Found problems: 5923
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.
Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.
Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.
How many ordered sequences of $36$ digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from $0$ to $9$.)
The bell number $b_n$ is the number of ways to partition the set $\{1,2,\ldots,n\}$. For example $b_3=5$. Find a recurrence for $b_n$ and show that $b_n=e^{-1}\sum_{k\geq 0} \frac{k^n}{k!}$. Using a combinatorial proof show that the number of ways to partition $\{1,2,\ldots,n\}$, such that now two consecutive numbers are in the same block, is $b_{n-1}$.
Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.
Show that the number $x$ is rational if and only if three distinct terms that form a geometric progression can be chosen from the sequence
\[x, ~ x+1, ~ x+2,~ x+3,\ldots . \]
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together.
For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$.
Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have.
[i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]
A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that
\[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \]
Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]
An equilateral triangle with side $n$ is built with $n^{2}$ [i]plates[/i] - equilateral triangles with side $1$. Each plate has one side black, and the other side white. We name [i]the move[/i] the following operation: we choose a plate $P$, which has common sides with at least two plates, whose visible side is the same color as the visible side of $P$. Then, we turn over plate $P$.
For any $n\geq 2$ decide whether there exists an innitial configuration of plates permitting for an infinite sequence of moves.
Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
Let $N$ be the number of sequences of positive integers greater than $ 1$ where the product of all of the terms of the sequence is $12^{64}$. If $N$ can be expressed as $a(2^b)$ ), where $a$ is an odd positive integer, determine $b$.
Suppose $P(x)$ is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial $P(x)$ on a board. At every step, if the polynomial on the board is $f(x)$, he can replace it with
1. $f(x)+c$ for a real number $c$, or
2. the polynomial $P(f(x))$.
Can he always find a finite sequence of steps so the final polynomial on the board has exactly $2020$ real roots? What about $2021$?
[i]~Sutanay Bhattacharya[/i]
Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$
$$A_{n+1}=\frac{A_n+2}{A_n +1}.$$
Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$
$$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$
Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.
Let $\{a_n\}$ be a nonnegative real sequence. Define
$$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$
Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$;
(b) $f\left(t_0\right)=1 / 2$;
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$;
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?
Construct a geometric gure in a sequence of steps. In step $1$, begin with a $4\times 4$ square. In step $2$, attach a $1\times 1$ square onto the each side of the original square so that the new squares are on the outside of the original square, have a side along the side of the original square, and the midpoints of the sides of the original square and the attached square coincide. In step $3$, attach a $\frac14\times \frac14$ square onto the centers of each of the $3$ exposed sides of each of the $4$ squares attached in step $2$. For each positive integer $n$, in step $n + 1$, attach squares whose sides are $\frac14$ as long as the sides of the squares attached in step n placing them at the centers of the $3$ exposed sides of the squares attached in step $n$. The diagram shows the gure after step $4$.
If this is continued for all positive integers $n$, the area covered by all the squares attached in all the steps is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
[img]https://cdn.artofproblemsolving.com/attachments/2/1/d963460373b56906e93c4be73bc6a15e15d0d6.png[/img]
Let $ a_0,a_1,\dots,a_{n \plus{} 1}$ be natural numbers such that $ a_0 \equal{} a_{n \plus{} 1} \equal{} 1$, $ a_i>1$ for all $ 1\leq i \leq n$, and for each $ 1\leq j\leq n$, $ a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}$. Prove that there exist one $ 2$ in the sequence.
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]