Olympiad problems

1985 IMO Shortlist, 17

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 $n \geq 1.$

2015 Saudi Arabia BMO TST, 1

Tags: recurrence relation , sequence

Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$. Nguyễn Duy Thái Sơn

1973 Spain Mathematical Olympiad, 3

Tags: complex analysis , analysis , recurrence relation , recurrence , sequence

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 $1.15$.

2011 China Northern MO, 1

Tags: recurrence relation , sequence , algebra

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 $a_{2011}$.

2009 China Northern MO, 1

Tags: recurrence relation , algebra , sequence

Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ ) Find the general term of {$x_n$}

1985 IMO Longlists, 63

Tags: algebra , inequalities , sequence , inequality , calculus , recurrence relation

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

1990 IMO Shortlist, 18

Tags: recursion , function , algebra , recurrence relation

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.$

2010 NZMOC Camp Selection Problems, 1

Tags: algebra , recurrence relation , sequence

For any two positive real numbers $x_0 > 0$, $x_1 > 0$, a sequence of real numbers is defined recursively by $$x_{n+1} =\frac{4 \max\{x_n, 4\}}{x_{n-1}}$$ for $n \ge 1$. Find $x_{2010}$.

1996 Austrian-Polish Competition, 3

Tags: polynomial , algebra , recurrence relation

The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.

1998 Moldova Team Selection Test, 10

Tags: bounded , sequence , unbounded , digit , product of digits , recurrence relation , number theory

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

1995 IMO Shortlist, 5

Tags: arithmetic sequence , recurrence relation , algebra , function

For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which \[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\] Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$

2009 Postal Coaching, 3

Tags: function , sum , combination , recurrence relation

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 $$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$

2016 Switzerland - Final Round, 6

Tags: sequence , number theory , recurrence relation

Let $a_n$ be a sequence of natural numbers defined by $a_1 = m$ and for $n > 1$. We call apair$ (a_k, a_{\ell })$ [i]interesting [/i] if (i) $0 < \ell - k < 2016$, (ii) $a_k$ divides $a_{\ell }$. Show that there exists a $m$ such that the sequence $a_n$ contains no interesting pair.

2023 Brazil Undergrad MO, 5

Tags: generating functions , recurrence relation , modular arithmetic , algebra

A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.

1999 Brazil Team Selection Test, Problem 3

Tags: sequence , recurrence relation , algebra

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 $n\ge0$.

2011 Grand Duchy of Lithuania, 2

Tags: algebra , inequalities , recurrence relation

Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality $(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$

1984 Bundeswettbewerb Mathematik, 3

Tags: algebra , sequence , recurrence relation

The sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,... $suffices for all positive integers $n$ of the following recursion: $a_{n+1} = a_n - b_n$ and $b_{n+1} = 2b_n$, if $a_n \ge b_n$, $a_{n+1} = 2a_n$ and $b_{n+1} = b_n - a_n$, if $a_n < b_n$. For which pairs $(a_1, b_1)$ of positive real initial terms is there an index $k$ with $a_k = 0$?

2021 Regional Olympiad of Mexico West, 3

Tags: sequence , recurrence relation , number theory

The sequence of real numbers $a_1, a_2, a_3, ...$ is defined as follows: $a_1 = 2019$, $a_2 = 2020$, $a_3 = 2021$ and for all $n \ge 1$ $$a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.$$ Show that this sequence does not contain numbers of the form $m^6$ where $m$ is a positive integer.

1998 Portugal MO, 6

Tags: algebra , inequalities , sequence , recurrence relation

Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.

1996 VJIMC, Problem 2

Tags: sequence , summation , binomial , algebra , recurrence relation

Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether $$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$

2008 Federal Competition For Advanced Students, P1, 3

Tags: algebra , inequalities , sequence , recurrence relation

Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$. Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.

1992 IMO Shortlist, 2

Tags: recurrence relation , functional equation , quadratic , algebra

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.

2019 Federal Competition For Advanced Students, P1, 1

Tags: algebra , sequence , recurrence , recurrence relation

We consider the two sequences $(a_n)_{n\ge 0}$ and $(b_n) _{n\ge 0}$ of integers, which are given by $a_0 = b_0 = 2$ and $a_1= b_1 = 14$ and for $n\ge 2$ they are defined as $a_n = 14a_{n-1} + a_{n-2}$ , $b_n = 6b_{n-1}-b_{n-2}$. Determine whether there are infinite numbers that occur in both sequences

1965 Swedish Mathematical Competition, 4

Tags: recurrence relation , algebra , sequence , function

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 $f(a_0), f(a_1), ...$.

1991 Romania Team Selection Test, 2

Tags: recurrence relation , sequence , coprime , number theory

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.