Found problems: 837
2007 Vietnam National Olympiad, 2
Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that:
$f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$
1999 Vietnam Team Selection Test, 1
Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying:
\[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\]
Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3
Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.
2005 Today's Calculation Of Integral, 69
Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$.
Find $\lim_{n\to\infty} f_n(x)$
1980 IMO, 4
Given a real number $x>1$, prove that there exists a real number $y >0$ such that
\[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]
2010 Contests, 3
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
2013 Brazil National Olympiad, 3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
2006 Stanford Mathematics Tournament, 8
Evaluate: $\lim_{n\rightarrow\infty}\sum_{k=n^2}^{(n+1)^2} \dfrac{1}{\sqrt{k}}$
2003 Romania National Olympiad, 4
$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that
[b]a)[/b] $ i\left( \mathbb{Z}_{12} \right) =4. $
[b]b)[/b] $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $
[b]c)[/b] there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that
$$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$
and
$$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$
[i]Barbu Berceanu[/i]
2011 Today's Calculation Of Integral, 753
Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$
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 1, 10
For natural number $t$, the repeating base-$t$ representation of the (base-ten) rational number $\frac{7}{51}$ is $0.\overline{23}_t=0.232323..._t$. What is $t$ ?
2004 VTRMC, Problem 7
Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.
2019 Jozsef Wildt International Math Competition, W. 7
If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$
ICMC 7, 6
Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection of the positive integers. Prove that at least one of the following limits is true: \[\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.\][i]Proposed by Dylan Toh[/i]
Today's calculation of integrals, 857
Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$
(1) Find $f(x).$
(2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.
1954 Putnam, B7
Let $a>0$. Show that
$$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$
lies between $e^a$ and $e^{a+1}.$
Today's calculation of integrals, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
2009 Today's Calculation Of Integral, 401
For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.
2001 Bulgaria National Olympiad, 3
Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform
($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $
into
$ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $
by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$
2009 Putnam, B2
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$
2007 Today's Calculation Of Integral, 229
Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.
2008 All-Russian Olympiad, 4
The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]
Show that $ a_{2008} < 5$.
2020 LIMIT Category 2, 18
Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$
(A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$
(B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$
(C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$
(D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$
1998 AIME Problems, 8
Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?