Found problems: 1239
2008 SEEMOUS, Problem 2
Let $P_0,P_1,P_2,\ldots$ be a sequence of convex polygons such that, for each $k\ge0$, the vertices of $P_{k+1}$ are the midpoints of all sides of $P_k$. Prove that there exists a unique point lying inside all these polygons.
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
2008 Federal Competition For Advanced Students, P1, 3
Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$.
Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.
2012 German National Olympiad, 1
Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.
2013 Singapore MO Open, 1
Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$-th powers of the digits of $a_n$. Do there exist distinct positive integers $i$, $j$ such that $a_i=a_j$?
2018 IMO Shortlist, A4
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 $a_{2018}-a_{2017}$.
2020 Romanian Master of Mathematics Shortlist, C4
A ternary sequence is one whose terms all lie in the set $\{0, 1, 2\}$. Let $w$ be a length $n$ ternary sequence $(a_1,\ldots,a_n)$. Prove that $w$ can be extended leftwards and rightwards to a length $m=6n$ ternary sequence \[(d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), \quad p,q\geqslant 0,\]containing no length $t > 2n$ palindromic subsequence.
(A sequence is called palindromic if it reads the same rightwards and leftwards. A length $t$ subsequence of $(d_1,\ldots,d_m)$ is a sequence of the form $(d_{i_1},\ldots,d_{i_t})$, where $1\leqslant i_1<\cdots<i_t \leqslant m$.)
2023 SEEMOUS, P2
For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]
VI Soros Olympiad 1999 - 2000 (Russia), 9.5
Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ .
(In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)
2019 Durer Math Competition Finals, 1
Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$
1958 November Putnam, A2
Let $R_1 =1$ and $R_{n+1}= 1+ n\slash R_n$ for $n\geq 1.$ Show that for $n\geq 1,$
$$ \sqrt{n} \leq R_n \leq \sqrt{n} +1.$$
2012 Dutch Mathematical Olympiad, 5
The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it.
How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?
2015 Saudi Arabia GMO TST, 2
Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even.
Lê Anh Vinh
1983 IMO Shortlist, 7
Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and
\[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\]
Show that for each positive integer $n$, $a_n$ is a positive integer.
1987 Bulgaria National Olympiad, Problem 4
The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.
1976 IMO Shortlist, 4
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$
1986 Miklós Schweitzer, 3
(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$.
(b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition.
[A. Balogh, I. Z. Ruzsa]
2005 Grigore Moisil Urziceni, 3
Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality
$$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$
for all natural numbers $ n. $
[b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $
[b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $
1977 All Soviet Union Mathematical Olympiad, 239
Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.
1986 Bulgaria National Olympiad, Problem 6
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.
2024 Middle European Mathematical Olympiad, 8
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that
\[a_ia_{i+1} \mid k-a_i^2\]
for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all
integers $n \ge M$.
1953 Miklós Schweitzer, 3
[b]3.[/b] Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$, where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$, prove that
$(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$.
[b](S. 24)[/b]
1997 Czech And Slovak Olympiad IIIA, 4
Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.
2009 239 Open Mathematical Olympiad, 1
In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.
1980 IMO Shortlist, 2
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]