Olympiad problems

1999 Czech and Slovak Match, 5

Find all functions $f: (1,\infty)\text{to R}$ satisfying $f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$. [hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution. I have also solved by using this method.By finding $f(x^5)$ in two ways I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]

Today's calculation of integrals, 851

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

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

2021 Brazil Undergrad MO, Problem 4

Tags: limit , exponent , mean , real analysis

For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$. Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

1989 Romania Team Selection Test, 1

Tags: algebra , inequalities , induction , limit

Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.

2011 Bulgaria National Olympiad, 3

Tags: limit , function , geometry

Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.

1986 Iran MO (2nd round), 1

Tags: limit , function , trigonometry , algebra

Let $f$ be a function such that \[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\] Find the limit of $f$ in the point $x_0=1.$

2010 VJIMC, Problem 2

Tags: limit , sequence

Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.

2012 District Olympiad, 1

Tags: sequence , real analysis , limit

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

MathLinks Contest 7th, 3.3

Tags: function , inequalities , limit

Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$: \[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right). \]

1985 IMO Longlists, 96

Tags: limit , function , algebra

Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions: (a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and (b) $\lim_{x\to \infty} f(x) = 0$.

1968 Miklós Schweitzer, 7

Tags: limit , advanced fields , function

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2012 Today's Calculation Of Integral, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2013 USAMTS Problems, 3

Tags: limit , arithmetic sequence , geometry , induction , inequalities , ratio

An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?

2012 Today's Calculation Of Integral, 818

Tags: logarithm , absolute value , function , calculus , integration , limit

For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

1992 IMO Longlists, 78

Tags: fibonacci , fibonacci sequence , sequence , limit , calculus

Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that \[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\] for $k \geq 1$, where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?

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 $n \geq 1.$

1985 IMO, 6

Tags: fixed point , sequence , calculus , polynomial , limit

For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.

1951 Miklós Schweitzer, 3

Tags: limit , real analysis

Consider the iterated sequence (1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$, where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).

1999 Putnam, 3

Tags: modular arithmetic , polynomial , limit , algebra

Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

2020 LIMIT Category 2, 5

Tags: coordinates , limit , geometry

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area? (A)$20\sqrt{3}$ (B)$20\sqrt{2}$ (C)$25\sqrt{3}$ (D)None of these

1981 Putnam, A3

Tags: double integral , limit

Find $$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$ or show that the limit does not exist.

2011 Today's Calculation Of Integral, 733

Tags: calculus computations , limit , calculus , integration

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

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

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

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2017 Mathematical Talent Reward Programme, MCQ: P 2

Tags: limit , calculus

$\lim \limits_{x\to \infty} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}=$ [list=1] [*] $\sqrt{e}$ [*] $\infty$ [*] Does not exists [*] None of these [/list]

2012 Putnam, 3

Tags: topology , algebra , limit , function , functional equation

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.