Found problems: 837
2013 VJIMC, Problem 1
Let $f:[0,\infty)\to\mathbb R$ be a differentiable function with $|f(x)|\le M$ and $f(x)f'(x)\ge\cos x$ for $x\in[0,\infty)$, where $M>0$. Prove that $f(x)$ does not have a limit as $x\to\infty$.
1951 Miklós Schweitzer, 3
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).
2011 Tokyo Instutute Of Technology Entrance Examination, 1
Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$.
For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$
1957 Putnam, A6
Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that
$$ \lim_{n \to \infty} S_n = a-1.$$
2020 LIMIT Category 2, 7
A circle $\mathfrak{D}$ is drawn through the vertices $A$ and $B$ of $\triangle ABC$. If $\mathfrak{D}$ intersects $AC$ at a point $M$ and $BC$ at $P$ and $MP$ contains the incenter of $\triangle ABC$, then the length $MP$ is (in standard notation, where $t=\frac{1}{a+b+c}$):
(A)$at(b+c)$
(B)$ct(b+a)$
(C)$bct$
(D)$abt$
Today's calculation of integrals, 896
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
2012 Today's Calculation Of Integral, 851
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$
Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$
2019 Jozsef Wildt International Math Competition, W. 47
[list=1]
[*] If $a$, $b$, $c$, $d > 0$, show inequality:$$\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)$$
[*] Calculate $$\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)$$where $\alpha \in \mathbb{R}$
[/list]
2004 Putnam, B6
Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.
2005 Today's Calculation Of Integral, 23
Evaluate
\[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]
1989 APMO, 5
Determine all functions $f$ from the reals to the reals for which
(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$,
where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)
1990 Flanders Math Olympiad, 4
Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function.
(a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$
Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$
(b) Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.
2000 Putnam, 3
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$.
Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]
2012 Pre - Vietnam Mathematical Olympiad, 2
Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$
2009 District Olympiad, 3
Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.
2002 District Olympiad, 1
a) Evaluate
\[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\]
with $a,b>0$.
b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and
\[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\]
Prove that:
1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded.
2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent.
[i]Valentin Matrosenco[/i]
2014 SEEMOUS, Problem 2
Consider the sequence $(x_n)$ given by
$$x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.$$Prove that the sequence $y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1$ is convergent and find its limit.
2005 IberoAmerican Olympiad For University Students, 1
Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find
\[\lim_{n\to\infty}f(n)^{1/n}\]
2011 Today's Calculation Of Integral, 708
Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.
2012 Today's Calculation Of Integral, 819
For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$
Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$
PEN E Problems, 24
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
1996 Tuymaada Olympiad, 6
Given the sequence
$f_1(a)=sin(0,5\pi a)$
$f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$
$...$
$f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number.
What limit aspire the members of this sequence as $n \to \infty$?
1999 IMC, 4
Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.
2010 Polish MO Finals, 3
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
1981 Spain Mathematical Olympiad, 5
Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this:
$$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$
a) Represent the function graphically.
b) Calculate $A_n =\int_0^1 f_n(x) dx$.
c) Find, if it exists, $\lim_{n\to \infty} A_n$ .