Found problems: 5923
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.
In an arithmetic sequence, the di fference of consecutive terms in constant. We consider sequences of integers in which the di fference of consecutive terms equals the sum of the differences of all preceding consecutive terms.
Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?
A sequence $a_n$ of real numbers satisfies $a_1=1$, $a_2=0$, and $a_n=(S_{n-1}+1)S_{n-2}$ for all integers $n\geq3$, where $S_k=a_1+a_2+\dots+a_k$ for positive integers $k$. What is the smallest integer $m>2$ such that $127$ divides $a_m$?
[i]2020 CCA Math Bonanza Individual Round #9[/i]
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$.
Prove that the set $S$ is finite.
[i]Proposed by F. Bakharev[/i]
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 $$n|pa_{n-1}+1$$
Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by
\[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\]
Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.
Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$.
[i]Proposed by Mark Sellke[/i]
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for
which the following property holds:
If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$.
Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]
A sequence $ (a_n)$ of positive integers satisfies: $ a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.
Given a sequence $ (c_n) $ of natural numbers defined recursively: $ c_1 = 2 $, $ c_{n+1} = \left[ \frac{3}{2}c_n\right] $. Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence.
For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies:
$$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$
It is further given that $a_4 = 194$. Calculate $a_5$.
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
Let $a_1,a_2,\ldots,a_6$ be a sequence of integers such that for all $1 \le i \le 5$,$$a_{i+1}=\frac{a_i}{3} \quad \text{or} \quad a_{i+1}={-}2a_i.$$Find the number of possible positive values of $a_1+a_2+\cdots+a_6$ less than $1000$.
[i]Proposed by [b]stayhomedomath[/b][/i]
Consider a recursively defined sequence $a_n$ with $a_1=1$ such that, for $n \ge 2,$ $a_n$ is formed by appending the last digit of $n$ to the end of $a_{n-1}.$ For a positive integer $m,$ let $\nu_3(m)$ be the largest integer $t$ such that $3^t \mid m.$ Compute $$\sum_{n=1}^{810} \nu_3(a_n).$$
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?
Let $x_0 = 1$ and $x_1 = 2003$ and define the sequence $(x_n)_{n \ge 0}$ by: $x_{n+1} =\frac{x^2_n + 1}{x_{n-1}}$ , $\forall n \ge 1$
Prove that for every $n \ge 2$ the denominator of the fraction $x_n$, when $x_n$ is expressed in lowest terms is a power of $2003$.
Three numbers are initially written on the board: 2023, 2024, and 2025. In each move, you can increase any two of these numbers by 1 and decrease the third one by 2.
a) Determine whether it is possible to perform a sequence of operations such that the board eventually contains two numbers that are equal.
b) Calculate the number of all possible ordered triples of positive integers that can be obtained by performing such operations some number of times.
[i]Proposed by Giorgi Arabidze, Georgia [/i]