Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

We say that two sequences $x,y \colon \mathbb{N} \to \mathbb{N}$ are [i]completely different[/i] if $x_n \neq y_n$ holds for all $n\in \mathbb{N}$. Let $F$ be a function assigning a natural number to every sequence of natural numbers such that $F(x)\neq F(y)$ for any pair of completely different sequences $x$, $y$, and for constant sequences we have $F \left((k,k,\dots)\right)=k$. Prove that there exists $n\in \mathbb{N}$ such that $F(x)=x_{n}$ for all sequences $x$.

A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations. (M Vyalyi)

[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer. [b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.) [b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality: (a) $xy \le \left(\frac{x+y}{2}\right)^2$ (b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$ [b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img] [b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

Prove that for all integers $n > 1$, $$\frac{1}{n}+\frac{1}{n + 1}+ ...+\frac{1}{n^2- 1}+\frac{1}{n^2} > 1$$

Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ ,such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square,for all $k\in\mathbb{N}$.

How many $2$-digit factors does $555555$ have?

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

Show that there are only finitely many integral solutions to \[ 3^m - 1 = 2^n \] and find them.

Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$.

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.

Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions: $\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$. $\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$. $\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$. Note: One or more of the sets may be empty.

Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.

Let $a, b, c, d, e$ be distinct positive integers such that $$a^4 + b^4 = c^4 + d^4 = e^5.$$ Show that $ac + bd$ is composite.

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

The currency exchange trades dinars (D), guilders (G), reals (R) and thalers (T). Exchange players have the right to make a purchase and sale transaction with each pair of currencies no more than once a day. The exchange rates are as follows: $D = 6G$,; $D=25R$, $D=120T$,$G = 4R$; $G=21T$, $R = 5T$. For example, the entry $D = 6G$ means that $1$ dinar can be bought for $6$ guilders (or $6$ guilders can be sold for $1$ dinar). In the morning the player had $32$ dinars. What is the maximum number that he can receive by evening a) in dinars? b) in thalers ?

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.