Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it covers a total area of $\frac{23}{3}$. [asy] fill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey); fill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey); fill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey); fill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey); draw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4)); draw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4)); draw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4)); draw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4)); draw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4)); draw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4)); [/asy] Find, with proof, the minimum $n$ such that the figure covers an area of at least $\sqrt{63}$.

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

Let $ABC$ be such a triangle, that $AB = 3\cdot BC$. Points $P$ and $Q$ lies on the side $AB$ and $AP = PQ = QB$. A point $M$ is the midpoint of the side $AC$. Prove that $\sphericalangle PMQ = 90^{\circ}$.

Find all positive integers $n$ such that: \[ \dfrac{n^3+3}{n^2+7} \] is a positive integer.

Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$.

Six young people - Antonio, Bernardo, Carlos, Diego, Eduardo, and Francisco, attended a meeting in vests of different colors. After the meeting, they decided to exchange the vests as souvenir. $1)$. Each of them came out of the meeting room, wearing a vest with color different from the one with which they went into the meeting room. $2)$. The vest with which Antonio came out of the meeting room was belong to the young man who came out with Bernardo's vest. $3)$. The owner of the vest with which Carlos came out of the meeting room, came out with the vest that was belong to the young man who came out with Diego's vest. $4)$. The one who came out of the meeting room with Eduardo's vest was not the owner of the vest with which Francisco came out. Determine who came out of the meeting room with Antonio's vest, and who owns the vest with which Antonio came out. [hide=original wording]Seis jovenes que asistieron a una reunion vistiendo chalecos de distintos colores, decidieron intercambiarlos y salieron vistiendo todos de color diferente a aquel con que llegaron. El chaleco con que salio Antonio perteneca al joven que salio con el chaleco de Bernardo. El dueno del chaleco con que salio Carlos, salio con el chaleco que perteneca al joven que se llevo el de Diego. Quien se llevo el chaleco de Eduardo no era el dueno del que se llevo Francisco. Determine quien salio con el chaleco de Antonio, y quien es el dueno del chaleco que se llevo Antonio.[/hide]

$\text{Let} S_1,S_2,...S_{2011}$ $\text{be nonempty sets of consecutive integers such that any}$ $2$ $\text{of them have a common element. Prove that there is a positive integer that belongs to every}$ $S_i, i=1,...,2011$ (For example, ${2,3,4,5}$ is a set of consecutive integers while ${2,3,5}$ is not.)

For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ? $\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$

For a positive integer $n$, let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$-$y$ plane. Denote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\ y$ coordinates of 4 vertices are integers, and the vertices are included in $C_n$. Prove that $\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi$.

On the plane there are two isosceles non-intersecting right triangles $ABC$ and $DEC$ ($AB$ and $DE$ are the hypotenuses,$ ABDE$ is a convex quadrilateral), and $AB = 2 DE$. Let's construct two more isosceles right triangles: $BDF$ (with hypotenuse $BF$ located outside triangle $BDC$) and $AEG$ (with hypotenuse $AG$ located outside triangle $AEC$). Prove that the line $FG$ passes through a point $N$ such that $DCEN$ is a square.

Let there be 50 natural numbers $ a_i$ such that $ 0 < a_1 < a_2 < ... < a_{50} < 150$. What is the greatest possible sum of the differences $ d_j$ where each $ d_j \equal{} a_{j \plus{} 1} \minus{} a_j$?

Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions. In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.

Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) [i](6 points)[/i]

Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation. [b][Weightage 17/100][/b]

On a circle the numbers from 1 to 6 are written in order, as depicted in the picture. In each move, Lior picks a number $a$ on the circle whose neighbors are $b$ and $c$ and replaces it by the number $\frac{bc}{a}$. Can Lior reach a state in which the product of the numbers on the circle is greater than $10^{100}$ in [b]a)[/b] at most 100 moves [b]b)[/b] at most 110 moves

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let the lengths of three sides of a triangle be $l, m, n(l>m>n)$. If $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, find the minimum value of the perimeter of the triangle. Note: $\{x\}=x-[x]$ and $[x]$ denotes the integral part of number $x$.

Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1. [Thanks Grobber for the correction]

Prove that for any positive real number $\lambda$,there are $n$ positive numbers $a_1,a_2,\cdots,a_n(n\geq 2)$,so that $a_1<a_2<\cdots<a_n<2^n\lambda$ and for any $k=1,2,\cdots,n$ we have \[\gcd(a_1,a_k)+\gcd(a_2,a_k)+\cdots+\gcd(a_n,a_k)\equiv 0\pmod{a_k}\]

Solve the equation $$\left| ... \left|\left||x^2-x| -1\right|-1 \right|...-1\right|=x^2-2x-14.$$ (There are $11$ units on the left side.)

How many pairs of positive integers $(a, b)$ satisfy $\frac1a + \frac1b = \frac1{2004}$?

The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("$1$",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("$2$",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("$3$",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("$4$",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("$5$",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("$6$",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("$7$",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("$8$",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("$9$",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities? $$a^2 + b^2 < 16$$ $$a^2 + b^2 < 8a$$ $$a^2 + b^2 < 8b$$