Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.

Let $a, b,$ and $c$ be non-zero real numbers and let $a \ge b \ge c$. Prove the inequality $\frac{a^3 - c^3}{3} \ge abc (\frac{a- b}{c}+ \frac{b- c}{a})$ . When does equality hold?

Given the function $$y =|x^2 - 4x + 3|.$$ Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.

Compute the maximum value attained by $f(x)=x^{1/x^2}$.

Triangle $ABC$ has median $AF$, and $D$ is the midpoint of the median. Line $CD$ intersects $AB$ in $E$. Prove that $BD = BF$ implies $AE = DE$!

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions: (i) $ f(0, 0, 0) = 1;$ (ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$ (iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$ Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$

Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$

In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules: (i) The only word of length $1$ is $A$; (ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$. Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$

In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.

How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.) [i]Proposed by Miles Yamner and Andrew Wu[/i] (Note: wording changed from original to clarify)

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.

Let $ABC$ be a triangle in which $AB=AC$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$.

Show that if $(a_n)$ is an infinite sequence of distinct positive integers, neither of which contains digit $0$ in the decimal expansion, then $$\sum_{n=1}^{\infty} \frac{1}{a_n}< 29.$$

James has a red jar, a blue jar and a pile of $100$ pebbles. Initially both jars are empty. A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile. The numbers of pebbles in the red and blue jars determine the state of the game. The followwing conditions must be satisfied: (a) The red jar may never contain fewer pebbles than the blue jar; (b) The game may never be returned to a previous state. What is the maximum number of moves that James can make?

Find the number of integers between $1$ and $200$ inclusive whose distinct prime divisors sum to $16$. (For example, the sum of the distinct prime divisors of $12$ is $2 + 3 = 5$.) In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$

Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.

Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.

For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.