Olympiad problems

2015 BMT Spring, 9

Find $$\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).$$

2010 Romania National Olympiad, 4

Tags: real analysis , logarithm , limit

Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

2012 Grigore Moisil Intercounty, 3

Tags: real analysis , limit , riemann sum , definite integral , claculus

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

2019 Putnam, B2

Tags: limit

For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.

2010 Romania National Olympiad, 1

Tags: limit , calculus , derivative , real analysis , function , integration

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

2009 IMS, 4

Tags: limit , algebra , polynomial , absolute value , vector , induction , ratio

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

1950 Miklós Schweitzer, 4

Tags: limit , vector , linear algebra , matrix

Put $ M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix}$ where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that $ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$

1988 IMO Longlists, 9

Tags: limit , quadratic , algebra

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by: \[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \] Show that there exist a real number $a > 0$ such that: [b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$ [b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$

1949 Miklós Schweitzer, 2

Tags: trigonometry , limit , real analysis , integration

Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .

2020 LIMIT Category 1, 1

Tags: limit , irrational

If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false? (A)$ab^2$ is irrational (B)$a^2b$ is rational (C)$\sqrt{ab}$ is rational (D)$a+b$ is irrational

PEN E Problems, 24

Tags: limit , inequalities , logarithm , prime number , number theory , floor function

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

2012 Today's Calculation Of Integral, 830

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

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

2007 Tuymaada Olympiad, 4

Tags: logarithm , number theory , limit , greatest common divisor , prime number , probability , ratio

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

2010 Today's Calculation Of Integral, 621

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

Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

2006 Cezar Ivănescu, 1

Tags: limit , factorial , harmonic series , newton s binom , integral , calculus

[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $ [b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $

2020 LIMIT Category 2, 13

Tags: sum of digits , limit

For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.

2004 Alexandru Myller, 4

Tags: limit , analysis , real analysis , definite integral , factorial , function

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

2007 Iran MO (3rd Round), 8

Tags: limit , floor function , logarithm , factorial , trigonometry , ceiling function

In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

PEN R Problems, 7

Tags: limit , integration , 3d geometry , triangle inequality , geometry , calculus , inequalities

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

1954 Putnam, B5

Tags: differentiability , limit , function

Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$ Prove that $$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$

PEN G Problems, 27

Tags: integration , calculus , irrational number , derivative , limit , function

Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

2009 Putnam, A3

Tags: trigonometry , induction , linear algebra , limit , matrix , function

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

1997 VJIMC, Problem 2

Tags: real analysis , sequence , limit

Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality $$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.

2004 VJIMC, Problem 3

Tags: real analysis , limit , convergence , sequence

Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series $$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.

1972 Putnam, A3

Tags: limit , sequence

A sequence $(x_{i})$ is said to have a [i]Cesaro limit[/i] exactly if $\lim_{n\to\infty} \frac{x_{1}+\ldots+x_{n}}{n}$ exists. Find all real-valued functions $f$ on the closed interval $[0, 1]$ such that $(f(x_i))$ has a Cesaro limit if and only if $(x_i)$ has a Cesaro limit.