You start with a number. Every second, you can add or subtract any number of the form $n!$ to your current number to get a new number. In how many ways can you get from $0$ to $100$ in $4$ seconds? ($n!$ is dened as $n\times (n -1)\times(n - 2) ... 2\times1$, so $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, etc.)

Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$. Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist infinitely many primes that are not contained in $S_{a}$.

Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2\plus{}ab\plus{}b^2)|ab(a\plus{}b)$. Prove that: $ |a\minus{}b|>\sqrt [3] {ab}$.

Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two. (N. Agakhanov)

Find all positive rational numbers $r\not=1$ such that $r^{\frac{1}{r-1}}$ is rational.

Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.

We call a set of integers $\textit{special}$ if it has $4$ elements and can be partitioned into $2$ disjoint subsets $\{ a,b \}$ and $\{ c, d \}$ such that $ab - cd = 1$. For every positive integer $n$, prove that the set $\{ 1, 2, \dots, 4n \}$ cannot be partitioned into $n$ disjoint special sets. [i]Proposed by Mohsen Jamali, Iran[/i]

Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.

Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.

Let $n$ be any integer $\ge 2$. Prove that $\sum 1/pq = 1/2$, where the summation is over all integers$ p, q$ which satisfy $0 < p < q \le n$,$ p + q > n$, $(p, q) = 1$.

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$. [i]Proposed by Andrew Wu[/i]

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

Let $(G, \cdot)$ a finite group with order $n \in \mathbb{N}^{*},$ where $n \geq 2.$ We will say that group $(G, \cdot)$ is arrangeable if there is an ordering of its elements, such that \[ G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}. \] a) Determine all positive integers $n$ for which the group $(Z_n, +)$ is arrangeable. b) Give an example of a group of even order that is arrangeable.

Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$

Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $? [b](A)[/b] the set of positive integers [b](B)[/b] the set of composite positive integers [b](C)[/b] the set of even positive integers [b](D)[/b] the set of integers greater than 3 [b](E)[/b] the set of integers greater than 4

Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.