Olympiad problems

2009 Miklós Schweitzer, 3

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]

1975 Spain Mathematical Olympiad, 1

Tags: algebra , limit , analysis

Calculate the limit $$\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).$$ (For the calculation of the limit, the integral construction procedure can be followed).

2020 LIMIT Category 1, 17

Tags: limit , algebra

The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is? (A)$89$ (B)$179$ (C)$168$ (D)$119$

2007 Today's Calculation Of Integral, 189

Tags: limit , calculus , integration , 3d geometry , calculus computations , geometry

Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$

2013 Today's Calculation Of Integral, 863

Tags: inequalities , derivative , trigonometry , limit , calculus , integration , 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 $F(t)\geq 1.$

2012 Putnam, 4

Tags: calculus , inequalities , integration , induction , limit , logarithm

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

2025 District Olympiad, P1

Tags: limit , real analysis

Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$ [i]Mathematical Gazette[/i]

2013 ISI Entrance Examination, 2

Tags: trigonometry , limit

For $x\ge 0$, define \[f(x)=\frac1{x+2\cos x}\] Find the set $\{ y \in \mathbb{R}: y=f(x), x\ge 0\}$

2003 VJIMC, Problem 3

Tags: real analysis , limit , nested radical

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

2010 Today's Calculation Of Integral, 522

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

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

1997 Traian Lălescu, 4

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

Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].

1998 VJIMC, Problem 2

Tags: real analysis , limit

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

1952 Miklós Schweitzer, 9

Tags: real analysis , limit , function , integration

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)

1977 All Soviet Union Mathematical Olympiad, 239

Tags: limit , sequence , algebra

Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.

2020 LIMIT Category 1, 3

Tags: algebra , limit

How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$ (A)$20$ (B)$15$ (C)$9$ (D)$8$

2020 LIMIT Category 1, 2

Tags: geometry , limit

In a square $ABCD$ of side $2$ units, $E$ is the midpoint of $AD$ and $F$ on $BE$ such that $CF\perp BE$, then the quadrilateral $CDEF$ has an area of (A)$2$ (B)$2.2$ (C)$\sqrt{5}$ (D)None of these

2013 China Team Selection Test, 2

Tags: limit , number theory , modular arithmetic , quadratic

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

2014 JBMO Shortlist, 3

Tags: logarithm , combinatorics , modular arithmetic , floor function , limit

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2012 Vietnam National Olympiad, 3

Tags: algebra , limit , function

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 Gheorghe Vranceanu, 3

Tags: limit , calculus , binom

$ \lim_{n\to\infty } \sqrt[n]{\sum_{i=0}^n\binom{n}{i}^2} $

2011 Tokio University Entry Examination, 3

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

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2012 Today's Calculation Of Integral, 851

Tags: calculus computations , limit , function , calculus , integration , geometric series , trigonometry

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2012 China Western Mathematical Olympiad, 2

Tags: algebra , limit , 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)

2007 District Olympiad, 3

Tags: floor function , limit , 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 $A$.

1997 Flanders Math Olympiad, 3

Tags: trigonometry , ratio , geometry , limit , geometric sequence , inequalities

$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$. Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there) Show $A_k < S$ for every positive integer $k$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]