Found problems: 307
2021 Olimphíada, 1
The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence:
$$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.
2020 Regional Olympiad of Mexico Northeast, 1
Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$?
Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.
2008 China Team Selection Test, 2
The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.
1979 IMO Shortlist, 9
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]
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 $b_m > a_{100}.$
1973 Dutch Mathematical Olympiad, 5
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
1989 Austrian-Polish Competition, 7
Functions $f_0, f_1,f_2,...$ are recursively defined by
$f_0(x) = x$ and $f_{2k+1} (x) = 3^{f_{2k}(x)}$ and $f_{2k+2} = 2^{f_{2k+1}(x)}$, $k = 0,1,2,...$ for all $x \in R$.
Find the greater one of the numbers $f_{10}(1)$ and $f_9(2)$.
2024 Canadian Mathematical Olympiad Qualification, 4
A sequence $\{a_i\}$ is given such that $a_1 = \frac13$ and for all positive integers $n$
$$a_{n+1} =\frac{a^2_n}{a^2_n - a_n + 1}.$$
Prove that $$\frac12 - \frac{1}{3^{2^{n-1}}} < a_1 + a_2 +... + a_n <\frac12 - \frac{1}{3^{2^n}} ,$$
for all positive integers $n$.
2006 IMO Shortlist, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
2018 IFYM, Sozopol, 7
The rows $x_n$ and $y_n$ of positive real numbers are such that:
$x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$
for each positive integer $n$.
Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9
2015 Saudi Arabia GMO TST, 1
Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$
Trần Nam Dũng
2011 China Northern MO, 1
It is known that the general term $\{a_n\}$ of the sequence is $a_n =(\sqrt3 +\sqrt2)^{2n}$ ($n \in N*$), let $b_n= a_n +\frac{1}{a_n}$ .
(1) Find the recurrence relation between $b_{n+2}$, $b_{n+1}$, $b_n$.
(2) Find the unit digit of the integer part of $a_{2011}$.
1981 Austrian-Polish Competition, 2
The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.
2018 Istmo Centroamericano MO, 1
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.
2020 Dutch Mathematical Olympiad, 2
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
1986 Czech And Slovak Olympiad IIIA, 5
A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$.
Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$.
2022 Brazil EGMO TST, 5
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
1984 All Soviet Union Mathematical Olympiad, 389
Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.
2017 Korea USCM, 3
Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.
1994 IMO Shortlist, 6
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
1987 Yugoslav Team Selection Test, Problem 1
Let $x_0=a,x_1=b$ and $x_{n+1}=2x_n-9x_{n-1}$ for each $n\in\mathbb N$, where $a,b$ are integers. Find the necessary and sufficient condition on $a$ and $b$ for the existence of an $x_n$ which is a multiple of $7$.
1958 November Putnam, A1
Let $f(m,1)=f(1,n)=1$ for $m\geq 1, n\geq 1$ and let $f(m,n)=f(m-1, n)+ f(m, n-1) +f(m-1 ,n-1)$ for $m>1$ and $n>1$. Also let
$$ S(n)= \sum_{a+b=n} f(a,b) \,\,\;\; a\geq 1 \,\, \text{and} \,\; b\geq 1.$$
Prove that
$$S(n+2) =S(n) +2S(n+1) \,\, \; \text{for} \, \, n \geq 2.$$
2005 Czech And Slovak Olympiad III A, 1
Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.
1991 Tournament Of Towns, (307) 4
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
1984 IMO Longlists, 14
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$