Found problems: 837
2014 Bulgaria National Olympiad, 2
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:
\[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\]
[i]Proposed by Nikolay Nikolov[/i]
2012 Today's Calculation Of Integral, 837
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$
Find $\lim_{n\to\infty} f_n(1).$
1953 Putnam, A6
Show that the sequence
$$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$
converges and evaluate the limit.
1992 Vietnam National Olympiad, 3
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.
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 AMC 10, 13
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{4}{9} \qquad
\textbf{(D)}\ \frac{5}{9} \qquad
\textbf{(E)}\ \frac{2}{3} $
2017 Korea USCM, 5
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
2013 Today's Calculation Of Integral, 872
Let $n$ be a positive integer.
(1) For a positive integer $k$ such that $1\leq k\leq n$, Show that :
\[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\]
(2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$
If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$
(3) Find $\lim_{n\to\infty} S_n.$
2006 IMS, 5
Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.
1965 AMC 12/AHSME, 10
The statement $ x^2 \minus{} x \minus{} 6 < 0$ is equivalent to the statement:
$ \textbf{(A)}\ \minus{} 2 < x < 3 \qquad \textbf{(B)}\ x > \minus{} 2 \qquad \textbf{(C)}\ x < 3$
$ \textbf{(D)}\ x > 3 \text{ and }x < \minus{} 2 \qquad \textbf{(E)}\ x > 3 \text{ and }x < \minus{} 2$
1971 IMO Longlists, 53
Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.
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)$.
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]
2005 Morocco TST, 1
Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying :
$(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$
1990 IMO Longlists, 56
For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows:
$K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$
(i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$
(ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.
1984 Iran MO (2nd round), 5
Suppose that
\[S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}\]
Find $\lim_{n \to \infty} S_n.$
2005 France Pre-TST, 8
Let $f$ be a function from the set $Q$ of the rational numbers onto itself such that $f(x+y)=f(x)+f(y)+2547$ for all rational numbers $x,y$.
Moreover $f(2004) = 2547$.
Determine $f(2547).$
Pierre.
1966 Miklós Schweitzer, 3
Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\]
[i]P. Erdos[/i]
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
2008 Moldova MO 11-12, 6
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
2007 Moldova National Olympiad, 11.2
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.
Find $\lim_{n\rightarrow\infty}a_{n}$.
2004 Romania Team Selection Test, 4
Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.
2012 Today's Calculation Of Integral, 832
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
2012 Pre-Preparation Course Examination, 2
Suppose that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$. Prove that
$\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab$.
2014 Olympic Revenge, 4
Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that
\[n \mid a^{f(n)}-1.\]
Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.