Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that \[ P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$}, \] and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.

How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four? $\text{(A)} \ 60 \qquad \text{(B)} \ 64 \qquad \text{(C)} \ 72 \qquad \text{(D)} \ 76 \qquad \text{(E)} \ 100$

Given an integer \( n \geq 1 \), Jo-Ané alternately writes crosses (\( \mathcal{X} \)) and circles (\( \mathcal{O}\)) in the cells of a square grid with \( 2n + 1 \) rows and \( 2n + 1 \) columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum \( \mathcal{X}+ \mathcal{O} \), where \( \mathcal{X} \) is the number of rows containing more crosses than circles and \( \mathcal{O} \) is the number of columns containing more circles than crosses. Determine, in terms of \( n \), the highest possible score that Jo-Ané can obtain..

In rectangle $ ABCD$, we have $ AB\equal{}8$, $ BC\equal{}9$, $ H$ is on $ \overline{BC}$ with $ BH\equal{}6$, $ E$ is on $ \overline{AD}$ with $ DE\equal{}4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$. [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep); draw(A--G); label("$F$",F,W); label("$G$",G,W); label("$C$",C,WSW); label("$H$",H,NNE); label("$6$",(6,8),N); label("$B$",B,NE); label("$A$",A,SW); label("$E$",Ep,S); label("$4$",(2,0),S); label("$D$",D,S);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$

For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that \[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]

Find all the odd naturals whose indicator is the same as $1990$. We clarify that, if a natural decomposes into prime factors in the form $\Pi_{j=1}^r p_j^{a_j}$, define the [i]indicator [/i] as : $\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)$. [hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.[/hide]

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?

On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

The sides of an equilateral triangle are divided into n equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^{\circ}$ or by $120^{\circ}$ . a) Prove that, if $n \geq 7$, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle. b) Determine all the values of $n \geq 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.

$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

On a non-rhombus parallelogram $ ABCD, $ the vertex $ B $ is projected on $ AC $ in the point $ E. $ The perpendicular on $ BD $ thru $ E $ intersects the lines $ BC $ and $ AB $ in $ F $ and $ G, $ respectively. Show that $ EF=EG $ if and only if $ \angle ABC=90^{\circ } . $ [i]Mircea Becheanu[/i]

Let $A$, $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$. Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

The set of $ x$-values satisfying the equation $ x^{\log_{10} x} \equal{} \frac{x^3}{100}$ consists of: $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \text{10, only} \qquad \textbf{(C)}\ \text{100, only} \qquad \textbf{(D)}\ \text{10 or 100, only} \qquad \textbf{(E)}\ \text{more than two real numbers.}$

Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$