Olympiad problems

2013 BMT Spring, 3

Evaluate $$\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}$$

1975 Canada National Olympiad, 7

Tags: trigonometry , limit , derivative , function , quadratic , calculus , algebra

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

1994 Putnam, 1

Tags: limit , inequalities

Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.

2007 Today's Calculation Of Integral, 228

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

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

1965 Putnam, A3

Tags: limit

Show that, for any sequence $a_1,a_2,\ldots$ of real numbers, the two conditions \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha \] and \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha \] are equivalent.

2009 District Olympiad, 3

Tags: real analysis , limit , induction

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

2004 Iran Team Selection Test, 6

Tags: polynomial , limit , algebra , number theory

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2013 Today's Calculation Of Integral, 882

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

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2001 Romania National Olympiad, 4

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

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2004 Nicolae Coculescu, 1

Tags: limit , sequence , series , harmonic series , calculus

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

2007 Today's Calculation Of Integral, 222

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

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

2012 IMC, 2

Tags: limit

Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and \[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\] Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value. [i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i]

2005 Today's Calculation Of Integral, 48

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

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

1973 Miklós Schweitzer, 4

Tags: limit , number theory , logarithm

Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\] [i]P. Erdos[/i]

1967 Miklós Schweitzer, 7

Tags: geometry , linear algebra , 3d geometry , limit , sphere , matrix , probability

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

2011 Turkey Team Selection Test, 3

Tags: floor function , limit , greatest common divisor , prime factorization , number theory , induction

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

2009 Today's Calculation Of Integral, 424

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

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

2019 Jozsef Wildt International Math Competition, W. 43

Tags: integration , algebra , sequence , limit , polynomial

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

2000 Romania National Olympiad, 4

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

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?

2016 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: limit , sequence , algebra

Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that $a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$ $b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$

1974 Poland - Second Round, 5

Tags: algebra , sequence , limit

The given numbers are real numbers $ q,t \in \langle \frac{1}{2}; 1) $, $ t \in (0; 1 \rangle $. Prove that there is an increasing sequence of natural numbers $ {n_k} $ ($ k = 1,2, \ldots $) such that $$ t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.$$

2011 Miklós Schweitzer, 9

Tags: real analysis , differential equation , limit

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

2010 Moldova National Olympiad, 11.4

Tags: trigonometry , number theory , limit

Let $ a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2}$ Find $ \lim_{n\to\infty}a_n$

2001 District Olympiad, 4

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

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]

2003 VJIMC, Problem 3

Tags: sequence , real analysis , limit

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$