Found problems: 963
1954 Putnam, A6
Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that
$$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$
Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.
2019 IFYM, Sozopol, 1
We define the sequence $a_n=(2n)^2+1$ for each natural number $n$. We will call one number [i]bad[/i], if there don’t exist natural numbers $a>1$ and $b>1$ such that $a_n=a^2+b^2$. Prove that the natural number $n$ is [i]bad[/i], if and only if $a_n$ is prime.
2021 Kyiv City MO Round 1, 10.5
The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where
$a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$.
[i]Proposed by Kyrylo Holodnov[/i]
2024 Rioplatense Mathematical Olympiad, 4
Let $N$ be a positive integer. A non-decreasing sequence $a_1 \le a_2 \le \dots$ of positive integers is said to be $N$-rioplatense if there exists an index $i$ such that $N = \frac{i}{a_i}$. Show that every sequence $2024$-rioplatense is $k$-rioplatense for $k=1, 2, 3, \dots, 2023$.
1974 Chisinau City MO, 73
For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold:
1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$
2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$
Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$
2021 Saudi Arabia IMO TST, 10
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2001 Rioplatense Mathematical Olympiad, Level 3, 6
For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let $f(m)=m+S(m)$.
Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$
2015 Belarus Team Selection Test, 3
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
2019 Tournament Of Towns, 5
Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
2016 Iran MO (3rd Round), 1
The sequence $(a_n)$ is defined as:
$$a_1=1007$$
$$a_{i+1}\geq a_i+1$$
Prove the inequality:
$$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$
1985 Tournament Of Towns, (102) 6
The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$
{A. Andjans, Riga)
2005 Slovenia Team Selection Test, 4
Find the number of sequences of $2005$ terms with the following properties:
(i) No three consecutive terms of the sequence are equal,
(ii) Every term equals either $1$ or $-1$,
(iii) The sum of all terms of the sequence is at least $666$.
2023 Belarus Team Selection Test, 3.2
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
2022 Bosnia and Herzegovina IMO TST, 3
An infinite sequence is given by $x_1=2, x_2=7, x_{n+1} = 4x_n - x_{n-1}$ for all $n \geq 2$. Does there exist a perfect square in this sequence?
[hide="Remark"]During the test the initial value of $x_1$ was given as $1$, thus the problem was not graded[/hide]
2024 Taiwan TST Round 3, 6
Find all positive integers $n$ and sequence of integers $a_0,a_1,\ldots, a_n$ such that the following hold:
1. $a_n\neq 0$;
2. $f(a_{i-1})=a_i$ for all $i=1,\ldots, n$, where $f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0$.
[i]
Proposed by usjl[/i]
2021 Iran Team Selection Test, 5
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
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 $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
2001 Austrian-Polish Competition, 6
Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.
1985 All Soviet Union Mathematical Olympiad, 402
Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that
a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$
b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$
1992 All Soviet Union Mathematical Olympiad, 570
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
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$.
2020 IMO Shortlist, N7
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
2019 Taiwan APMO Preliminary Test, P4
We define a sequence ${a_n}$:
$$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$
(1)Find $\lfloor a_{2019}\rfloor$
(2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$
2017 Puerto Rico Team Selection Test, 4
We define the sequences $a_n =\frac{n (n + 1)}{2}$ and $b_n = a_1 + a_2 +… + a_n$.
Prove that there is no integer $n$ such that $b_n = 2017$.
2010 Indonesia TST, 2
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]