Found problems: 963
2017 Azerbaijan Senior National Olympiad, C3
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{17}+2016)$ on the board? \\
(Explain your answer) \\
[hide=Note]This type of the question is well known but I am going to make a collection so, :blush: [/hide]
1992 Putnam, B3
For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows:
$$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$
Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.
1977 Vietnam National Olympiad, 5
The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.
2022 USA TSTST, 9
Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties:
[list=disc]
[*]Each element of the sequence is between $1$ and $n$, inclusive.
[*]For any two different contiguous subsequence of the sequence with length between $2$ and $k$ inclusive, the multisets of values in those two subsequences is not the same.
[*]The sequence has length at least $0.499n^2$
[/list]
2023 Indonesia TST, 1
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 $a_{2022}\leq 1$.
2017 Hanoi Open Mathematics Competitions, 10
Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
2018 Poland - Second Round, 6
Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let
$b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$
for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.
2021 Indonesia MO, 3
A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.
1980 Bundeswettbewerb Mathematik, 4
A sequence of integers $a_1,a_2,\ldots $ is defined by $a_1=1,a_2=2$ and for $n\geq 1$,
$$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. $$
(a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms.
(b) Prove that no term of the sequence is zero.
(c) Show that if $n = 2^k - 1$ for $k\geq 2$, then $a_n$ is divisible by $7$.
2022 Nigerian MO round 3, Problem 1
Integer sequence $(x_{n})$ is defined as follows;
$x_{1} = 1$, and for each integer $n \geq 1$, $x_{n+1}$ is equal to the largest number that can be obtained by permutation of the digits of $x_{n}+2$. Find the smallest $n$ for which the decimal representation of $x_{n}$ contains exactly $2022$ digits
1991 IMO Shortlist, 28
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
2015 Hanoi Open Mathematics Competitions, 1
What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$?
(A) $ -1$ (B) $ -2$ (C) $-3$ (D) $-4$ (E) None of the above
2025 Turkey Team Selection Test, 8
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]
Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]
is satisfied. Prove that this sequence must be eventually constant.
2005 Germany Team Selection Test, 1
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
2002 German National Olympiad, 4
Given a positive real number $a_1$, we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let
$$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$
Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.
2020 Canadian Junior Mathematical Olympiad, 1
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
1988 Swedish Mathematical Competition, 6
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$.
Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.
1999 Mongolian Mathematical Olympiad, Problem 3
Let $(a_n)^\infty_{n=1}$ be a non-decreasing sequence of natural numbers with $a_{20}=100$. A sequence $(b_n)$ is defined by $b_m=\min\{n|an\ge m\}$. Find the maximum value of $a_1+a_2+\ldots+a_{20}+b_1+b_2+\ldots+b_{100}$ over all such sequences $(a_n)$.
2019 IMC, 4
Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers
1994 Tournament Of Towns, (427) 4
From the sequence $1,\frac12, \frac13, ...$ can one choose
(a) a subsequence of $100$ different numbers,
(b) an infinite subsequence
such that each number (beginning from the third) is equal to the difference between the two preceding numbers ($a_k=a_{k-2}-a_{k-1}$)?
(SI Tokarev)
2015 Taiwan TST Round 3, 3
Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ .
[i]Proposed by Austria[/i]
2010 Junior Balkan Team Selection Tests - Romania, 2
Show that:
a) There is a sequence of non-zero natural numbers $a_1, a_2, ...$ uniquely determined, so that:
$n = \sum _ {d | n} a _ d$ for whatever $n \in N ^ {*}$ .
b) There is a sequence of non-zero natural numbers $b_1, b_2, ...$ uniquely determined, so that:
$n = \prod _ {d | n} b _ d$ for whatever $n \in N ^ {*}$ .
Note: The sum from a), respectively the product from b), are made after all the natural divisors $d$ of the number $n$ , including $1$ and $n$ .
2012 Korea Junior Math Olympiad, 6
$p > 3$ is a prime number such that $p|2^{p-1} - 1$ and $p \nmid 2^x - 1$ for $x = 1, 2,...,p-2$. Let $p = 2k + 3$. Now we define sequence $\{a_n\}$ as $$a_i = a_{i+k} = 2^i \,\, (1 \le i \le k ), \,\,\,\, a_{j+2k} = a_ja_{j+k} \,\, (j \le 1)$$
Prove that there exist $2k$ consecutive terms of sequence $a_{x+1},a_{x+2},..., a_{x+2k}$ such that $a_{x+i } \not\equiv a_{x+j}$ (mod $p$) for all $1 \le i < j \le 2k$ .
2021 SAFEST Olympiad, 2
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]