Found problems: 1239
2019 Jozsef Wildt International Math Competition, W. 32
Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
1996 IMO Shortlist, 2
Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$
\[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\]
Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that
\[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$
2015 APMO, 5
Determine all sequences $a_0 , a_1 , a_2 , \ldots$ of positive integers with $a_0 \ge 2015$ such that for all integers $n\ge 1$:
(i) $a_{n+2}$ is divisible by $a_n$ ;
(ii) $|s_{n+1} - (n + 1)a_n | = 1$, where $s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0$ .
[i]Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand[/i]
2017 Bundeswettbewerb Mathematik, 4
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$:
(a) The number $a_n$ is a positive integer.
(b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$.
(c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.
Revenge EL(S)MO 2024, 3
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
2010 Saudi Arabia BMO TST, 4
Let $f : N \to [0, \infty)$ be a function satisfying the following conditions:
a) $f(4)=2$
b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$.
Find $f(n)$ in closed form.
1980 IMO Longlists, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2017 IFYM, Sozopol, 6
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
2012 Bogdan Stan, 2
Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence:
$$ x_{n+3} =\sqrt[3]{3x_n-2} . $$
Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[i]Cristinel Mortici[/i]
2018 Estonia Team Selection Test, 10
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 $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
2018 Vietnam National Olympiad, 1
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
1993 Romania Team Selection Test, 1
Define the sequence ($x_n$) as follows: the first term is $1$, the next two are $2,4$, the next three are $5,7,9$, the next four are $10,12,14,16$, and so on. Express $x_n$ as a function of $n$.
1998 Bosnia and Herzegovina Team Selection Test, 6
Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$
$a)$ Find $u_{1998}$
$b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$
2023 Brazil National Olympiad, 6
For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties:
• $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$;
• $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$.
Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.
1991 ITAMO, 6
We say that each positive number $x$ has two sons: $x+1$ and $\frac{x}{x+1}$. Characterize all the descendants of number $1$.
1996 Tuymaada Olympiad, 6
Given the sequence
$f_1(a)=sin(0,5\pi a)$
$f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$
$...$
$f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number.
What limit aspire the members of this sequence as $n \to \infty$?
1982 Polish MO Finals, 5
Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.
2015 APMO, 3
A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.
(i) The value of $a_0$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $
(iii) There exists a positive integer $k$ such that $a_k = 2014$.
Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$.
[i]Proposed by Wang Wei Hua, Hong Kong[/i]
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]
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 $n \ge 0$.
2001 BAMO, 5
For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$.
Prove that $3^{334}$ divides $a_{2001}$.
(A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and
onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)
2022 Bosnia and Herzegovina BMO TST, 1
Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold:
$i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$;
$ii)$ $\sum_{j=1}^{k} a_j \leq 1$.
Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$
2024 SG Originals, Q4
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
[i]Proposed by DVDthe1st[/i]
2010 Indonesia TST, 1
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and
$$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$
Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
2016 Tuymaada Olympiad, 1
The sequence $(a_n)$ is defined by $a_1=0$,
$$
a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.
$$
Prove that $a_{2016}>{1\over 2}+a_{1000}$.