Found problems: 1239
Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying
\[ x_{n+1}=A-\frac{1}{x_n} \]
for every integer $n \ge 1$, has only finitely many negative terms.
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 $n$.
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
Let $c_1, \cdots, c_n \ (n \geq 2)$ be real numbers such that $0 \leq \sum c_i \leq n$. Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$
Let $a_1 \le a_2 \le ... \le a_n$ be real numbers for which $$\sum_{i=1}^{n} a_i^{2k+1} = 0$$ holds for all integers $0 \le k < n$. Show that in this case, $a_i = -a_{n+1-i}$ holds for all $1 \le i \le n$.
Determine how many $100$-positive integer sequences satisfy the two conditions following:
- At least one term of the sequence is equal to $4$ or $5$.
- Any two adjacent terms differ as a maximum in $2$.
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]
Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
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]
Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers.
Define the sequence $f(A)$ to be
$$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$
and for $k$ a positive integer define $f^k (A)$ to be$ f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$)
Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?
Consider the sequence $\{1,3,13,31,...\}$ that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the $100$th position of that sequence.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/3531353472a748e3e0b1497a088472691f67fd.png[/img]
Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and
\[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\]
for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent.
Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $
For any positive integer $x$, we set
$$
g(x) = \text{ largest odd divisor of } x,
$$
$$
f(x) = \begin{cases}
\frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}
\end{cases}
$$
Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
(a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$.
(b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
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 $a_{2022}\leq 1$.
Let $u_1=1,u_2=1$ and for all $k \geq 1$'s
$$u_{k+2}=u_{k+1}+u_{k}$$
Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$