Found problems: 307
2004 Thailand Mathematical Olympiad, 7
Let f be a function such that $f(0) = 0, f(1) = 1$, and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$. Find f(n) in terms of $n$.
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 $a_n$ .
1983 IMO Longlists, 19
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.
1971 IMO Shortlist, 9
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 $x.$
2019 Saint Petersburg Mathematical Olympiad, 1
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
1992 Tournament Of Towns, (354) 3
Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
2009 Postal Coaching, 3
Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions
(i) $f(0, 0) = 1$, $f(0, 1) = 1$
(ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and
(iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of
$$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$
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 $x$.
2010 Morocco TST, 1
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 ?
2001 Moldova National Olympiad, Problem 2
Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.
1996 Greece National Olympiad, 1
Let $a_n$ be a sequence of positive numbers such that:
i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$
ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$.
(a) Prove that $(a_n)$ is a geometric progression.
(n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$
1992 IMO Longlists, 55
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.
2022 Assara - South Russian Girl's MO, 6
There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$.
1969 IMO Shortlist, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
1990 Tournament Of Towns, (271) 5
The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$.
(M Kontsevich, Moscow)
1983 Austrian-Polish Competition, 8
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n
(b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.
2021 Saudi Arabia JBMO TST, 1
Let $(a_n)_{n\ge 1}$ be a sequence given by $a_1 = 45$ and $$a_n = a^2_{n-1} + 15a_{n-1}$$ for $n > 1$. Prove that the sequence contains no perfect squares.
2014 Federal Competition For Advanced Students, 3
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
V Soros Olympiad 1998 - 99 (Russia), 10.6
Find the formula for the general term of the sequence an, for which $a_1 = 1$, $a_2 = 3$, $a_{n+1} = 3a_n-2a_{n-1}$ (you need to express an in terms of $n$).
1991 Romania Team Selection Test, 2
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
2008 Postal Coaching, 1
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{
\begin{array}{ll}
x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\
& \\
x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\
\end{array}
\right. $$
where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.
2012 China Northern MO, 5
Let $\{a_n\}$ be the sequance with $a_0=0$, $a_n=\frac{1}{a_{n-1}-2}$ ($n\in N_+$). Select an arbitrary term $a_k$ in the sequence $\{a_n\}$ and construct the sequence $\{b_n\}$: $b_0=a_k$, $b_n=\frac{2b_{n-1}+1} {b_{n-1}}$ ($n\in N_+$) . Determine whether the sequence $\{b_n\}$ is a finite sequence or an infinite sequence and give proof.
1996 IMO Shortlist, 3
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$
b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$
2014 Costa Rica - Final Round, 5
Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$
1964 Vietnam National Olympiad, 4
Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$. Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$, where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$.