Found problems: 280
Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
1998 Romania National Olympiad, 2
Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1]
[*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent;
[*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded.
[/list]
Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent.
Note: $\{ x \}$ denotes the fractional part of $x.$
1989 Bulgaria National Olympiad, Problem 2
Prove that the sequence $(a_n)$, where
$$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.
1954 Miklós Schweitzer, 1
[b]1.[/b] Given a positive integer $r>1$, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers $S_{1},S_{2},\dots,S_{r}$ such that $S_{1}+S_{2}+\dots+S_{r}=1$, any of these infinite geometrical series can be divided into $r$ infinite series(not necessarily geometrical) having the sums $S_{1},S_{2},\dots,S_{r}$, respectively. [b](S. 6)[/b]
2004 Gheorghe Vranceanu, 1
Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as
$$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$
Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.
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.
2021 Serbia National Math Olympiad, 6
A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$).
Show that the sequence contains less than $n$ repetitions.
Russian TST 2019, P2
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
1978 Putnam, B3
The sequence $(Q_{n}(x))$ of polynomials is defined by
$$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$
and for $m \geq 1 $ by
$$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$
$$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$
Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$
1985 Bulgaria National Olympiad, Problem 6
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.
1990 French Mathematical Olympiad, Problem 1
Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.
2016 Saudi Arabia IMO TST, 1
Let $ n \geq 3 $ be an integer and let
\begin{align*}
x_1,x_2, \ldots, x_n
\end{align*}
be $ n $ distinct integers. Prove that
\begin{align*}
(x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6.
\end{align*}
1984 Putnam, A6
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$.
$(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$.
$(\text b)$ Assuming part $(\text a)$, we can define
$$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$where $s=a_1+a_2+\ldots+a_k$. Find the least positive integer $p$ for which
$$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$or show that no such $p$ exists.
1979 IMO Longlists, 54
Consider the sequences $(a_n), (b_n)$ defined by
\[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \]
Find the smallest integer $m$ for which $b_m > a_{100}.$
2008 Alexandru Myller, 3
Let be a $ \beta >1. $ Calculate $ \lim_{n\to\infty} \frac{k(n)}{n} ,$ where $ k(n) $ is the smallest natural number that satisfies the inequality $ (1+n)^k\ge n^k\beta . $
[i]Neculai Hârţan[/i]
2004 VJIMC, Problem 3
Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series
$$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.
2017 VJIMC, 2
We say that we extend a finite sequence of positive integers $(a_1,\dotsc,a_n)$ if we replace it by
\[(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)\]
i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$
and he extends it $2017$ times. Then he chooses randomly one element of the resulting sequence. What is the
probability that the chosen element is $1$?
1998 Brazil Team Selection Test, Problem 5
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
2011 VTRMC, Problem 2
A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$, and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$. Find $a_{100}$.
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 $x$.
2021 SYMO, Q3
Let $a_1,a_2,a_3,\dots$ be an infinite sequence of non-zero reals satisfying \[a_{i} = \frac{a_{i-1}a_{i-2}-2}{a_{i-3}}\]for all $i\geq 4$. Determine all positive integers $n$ such that if $a_1,a_2,\dots,a_n$ are integers, then all elements of the sequence are integers.
2021-IMOC, N3
Define the function $f:\mathbb N_{>1}\to\mathbb N_{>1}$ such that $f(x)$ is the greatest prime factor of $x$. A sequence of positive integers $\{a_n\}$ satisfies $a_1=M>1$ and
$$a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases}$$
Show that for any positive integers $M,k$, the sequence $\{a_n\}$ is bounded.
(TAN768092100853)
2004 Germany Team Selection Test, 1
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
1988 Greece National Olympiad, 4
Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$.
a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$
b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$
2018 Iran MO (1st Round), 24
The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*}
$\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$