Olympiad problems

2013 Today's Calculation Of Integral, 899

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

1984 Iran MO (2nd round), 1

Tags: function , floor function , domain , limit , algebra

Let $f$ and $g$ be two functions such that \[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\] Find the domains of $f$ and $g$ and then prove that \[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]

2011 Today's Calculation Of Integral, 727

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

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

1987 Austrian-Polish Competition, 3

Tags: sequence , functional , functional equation , limit

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

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$.

1983 IMO Longlists, 46

Tags: function , limit , algebra , logarithm

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that \[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\] For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$

1984 IMO Longlists, 58

Tags: inequalities , limit , algebra

Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

2002 Austrian-Polish Competition, 8

Tags: trigonometry , limit , inequalities , algebra

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

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.

2020 Brazil Undergrad MO, Problem 1

Tags: limit , calculus , geometry

Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of $\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?

1988 IMO Longlists, 35

Tags: induction , limit , algebra

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

2013 Today's Calculation Of Integral, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2010 Contests, 3

Tags: function , limit , algebra

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

2020 LIMIT Category 2, 9

Tags: probability , limit

Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is $\frac{1}{K}$, $K\in\mathbb{N}$. Find $K$.

2006 Moldova National Olympiad, 11.6

Tags: limit , linear algebra , matrix , vector , algebra

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

2012 Online Math Open Problems, 28

Tags: modular arithmetic , real analysis , function , limit , number theory , logarithm

Find the remainder when \[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.) [i]Victor Wang.[/i]

2010 Today's Calculation Of Integral, 528

Tags: calculus , integration , function , geometry , geometric transformation , rotation , limit

Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers. (1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$. (2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtained by rotation of $ D_1$ about the $ x$-axis. Find $ \lim_{C\rightarrow \infty} V_1(C)$. (3) Let $ M$ be the maximum value of $ y\equal{}f(x)$ for $ x\geq 0$. Denote $ D_2$ the region bounded by $ y\equal{}f(x)$, the $ y$-axis and $ y\equal{}M$. Find the volume $ V_2$ obtained by rotation of $ D_2$ about the $ y$-axis.

2005 Today's Calculation Of Integral, 55

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

Evaluate \[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]

2014 SEEMOUS, Problem 4

Tags: limit , integration , calculus

a) Prove that $\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2$. b) Find the limit $\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right)$.

1989 Iran MO (2nd round), 1

Tags: floor function , function , limit , inequalities

[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]

Today's calculation of integrals, 864

Tags: calculus , integration , inequalities , limit , calculus computations

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2010 Tuymaada Olympiad, 4

Tags: floor function , algebra , polynomial , limit , number theory

Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

2012 Today's Calculation Of Integral, 784

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

Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$. (1) Find the local maximum $f_n(x)$. (2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized. (3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$. (4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$. Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$

2014 China Team Selection Test, 5

Tags: limit , combinatorics

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

2010 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , limit , trigonometry

Compute $\displaystyle\lim_{n\to\infty}\dfrac{\sum_{k=1}^n|\cos(k)|}{n}$.