Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$, $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$

Calculate the determinant of the $ n\times n $ complex matrix $ \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} $ defined by $$ a_j^i=\left\{\begin{matrix} 1+x^2,\quad i=j\\x,\quad |i-j|=1\\0,\quad |i-j|\ge 2\end{matrix}\right. , $$ where $ n $ is a natural number greater than $ 2. $

Find $m$ and $n$ such that the set of points whose coordinates $x$ and $y$ satisfy the equation $|y-2x|=x$, coincides with the set of points specified by the equation $|mx + ny| = y$.

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city? $ \textbf{(A)}\ 22\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$

A rhombus is inscribed in triangle $ ABC$ in such a way that one of its vertices is $ A$ and two of its sides lie along $ AB$ and $ AC$. If $ \overline{AC} \equal{} 6$ inches, $ \overline{AB} \equal{} 12$ inches, and $ \overline{BC} \equal{} 8$ inches, the side of the rhombus, in inches, is: $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 3 \frac {1}{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$

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$, $(p, q) = 1$.

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

Arnim and Brentano have a little vase with $1500$ candies on the table and a huge sack with spare candies under the table. They play a game taking turns, Arnim begins . At each move a player can either eat $7$ candies or take $6$ candies from under the table and add them to the vase. A player cannot go under the table in two consecutive moves. A player is declared the winner if he leaves the vase empty. In any other case, if a player cannot make a move in his turn, the game is declared a tie. Is there a winning strategy for one of the players?

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

Consider the curves with equations $x^n+y^n=1$ for $n=2,4,6,8,\dots$. Denote $L_{2k}$ the length of the curve with $n=2k$. Find $\lim_{k\rightarrow\infty}L_{2k}$.

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

Triangles $ABC$ and $ABD$ are isosceles with $AB =AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C + \angle D$ is [asy] size(130); defaultpen(linewidth(0.8) + fontsize(11pt)); pair A, B, C, D, E; real angle = 70; B = origin; A = dir(angle); D = dir(90-angle); C = rotate(2*(90-angle), A) * B; draw(A--B--C--cycle); draw(B--D--A); E = extension(B, D, C, A); draw(rightanglemark(B, E, A, 1.5)); label("$A$", A, dir(90)); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, dir(0)); label("$E$", E, 1.5*dir(340)); [/asy] $\textbf{(A)}\ 115^\circ \qquad \textbf{(B)}\ 120^\circ \qquad \textbf{(C)}\ 130^\circ \qquad \textbf{(D)}\ 135^\circ \qquad \textbf{(E)}\ \text{not uniquely determined}$

Prove that for any positive integer $n$ there are $n$ consecutive composite numbers all less than $4^{n+2}$.