chessboard , counting , limit

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?

2005 Today's Calculation Of Integral, 90

Tags: calculus , integration , limit , factorial , function , ratio , logarithm

Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2003 VJIMC, Problem 3

Tags: limit , nested radical , real analysis

Find the limit $$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$

2007 IMS, 5

Tags: limit , calculus , calculus computations

Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]

1970 IMO Longlists, 52

Tags: limit , inequality , calculus , integration , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

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.

1975 Miklós Schweitzer, 5

Tags: function , integration , limit , real analysis

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

2004 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , limit

Find \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]

2012 Vietnam National Olympiad, 3

Tags: function , limit , algebra

Find all $f:\mathbb{R} \to \mathbb{R}$ such that: (a) For every real number $a$ there exist real number $b$:$f(b)=a$ (b) If $x>y$ then $f(x)>f(y)$ (c) $f(f(x))=f(x)+12x.$

2007 District Olympiad, 3

Tags: limit , floor function , real analysis

Let $a,b\in \mathbb{R}$. Evaluate: \[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\] Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$. a) Prove that $\mathbb{Q}\cap [0,1]\subset A$. b) Find $A$.

2001 Bulgaria National Olympiad, 3

Tags: invariant , limit , induction , ceiling function , symmetry , combinatorics

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

2009 Today's Calculation Of Integral, 421

Tags: calculus , integration , limit , calculus computations

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.