Found problems: 837
1995 Putnam, 6
For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that
\[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \]
\[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]
1973 Poland - Second Round, 4
Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $.
Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.
2019 Jozsef Wildt International Math Competition, W. 8
Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.
2010 Contests, 3
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, 2
The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ :
(i) $g(1,0,0,1)=1$
(ii) $g(ea,b,ec,d)=eg(a,b,c,d)$
(iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$
(iv) $g(a,b,c,d)+g(b,a,d,c)=0$
is :
(A)$1$
(B)$0$
(C)$\text{infinitely many}$
(D)$\text{None of these}$
[Hide=Hint(given in question)]
Think of matrices[/hide]
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]
1981 IMO Shortlist, 16
A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$:
\[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\]
Describe, with proof, the behavior of $u_n$ as $n \to \infty.$
2010 ISI B.Stat Entrance Exam, 4
A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.
2007 Harvard-MIT Mathematics Tournament, 1
Compute: \[\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}\]
2012 Today's Calculation Of Integral, 784
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 $\lim_{x\to\infty} \frac{\ln x}{x}=0.$
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, $\lim_{n\to \infty} A_n$ .
1977 Miklós Schweitzer, 10
Let the sequence of random variables $ \{ X_m, \; m \geq 0\ \}, \; X_0=0$, be an infinite random walk on the set of nonnegative integers with transition probabilities \[ p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,\] \[ q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.\] Prove that for arbitrary $ k >0$ there is an $ \alpha_k > 1$ such that \[ P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)\] satisfies the limit relation \[ \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.\]
[i]J. Tomko[/i]
1994 Polish MO Finals, 3
$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
2005 Romania National Olympiad, 4
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function.
a) Prove that $f$ is continous;
b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]
2012 Today's Calculation Of Integral, 830
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$
1992 IMO Longlists, 78
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?
2020 LIMIT Category 2, 11
$\triangle PQR$ is isosceles and right angled at $R$. Point $A$ is inside $\triangle PQR$, such that $PA=11, QA=7$, and $RA=6$. Legs $\overline{PR}$ and $\overline{QR}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?
2010 District Olympiad, 4
Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that:
\[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]
2010 Today's Calculation Of Integral, 579
Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$
1983 Iran MO (2nd round), 5
Find the value of $S_n= \arctan \frac 12 + \arctan \frac 18+ \arctan \frac {1}{18} + \cdots + \arctan \frac {1}{2n^2}.$ Also find $\lim_{n \to \infty} S_n.$
1967 Miklós Schweitzer, 5
Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]
2004 Iran Team Selection Test, 6
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
2019 Jozsef Wildt International Math Competition, W. 9
Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$
1980 VTRMC, 3
Let $$a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}.$$
(a) Prove that $\lim_{n\to \infty}a_n$ exists.
(b) Show that $$a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}.$$
(c) Find $\lim_{n\to\infty}a_n$ and justify your answer