Olympiad problems

2013 Vietnam National Olympiad, 2

Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.

2015 Miklos Schweitzer, 4

Tags: limit , algebra and number theory , prime number , number theory

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

2005 Today's Calculation Of Integral, 69

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

Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

2012 Today's Calculation Of Integral, 799

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

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

2010 Today's Calculation Of Integral, 549

Tags: calculus , integration , function , algebra , polynomial , limit , calculus computations

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

2011 District Olympiad, 1

Tags: function , limit , continued fraction , real analysis

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

1984 Putnam, B6

Tags: limit , geometry

A sequence of convex polygons $(P_n),n\ge0,$ is defined inductively as follows. $P_0$ is an equilateral triangle with side length $1$. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Express $\lim_{n\to\infty}[P_n]$ in the form $\frac{\sqrt a}b$, where $a,b$ are integers.

2012 China Western Mathematical Olympiad, 2

Tags: limit , algebra , logarithm

Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)

2005 IMC, 5

Tags: limit , integration , calculus

5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$

2011 Today's Calculation Of Integral, 768

Tags: calculus , integration , limit , geometry , 3d geometry , sphere , symmetry

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2011 Indonesia TST, 2

Tags: binomial coefficient , limit , combinatorics

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

2006 ISI B.Stat Entrance Exam, 8

Tags: limit , algebra , 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.

2006 Pre-Preparation Course Examination, 1

Tags: limit , vector , probability , invariant , topology , real analysis , 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 $x_0$.

2011 Today's Calculation Of Integral, 733

Tags: calculus , integration , limit , calculus computations

Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

2011 Uzbekistan National Olympiad, 2

Tags: floor function , limit , inequalities , algebra

Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

1996 Turkey Team Selection Test, 3

Tags: limit , algebra

Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.

1984 Iran MO (2nd round), 2

Tags: function , trigonometry , floor function , limit , algebra

Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

1983 Putnam, A6

Tags: limit , calculus , integration , real analysis

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

2009 Today's Calculation Of Integral, 401

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

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

2007 Today's Calculation Of Integral, 219

Tags: calculus , integration , limit , algebra , function , domain , logarithm

Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$. Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.

2010 Today's Calculation Of Integral, 665

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

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

2008 All-Russian Olympiad, 4

Tags: limit , algebra

The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\] Show that $ a_{2008} < 5$.

2000 Turkey Team Selection Test, 3

Tags: function , limit , inequalities , algebra

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\] Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$

1998 VJIMC, Problem 2

Tags: limit , real analysis

Find the limit $$\lim_{n\to\infty}\left(\frac{\left(1+\frac1n\right)^n}e\right)^n.$$

1981 Putnam, B1

Tags: limit , summation

Find $$\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).$$