Olympiad problems

2020 Romania EGMO TST, P1

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

2018 Estonia Team Selection Test, 10

Tags: algebra , Sequence , recurrence relation , inequalities

A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations $b_1 = a_1$ , $b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ , $b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$. Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$

1988 All Soviet Union Mathematical Olympiad, 485

Tags: polynomial , Integer sequence , Sequence , greatest common divisor

The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.

2011 China Northern MO, 1

Tags: algebra , recurrence relation , Sequence

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 $a_{2011}$.

1969 IMO Shortlist, 28

Tags: polynomial , number theory , Sequence , Divisibility , recurrence relation , IMO Shortlist , IMO Longlist

$(GBR 5)$ 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 $a(u_b -1).$

2023 Thailand TST, 1

Tags: algebra , Sequence , ISL 2022

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

2022 Assara - South Russian Girl's MO, 6

Tags: algebra , Sequence , recurrence relation

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 $a_{2023} = a_1$, $a_{2024} = a_2$.

2021 Romania EGMO TST, P1

Tags: number theory , algebra , number theory unsolved , Sequence , algebra unsolved

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

2003 Estonia Team Selection Test, 4

Tags: Sequence , combinatorics

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

2005 Serbia Team Selection Test, 3

Tags: polynomial , algebra , Sequence

Find all polynomial with real coefficients such that: P(x^2+1)=P(x)^2+1

1976 IMO Longlists, 3

Tags: Inequality , Convexity , Sequence , recurrence relation , IMO Shortlist

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 $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

2020 Moldova EGMO TST, 3

Tags: Sequence , algebra unsolved

Let the sequence $a_n$, $n\geq2$, $a_n=\frac{\sqrt[3]{n^3+n^2-n-1}}{n} $. Find the greatest natural number $k$ ,such that $a_2 \cdot a_3 \cdot . . .\cdot a_k <8$

2003 Estonia Team Selection Test, 4

Tags: Sequence , combinatorics

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

2018 Vietnam Team Selection Test, 4

Tags: algebra , Sequence , inequalities

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.

2021 Azerbaijan EGMO TST, 2

Tags: Sequence , unbounded , algebra , AZE EGMO TST

Given a non-decreasing unbounded sequence $a_n,$ construct a new sequence $b_n$ as follows $$b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}$$ Prove that $b_n$ is also unbounded.

1988 Austrian-Polish Competition, 5

Tags: Perfect Square , number theory , recurrence relation , Sequence

Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.

2016 China Second Round Olympiad, 4

Tags: Sequence , number theory , China second round

Let $p>3$ and $p+2$ are prime numbers,and define sequence $$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$ show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$

1983 All Soviet Union Mathematical Olympiad, 356

Tags: Sequence , number theory , periodical , last digits , Digits , floor function

The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

2024 District Olympiad, P2

Tags: algebra , Sequence , fractional part

Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a] [*]Determine the general formula for $a_n.$ [*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$ [/list]

KoMaL A Problems 2019/2020, A. 776

Tags: Sequence , number theory

Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let $$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$ Prove that $f_n$ satisfy the following recursion: $$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$

2014 Taiwan TST Round 2, 2

Tags: function , combinatorics , Additive combinatorics , Sequence , IMO Shortlist

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

2020 Macedonia Additional BMO TST, 4

Tags: number theory , Sequence

Prove that for all $n\in \mathbb{N}$ there exist natural numbers $a_1,a_2,...,a_n$ such that: $(i)a_1>a_2>...>a_n$ $(ii)a_i|a^2_{i+1},\forall i\in\{1,2,...,n-1\}$ $(iii)a_i\nmid a_j,\forall i,j\in \{1,2,...,n\},i\neq j$

2019 District Olympiad, 3

Tags: Sequence , floor function , Arithmetic Progression , algebra

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$ $\textbf{a)}$ Prove that the given sequence is an arithmetic progression. $\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.

2011 Abels Math Contest (Norwegian MO), 3a

Tags: inequalities , algebra , Sequence

The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ . .

2009 239 Open Mathematical Olympiad, 1

Tags: number theory , Sequence

In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.