Found problems: 307
1990 IMO Longlists, 62
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by
\[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\
n \minus{} b, & \text{if } n >M. \end{cases}
\]
Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$
1976 Polish MO Finals, 2
Four sequences of real numbers $(a_n), (b_n), (c_n), (d_n)$ satisfy for all $n$,
$$a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n,$$
$$c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n.$$
Prove that if $a_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m$ for some $k\ge 1,n \ge 1$, then $a_2 = b_2 = c_2 = d_2 = 0$.
1971 IMO Shortlist, 1
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 $n \geq 1,$
\[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]
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 $b_m > a_{100}.$
1990 IMO Shortlist, 13
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
1992 IMO Longlists, 32
Let $S_n = \{1, 2,\cdots, n\}$ and $f_n : S_n \to S_n$ be defined inductively as follows: $f_1(1) = 1, f_n(2j) = j \ (j = 1, 2, \cdots , [n/2])$ and
[list]
[*][b][i](i)[/i][/b] if $n = 2k \ (k \geq 1)$, then $f_n(2j - 1) = f_k(j) + k \ (j = 1, 2, \cdots, k);$
[*][b][i](ii)[/i][/b] if $n = 2k + 1 \ (k \geq 1)$, then $f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1) \ (j = 1, 2,\cdots , k).$[/list]
Prove that $f_n(x) = x$ if and only if $x$ is an integer of the form
\[\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}\]
for some positive integer $d.$
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 $[178,9]\le[a,b,c,d]\le[198,9]$.
2012 Singapore Senior Math Olympiad, 4
Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$
Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$
1967 IMO Longlists, 24
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
2010 Saudi Arabia IMO TST, 3
Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$.
Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .
1984 Czech And Slovak Olympiad IIIA, 3
Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation
$$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$
Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation
$$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$
where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1).
Note: $[x]$ indicates the whole part of the number $x$.
1973 Swedish Mathematical Competition, 2
The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.
1976 IMO Shortlist, 2
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
1985 IMO Longlists, 78
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
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$.
2023 ISI Entrance UGB, 6
Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers defined as $u_1 = 1$ and
\[ u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}\]
Prove that $u_n \le \frac{3\sqrt{n}}{2}$ for all $n$.
1972 IMO Longlists, 15
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.
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.
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 $n$.
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 $1$.
V Soros Olympiad 1998 - 99 (Russia), 11.9
The sequence of $a_n$ is determined by the relation
$$a_{n+1}=\frac{k+a_n}{1-a_n}$$
where $k > 0$. It is known that $a_{13} = a_1$. What values can $k$ take?
1969 IMO Longlists, 41
$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]
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$.
2020 Spain Mathematical Olympiad, 2
Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as:
$\bullet$ $f(1) = 1$.
$\bullet$ $f(n) = f(n/2)$ if $n$ is even.
$\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$.
$\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$.
a) Compute $f(2^{2020}-1)$.
b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$.
1989 Dutch Mathematical Olympiad, 1
For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies:
$$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$
It is further given that $a_4 = 194$. Calculate $a_5$.