How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

[u]Round 1[/u] [b]p1.[/b] A positive integer is said to be transcendent if it leaves a remainder of $1$ when divided by $2$. Find the $1010$th smallest positive integer that is transcendent. [b]p2.[/b] The two diagonals of a square are drawn, forming four triangles. Determine, in degrees, the sum of the interior angle measures in all four triangles. [b]p3.[/b] Janabel multiplied $2$ two-digit numbers together and the result was a four digit number. If the thousands digit was nine and hundreds digit was seven, what was the tens digit? [u]Round 2[/u] [b]p4.[/b] Two friends, Arthur and Brandon, are comparing their ages. Arthur notes that $10$ years ago, his age was a third of Brandon’s current age. Brandon points out that in $12$ years, his age will be double of Arthur’s current age. How old is Arthur now? [b]p5.[/b] A farmer makes the observation that gathering his chickens into groups of $2$ leaves $1$ chicken left over, groups of $3$ leaves $2$ chickens left over, and groups of $5$ leaves $4$ chickens left over. Find the smallest possible number of chickens that the farmer could have. [b]p6.[/b] Charles has a bookshelf with $3$ layers and $10$ indistinguishable books to arrange. If each layer must hold less books than the layer below it and a layer cannot be empty, how many ways are there for Charles to arrange his $10$ books? [u]Round 3[/u] [b]p7.[/b] Determine the number of factors of $2^{2019}$. [b]p8.[/b] The points $A$, $B$, $C$, and $D$ lie along a line in that order. It is given that $\overline{AB} : \overline{CD} = 1 : 7$ and $\overline{AC} : \overline{BD} = 2 : 5$. If $BC = 3$, find $AD$. [b]p9.[/b] A positive integer $n$ is equal to one-third the sum of the first $n$ positive integers. Find $n$. [u]Round 4[/u] [b]p10.[/b] Let the numbers $a,b,c$, and $d$ be in arithmetic progression. If $a +2b +3c +4d = 5$ and $a =\frac12$ , find $a +b +c +d$. [b]p11.[/b] Ten people playing brawl stars are split into five duos of $2$. Determine the probability that Jeff and Ephramare paired up. [b]p12.[/b] Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and for all $n\ge 2$, $$F_n = \left \lceil \frac{F_{n-1}+F_{n-2}}{2} \right \rceil +1,$$ where $\lceil r \rceil$ denotes the least integer greater than or equal to $r$ . Find $F_{2019}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the units digit of the decimal expansion of \[(15 \plus{} \sqrt{220})^{19} \plus{} (15 \plus{} \sqrt{220})^{82}.\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ \text{none of these}$

Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day. Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.

For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$. They are defined inductively, by the following recurrences. $A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$. [hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

A sequence of integers fsng is defined as follows: fix integers $a$, $b$, $c$, and $d$, then set $s_1 = a$, $s_2 = b$, and $$s_n = cs_{n-1} + ds_{n-2}$$ for all $n \ge 3$. Create a second sequence $\{t_n\}$ by defining each $t_n$ to be the remainder when $s_n$ is divided by $2018$ (so we always have $0 \le t_n \le 2017$). Let $N = (2018^2)!$. Prove that $t_N = t_{2N}$ regardless of the choices of $a$, $b$, $c$, and $d$.