Found problems: 307
1979 IMO Shortlist, 19
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 $b_m > a_{100}.$
2024 Czech and Slovak Olympiad III A, 5
Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then
$$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$
Definitely all positive integers $n$ such that if $a_0 = 1/n$, then this sequence is constant after a certain term.
VMEO IV 2015, 10.1
Where $n$ is a positive integer, the sequence $a_n$ is determined by the formula $$a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.$$ Find the limit of the sequence $S_n$ defined by $S_n=a_1 + a_2 +... + a_n$.
2010 Saudi Arabia BMO TST, 3
Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?
1994 Austrian-Polish Competition, 2
The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$
Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$
1992 IMO Shortlist, 14
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
1971 IMO Longlists, 34
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and
\[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\]
Show that for all $k$,
\[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\]
where $[x]$ denotes the greatest integer not exceeding $x.$
1978 Polish MO Finals, 5
For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and
$$a_{n+1} =\begin{cases}
\dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\
0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$
Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.
1976 IMO Shortlist, 9
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
1973 Spain Mathematical Olympiad, 3
The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$
Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.
2001 Mongolian Mathematical Olympiad, Problem 1
Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies
$$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.
2005 VJIMC, Problem 4
Let $(x_n)_{n\ge2}$ be a sequence of real numbers such that $x_2>0$ and $x_{n+1}=-1+\sqrt[n]{1+nx_n}$ for $n\ge2$. Find
(a) $\lim_{n\to\infty}x_n$,
(b) $\lim_{n\to\infty}nx_n$.
1985 IMO Shortlist, 12
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
2021 Dutch IMO TST, 1
The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.
2019 Pan-African Shortlist, A1
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
1987 IMO Longlists, 19
How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one?
[i]Proposed by Germany, FR.[/i]
Kvant 2020, M2603
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
1989 French Mathematical Olympiad, Problem 4
For natural numbers $x_1,\ldots,x_k$, let $[x_k,\ldots,x_1]$ be defined recurrently as follows: $[x_2,x_1]=x_2^{x_1}$ and, for $k\ge3$, $[x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}$.
(a) Let $3\le a_1\le a_2\le\ldots\le a_n$be integers. For a permutation $\sigma$ of the set $\{1,2,\ldots,n\}$, we set $P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]$. Find the permutations $\sigma$ for which $P(\sigma)$ is minimal or maximal.
(b) Find all integers $a,b,c,d$, greater than or equal to $2$, for which $[178,9]\le[a,b,c,d]\le[198,9]$.
1979 Dutch Mathematical Olympiad, 3
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
2019 Nigerian Senior MO Round 4, 4
We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
1971 IMO Longlists, 5
Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds:
\[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\]
Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$
\[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]
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$.
2018 IFYM, Sozopol, 8
The row $x_1, x_2,…$ is defined by the following recursion
$x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n}$
Prove that
$\sum_{n=1}^{2018}{\frac{1}{x_n}}<3$.
1965 Swedish Mathematical Competition, 4
Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.
2016 Saudi Arabia IMO TST, 1
On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions:
$$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 $$ Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct.
Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.