Found problems: 112
2023 ISI Entrance UGB, 2
Let $a_0 = \frac{1}{2}$ and $a_n$ be defined inductively by
\[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$.} \]
[list=a]
[*] Show that for $n = 0,1,2, \ldots,$
\[a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$, }\]
and determine $\theta_n$.
[*] Using (a) or otherwise, calculate
\[ \lim_{n \to \infty} 4^n (1 - a_n).\]
[/list]
2004 Alexandru Myller, 4
For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $
[i]Gheorghe Iurea[/i]
1996 Romania National Olympiad, 4
Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.
2012 Centers of Excellency of Suceava, 2
Calculate $ \lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} , $ where $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ is an unbounded and nondecreasing function.
[i]Dan Popescu[/i]
2011 Laurențiu Duican, 4
[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded.
[b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that
$$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$
for any natural number $ n. $ Prove that the following relations are equivalent:
$\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $
$ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $
[i]Eugen Păltănea[/i]
2012 Grigore Moisil Intercounty, 3
$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $
2005 VJIMC, Problem 4
Let $(x_n)_{n\ge2}$ be a sequence of real numbers such that $x_2>0$ and $x_{n+1}=-1+\sqrt[n]{1+nx_n}$ for $n\ge2$. Find
(a) $\lim_{n\to\infty}x_n$,
(b) $\lim_{n\to\infty}nx_n$.
1982 Putnam, A6
Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties:
$(\text i)$ $|x_n|$ is a strictly decreasing function of $n$;
$(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$;
$(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$.
Prove or disprove that these conditions imply that
$$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$
2011 Bogdan Stan, 3
Let be a sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ chosen such that the limit of the sequence $ \left(
x_{n+2011}-x_n \right)_{n\ge 1} $ exists. Calculate $ \lim_{n\to\infty } \frac{x_n}{n} . $
[i]Cosmin Nițu[/i]
2004 Unirea, 3
[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality
$$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$
holds.
[b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?
2013 IPhOO, 9
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.
[i](B. Dejean, 6 points)[/i]
2020 Brazil Undergrad MO, Problem 1
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
$\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?
2004 Nicolae Coculescu, 1
Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $
2000 VJIMC, Problem 3
Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?
2023 OMpD, 2
Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$.
(a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times.
(b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.
2004 Nicolae Coculescu, 4
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $
[i]Florian Dumitrel[/i]
1985 Traian Lălescu, 1.2
Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $
2004 Nicolae Coculescu, 2
Let bet a sequence $\left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined as $ a_n=\sqrt[n]{1+na_{n-1}} . $
Show that $ \left( a_n \right)_{n\ge 1} $ is convergent and determine its limit.
[i]Florian Dumitrel[/i]
2013 BMT Spring, 10
Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms,
$$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$
2022 Miklós Schweitzer, 3
Original in Hungarian; translated with Google translate; polished by myself.
Let $f: [0, \infty) \to [0, \infty)$ be a function that is linear between adjacent integers, and for $n = 0, 1, \dots$ satisfies
$$f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}$$
Let $f^1(x) = f(x)$, and $f^k(x) = f(f^{k - 1}(x))$ for all integers $k \geq 2$. Determine the values of $\liminf\nolimits_{k\to\infty}f^k(x)$ and $\limsup\nolimits_{k\to\infty}f^k(x)$ for almost all $x \in [0, \infty)$ under Lebesgue measure.
(Not sure whether the last sentence translates correctly; the original:
Határozzuk meg Lebesgue majdnem minden $x\in [0, \infty)$-re a $\liminf\nolimits_{k\to\infty}f^k(x)$ és $\limsup\nolimits_{k\to\infty}f^k(x)$ értékét.)
2001 VJIMC, Problem 3
Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.
2006 Petru Moroșan-Trident, 1
What relationship should be between the positive real numbers $ a $ and $ b $ such that the sequence $ \left(\left( a\sqrt[n]{n} +b \right)^{\frac{n}{\ln n}}\right)_{n\ge 1} $ has a nonzero and finite limit? For such $ a,b, $ calculate the limit of this sequence.
[i]Ion Cucurezeanu[/i]
2010 Gheorghe Vranceanu, 2
Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as
$$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$
Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.
1986 Traian Lălescu, 2.3
Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $
1997 VJIMC, Problem 2
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.