What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}$

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.

Today there are $2^n$ species on the planet Kerbin, all of which are exactly n steps from an original species. In an evolutionary step, One species split into exactly two new species and died out in the process. There were already $2^n-1$ species in the past, which are no longer present today can be found, but are only documented by fossils. The famous space pioneer Jebediah Kerman once suggested reducing the biodiversity of a planet by doing this to measure how closely two species are on average related, with also already extinct species should be taken into account. The degree of relationship is measured two types, of course, by how many evolutionary steps before or back you have to do at least one to get from one to the other. What is the biodiversity of the planet Kerbin?

The number $N$ consists of $1999$ digits such that if each pair of consecutive digits in $N$ were viewed as a two-digit number, then that number would either be a multiple of $17$ or a multiple of $23$. THe sum of the digits of $N$ is $9599$. Determine the rightmost ten digits of $N$.

Triangle $ABC$ is given. Let $O$ be it's circumcenter. Let $I$ be the center of it's incircle.The external angle bisector of $A$ meet $BC$ at $D$. And $I_A$ is the $A$-excenter . The point $K$ is chosen on the line $AI$ such that $AK=2AI$ and $A$ is closer to $K$ than $I$. If the segment $DF$ is the diameter of the circumcircle of triangle $DKI_A$, then prove $OF=3OI$.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

All of David's telephone numbers have the form $ 555\minus{}abc\minus{}defg$, where $ a$, $ b$, $ c$, $ d$, $ e$, $ f$, and $ g$ are distinct digits and in increasing order, and none is either $ 0$ or $ 1$. How many different telephone numbers can David have? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order. [i]Proposed by Vincent Huang[/i]

Let $ABCD$ be a quadrilateral inscribed in circle of center $O$. The perpendicular on the midpoint $E$ of side $BC$ intersects line $AB$ at point $Z$. The circumscribed circle of the triangle $CEZ$, intersects the side $AB$ for the second time at point $H$ and line $CD$ at point $G$ different than $D$. Line $EG$ intersects line $AD$ at point $K$ and line $CH$ at point $L$. Prove that the points $A,H,L,K$ are concyclic, e.g. lie on the same circle.

Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.

Two distinct positive even integers sum to $8.$ Determine the larger of the $2$ integers.

Let $a,b\in \mathbb{C}$. Prove that $\left| az+b\bar{z} \right|\le 1$, for every $z\in \mathbb{C}$, with $\left| z \right|=1$, if and only if $\left| a \right|+\left| b \right|\le 1$.

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

Find all pairs of natural numbers $(a,b)$ satisfying the following conditions: [list] [*]$b-1$ is divisible by $a+1$ and [*]$a^2+a+2$ is divisible by $b$. [/list]

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

The expression \[ \frac{P+Q}{P-Q}-\frac{P-Q}{P+Q} \] where $P=x+y$ and $Q=x-y$, is equivalent to: ${ \textbf{(A)}\ \frac{x^2-y^2}{xy}\qquad\textbf{(B)}\ \frac{x^2-y^2}{2xy}\qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{x^2+y^2}{xy} \qquad\textbf{(E)}\ \frac{x^2+y^2}{2xy} } $

Determine all triples $(x,y,z)$ of integers greater than $1$ with the property that $x$ divides $yz-1$, $y$ divides $zx-1$ and $z$ divides $xy-1$.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\] for all positive integers $n$. Given this information, determine all possible values of $f(1000)$. [i]Proposed by Evan Chen[/i]

Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test has $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat gets the $n$th problem correct with probability $\frac{1}{n^2}$ while Q gets the $n$th problem correct with probability $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly. Just before turning in their test, the ducks and ducklings decide to share answers! On any question which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores? [i]Proposed by Evan Chen[/i]

Suppose that zeros and ones are written in the cells of an $n\times n$ board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed $\frac n2\left(1+\sqrt{4n-3}\right)$.

Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?