Found problems: 5923
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
We define the sequence $x_n$ so that
\[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\]
Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.
Consider a grid with $n{}$ lines and $m{}$ columns $(n,m\in\mathbb{N},m,n\ge2)$ made of $n\cdot m \; 1\times1$ squares called ${cells}$. A ${snake}$ is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the ${length}$ of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.
Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$.
$\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$
In each square of an $n\times{n}$ grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of $n$ the minimal number of touches that are necessary to turn on every lamp.
A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.
Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(9cm);
for (int x=0; x<=11; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (11, y), linewidth(.5pt));
filldraw((0,4)--(0,3)--(2,3)--(2,4)--cycle, gray, gray);
filldraw((1,1)--(1,2)--(3,2)--(3,1)--cycle, gray, gray);
filldraw((4,1)--(4,4)--(5,4)--(5,1)--cycle, gray, gray);
filldraw((7,0)--(7,3)--(6,3)--(6,0)--cycle, gray, gray);
filldraw((7,4)--(7,5)--(6,5)--(6,4)--cycle, gray, gray);
filldraw((8,1)--(8,2)--(10,2)--(10,1)--cycle, gray, gray);
filldraw((9,4)--(9,3)--(11,3)--(11,4)--cycle, gray, gray);
draw((0,0)--(11,0)--(11,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(1, 2, "10");
foo(4, 0, "31");
foo(5, 0, "26");
foo(10, 0, "59");
foo(0, 4, "3");
foo(7, 4, "59");
[/asy]
A sequence $\{an\}$ of positive integers is defined by
\[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\]
Determine the positive integers that occur in the sequence.
Let $ \left(a_{n} \right)_{n \equal{} 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n \plus{} 1} \equal{} a_{n} a_{n \plus{} 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$
Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.
Define $f(x, y)$ to be $\frac{|x|}{|y|}$ if that value is a positive integer, $\frac{|y|}{|x|}$ if that value is a positive integer, and zero otherwise. We say that a sequence of integers $\ell_1$ through $\ell_n$ is [i]good [/i] if $f(\ell_i, \ell_{i+1})$ is nonzero for all $i$ where $1 \le i \le n - 1$, and the score of the sequence is $\sum^{n-1}_{i=1} f(\ell_i, \ell_{i+1})$
Let $b$ and $c$ be real numbers not both equal to $1$ such that $1,b,c$ is an arithmetic progression and $1,c,b$ is a geometric progression. What is $100(b-c)$?
[i]Proposed by Noah Kravitz[/i]
Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$:
$$\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.$$
Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$.
[i]Remark:[/i] A sequence $x_1,x_2,x_3,\dotsc $ is [i]eventually periodic[/i] if there are positive integers $c$ and $t$ such that $x_n=x_{n+t}$ for all $n\geqslant c$.
[i]Proposed by Petar Nizié-Nikolac[/i]
Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by
$n.$
[b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$
Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and
\[
a_{n+1} =
\begin{cases}
a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\
a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$}
\end{cases}\qquad\text{for }n=1,2,\ldots.
\]
for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.
For each positive integer $k$, let $S_k$ be the set of real numbers that can be expressed in the form
\[\frac{1}{n_1}+\frac{1}{n_2}+\dots+\frac{1}{n_k},\]
where $n_1,n_2\dots,n_k$ are positive integers.
Prove that $S_k$ does not contain an infinite strictly increasing sequence.
An [i]augmentation[/i] on a graph $G$ is defined as doing the following:
- Take some set $D$ of vertices in $G$, and duplicate each vertex $v_i \in D$ to create a new vertex $v_i'$.
- If there's an edge between a pair of vertices $v_i, v_j \in D$, create an edge between vertices $v_i'$ and $v_j'$. If there's an edge between a pair of vertices $v_i \in D$, $v_j \notin D$, you can choose to create an edge between $v_i'$ and $v_j$ but do not have to.
A graph is called [i]reachable[/i] from $G$ if it can be created through some sequence of augmentations on $G$. Some graph $H$ has $n$ vertices and satisfies that both $H$ and the complement of $H$ are reachable from a complete graph of $2021$ vertices. If the maximum and minimum values of $n$ are $M$ and $m$, find $M+m$.
[i]Proposed by Oliver Hayman[/i]
For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies
$$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$
Determine the number of divisors of $2024^{2024}$ amongst the sequence.
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
For integers $m,n\geq 1$, let $A(n,m)$ be the number of sequences $(a_1,\cdots,a_{nm})$ of integers satisfying the following two properties:
[list=a]
[*]Each integer $k$ with $1\leq k\leq n$ occurs exactly $m$ times in the sequence $(a_1,\cdots,a_{nm})$.
[*]If $i,j,$ and $k$ are integers such that $1\leq i\leq nm$ and $1\leq j\leq k\leq n$, then $j$ occurs in the sequence $(a_1,\cdots,a_i)$ at least as many times as $k$ does.[/list]
For example, if $n=2$ and $m=5$, a possible sequence is $(a_1,\cdots,a_{10})=(1,1,2,1,2,2,1,2,1,2)$. On the other hand, the sequence $(a_1,\cdots,a_{10})=(1,2,1,2,2,1,1,1,2,2)$ does not satisfy property (2) for $i=5$, $j=1$, and $k=2$.
Prove that $A(n,m)=A(m,n)$.
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]
The function $f:\mathbb{N}\rightarrow \mathbb{N}$ is [b]peruvian[/b] if it satifies the following two properties:
$\triangleright f$ is strictly increasing.
$\triangleright$ The numbers $a_1,a_2,a_3,\dots$ where $a_1=f(1)$ and $a_{n+1}=f(a_n)$ for every $n\geq 1$, are in arithmetic progression.
Determine all peruvian functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(1)=3$.
Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations:
$1.$ Erase the written number $m$ and write it´s cube $m^3$.
$2.$ Erase the written number $m$ and write the product $2023m$.
Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.
Prove that there exist two infinite sequences $ \{a_n\}_{n\ge 1}$ and $ \{b_n\}_{n\ge 1}$ of positive integers such that the following conditions hold simultaneously:
$ (i)$ $ 0 < a_1 < a_2 < a_3 < \cdots$;
$ (ii)$ $ a_n < b_n < a_n^2$, for all $ n\ge 1$;
$ (iii)$ $ a_n \minus{} 1$ divides $ b_n \minus{} 1$, for all $ n\ge 1$
$ (iv)$ $ a_n^2 \minus{} 1$ divides $ b_n^2 \minus{} 1$, for all $ n\ge 1$
[19 points out of 100 for the 6 problems]
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?