Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

Which of the following statements are true? (A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

A positive integer $n$ is called [i]cute[/i] when there is a positive integer $m$ such that $m!$ ends in exactly $n$ zeros. a) Determine if $2019$ is cute. b) How many positive integers less than $2019$ are cute?

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

[b]p1.[/b] Compute $x$ such that $2009^{2010} \equiv x$ (mod $2011$) and $0 \le x < 2011$. [b]p2.[/b] Compute the number of "words" that can be formed by rearranging the letters of the word "syzygy" so that the y's are evenly spaced. (The $y$'s are evenly spaced if the number of letters (possibly zero) between the first $y$ and the second $y$ is the same as the number of letters between the second $y$ and the third $y$.) [b]p3.[/b] Let $A$ and $B$ be subsets of the integers, and let $A + B$ be the set containing all sums of the form $a + b$, where $a$ is an element of $A$, and $b$ is an element of $B$. For example, if $A = \{0, 4, 5\}$ and $B =\{-3,-1, 2, 6\}$, then $A + B = \{-3,-1, 1, 2, 3, 4, 6, 7, 10, 11\}$. If $A$ has $1955$ elements and $B$ has $1891$ elements, compute the smallest possible number of elements in $A + B$. [b]p4.[/b] Compute the sum of all integers of the form $p^n$ where $p$ is a prime, $n \ge 3$, and $p^n \le 1000$. [b]p5.[/b] In a season of interhouse athletics at Caltech, each of the eight houses plays each other house in a particular sport. Suppose one of the houses has a $1/3$ chance of beating each other house. If the results of the games are independent, compute the probability that they win at least three games in a row. [b]p6.[/b] A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers. [b]p7.[/b] Eight friends are playing informal games of ultimate frisbee. For each game, they split themselves up into two teams of four. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Determine the smallest number of games they need to play in order to achieve this. [b]p8.[/b] Compute the number of ways to choose five nonnegative integers $a, b, c, d$, and $e$, such that $a + b + c + d + e = 20$. [b]p9.[/b] Is $23$ a square mod $41$? Is $15$ a square mod $41$? [b]p10.[/b] Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute $ \sum_{d|15015} \phi (d)$. [b]p11.[/b] Compute the largest possible volume of an regular tetrahedron contained in a cube with volume $1$. [b]p12.[/b] Compute the number of ways to cover a $4 \times 4$ grid with dominoes. [b]p13.[/b] A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, and $E$ be five mutually equidistant points in four-dimensional space. Let $P$ be a point such that $AP = BP = CP = DP = EP = 1$. Compute the side length $AB$. [b]p14. [/b]Ten turtles live in a pond shaped like a $10$-gon. Because it's a sunny day, all the turtles are sitting in the sun, one at each vertex of the pond. David decides he wants to scare all the turtles back into the pond. When he startles a turtle, it dives into the pond. Moreover, any turtles on the two neighbouring vertices also dive into the pond. However, if the vertex opposite the startled turtle is empty, then a turtle crawls out of the pond and sits at that vertex. Compute the minimum number of times David needs to startle a turtle so that, by the end, all but one of the turtles are in the pond. [b]p15.[/b] The game hexapawn is played on a $3 \times 3$ chessboard. Each player starts with three pawns on the row nearest him or her. The players take turns moving their pawns. Like in chess, on a player's turn he or she can either $\bullet$ move a pawn forward one space if that square is empty, or $\bullet$ capture an opponent's pawn by moving his or her own pawn diagonally forward one space into the opponent's pawn's square. A player wins when either $\bullet$ he or she moves a pawn into the last row, or $\bullet$ his or her opponent has no legal moves. Eve and Fred are going to play hexapawn. However, they're not very good at it. Each turn, they will pick a legal move at random with equal probability, with one exception: If some move will immediately win the game (by either of the two winning conditions), then he or she will make that move, even if other moves are available. If Eve moves first, compute the probability that she will win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$ [i]Proposed by F. Petrov and K. Sukhov[/i]

Show that the product of three consecutive positive integers is never a perfect square.

Two players take turns writing down all proper non-decreasing fractions with denominators from $1 $ to $1999$ and at the same time writing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?

Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values ​​that $e$ can take.

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

Four digits $a, b, c, d$, different from each other and different from zero, are chosen and the list of all the four-digit numbers that are obtained by exchanging the digits $a, b, c, d$ is written. What digits must be chosen so that the list has the greatest possible number of four-digit numbers that are multiples of $36$?

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold: $\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$ $\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$. Find the number of good sequences (in terms of $n$).

The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$ Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.

Let $m,n$ be distinct positive integers. Prove that $$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$ Further, determine when equality holds.

Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?

Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ? (S. Fomin , Leningrad)

Given are a prime $p$ and a positive integer $a$. Let $q$ be a prime divisor of $\frac{a^{p^3}-1}{a^{p^2}-1}$ and $q\neq p$. Prove that $q\equiv 1 ( \mod p^3)$.