Found problems: 307
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.
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 $n \ge 0$.
2013 Saudi Arabia Pre-TST, 4.1
Let $a_1,a_2, a_3,...$ be a sequence of real numbers which satisfy the relation $a_{n+1} =\sqrt{a_n^2 + 1}$
Suppose that there exists a positive integer $n_0$ such that $a_{2n_0} = 3a_{n_0}$ . Find the value of $a_{46}$.
1990 IMO Longlists, 37
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.$
1994 IMO Shortlist, 4
Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$.
If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$.
If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$.
Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.
1992 IMO Longlists, 49
Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that
\[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\]
prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that
\[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]
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 $a_1$.
2009 Kyiv Mathematical Festival, 5
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be?
b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?
2020 Tournament Of Towns, 4
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
1999 Brazil Team Selection Test, Problem 3
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
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 $n \geq 1,$
\[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]
1976 IMO Longlists, 3
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 $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
2010 China Northern MO, 1
It is known that the sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}$, $(n \ge 2)$, find the general term of $a_n$.
1988 Dutch Mathematical Olympiad, 2
Given is a number $a$ with 0 $\le \alpha \le \pi$. A sequence $c_0,c_1, c_2,...$ is defined as
$$c_0=\cos \alpha$$
$$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...$$
Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$
Mathley 2014-15, 5
Given the sequence $(u_n)_{n=1}^{\infty}$, where $u_1 = 1, u_2 = 2$, and $u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2}$ for any positive integers $n$. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers $(u_n)_{n=1}^{\infty}$
Nguyen Duy Thai Son, The University of Danang, Da Nang.
1999 Austrian-Polish Competition, 5
A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$
Prove that there exists at most one $n$ for which $a_n$ is a perfect square.
1976 IMO Longlists, 13
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 $\leq x)$
2006 Cuba MO, 7
The sequence $a_1, a_2, a_3, ...$ satisfies that $a_1 = 3$, $a_2 = -1$, $a_na_{n-2} +a_{n-1} = 2$ for all $n \ge 3$. Calculate $a_1 + a_2+ ... + a_{99}$.
1989 ITAMO, 6
Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by
$f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$,
$f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$.
Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.
2001 IMO Shortlist, 5
Find all positive integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots +
\frac{a_{n-1}}{a_n},
\]
where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.
1966 Dutch Mathematical Olympiad, 2
For all $n$, $t_{n+1} = 2(t_n)^2 - 1$. Prove that gcd $(t_n,t_m) = 1$ if $n \ne m$.
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.
1978 Germany Team Selection Test, 3
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2013 Korea Junior Math Olympiad, 3
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$).
For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$
Prove that $b_n$ is positive integer.