The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$. For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$. [i]Proposed by Nikola Velov[/i]

An ant walks on the plane as follows: initially, it walks $1$ cm in any direction. After, at each step, it changes the trajectory direction by $60^o$ left or right and walks $1$ cm in that direction. It is possible that it returns to the point from which it started in (a) $2008$ steps? (b) $2009$ steps? [img]https://cdn.artofproblemsolving.com/attachments/8/b/d4c0d03c67432c4e790b465a74a876b938244c.png[/img]

The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$

Let $ABCD$ be an isosceles trapezoid with $\overline{BC}\parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$ [asy] size(6cm); usepackage("mathptmx"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label("2",C--Y,S); label("1",Y--X,S); label("3",X--A,S); label("$A$",A,E); label("$B$",B,N); label("$C$",C,W); label("$D$",D,S); label("$Y$",Y,sqrt(2)*NE); label("$X$",X,N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy] $\textbf{(A)}\: 15\qquad\textbf{(B)} \: 5\sqrt{11}\qquad\textbf{(C)} \: 3\sqrt{35}\qquad\textbf{(D)} \: 18\qquad\textbf{(E)} \: 7\sqrt{7}$

An investor has two rectangular pieces of land of size $120\times 100$. a. On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house. b. On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$.

Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.

$21$ numbers are written in a row. $u,v,w$ are three consecutive numbers so $v=\frac{2uw}{u+w}$ . The first number is $\frac{1}{100}$ , the last one is $\frac{1}{101}$ . Find the $15$th number.

The nonzero real numbers $a,b,c$ are such that: $a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3$. Compute the possible values of $a+b+c$.

Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.

Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ . Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$. Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$. Prove that: $(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

Celia chooses a number $n$ and writes the list of natural numbers from $1$ to $n$: $1, 2, 3, 4, ..., n-1, n.$ At each step, it changes the list: it copies the first number to the end and deletes the first two. After $n-1$ steps a single number will be written. For example, for $n=6$ the five steps are: $$ 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5$$ and the number $5$ is written. Celia chose a number $n$ between $1000$ and $3000$ and after $n-1$ steps the number $1$ was written. Determine all the values of $n$ that Celia could have chosen. Justify why those values work, and the others do not.

A circle is circumscribed about a triangle with sides $ 20$, $ 21$, and $ 29$, thus dividing the interior of the circle into four regions. Let $ A$, $ B$, and $ C$ be the areas of the non-triangular regions, with $ C$ being the largest. Then $ \textbf{(A)}\ A \plus{} B \equal{} C\qquad \textbf{(B)}\ A \plus{} B \plus{} 210 \equal{} C\qquad \textbf{(C)}\ A^2 \plus{} B^2 \equal{} C^2\qquad \\ \textbf{(D)}\ 20A \plus{} 21B \equal{} 29C\qquad \textbf{(E)}\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}$

For an integer $n$ with $b$ digits, let a [i]subdivisor[/i] of $n$ be a positive number which divides a number obtained by removing the $r$ leftmost digits and the $l$ rightmost digits of $n$ for nonnegative integers $r,l$ with $r+l<b$ (For example, the subdivisors of $143$ are $1$, $2$, $3$, $4$, $7$, $11$, $13$, $14$, $43$, and $143$). For an integer $d$, let $A_d$ be the set of numbers that don't have $d$ as a subdivisor. Find all $d$, such that $A_d$ is finite.

Find the number of pairs of elements, from a group of order $ 2011, $ such that the square of the first element of the pair is equal to the cube of the second element. [i]Teodor Radu[/i]

Let $ABC$ be a triangle with $AB=7, BC=8, CA=9.$ Denote by $D$ and $G$ the foot from $A$ to $BC$ and the centroid of $\triangle ABC,$ respectively. Let $M$ be the midpoint of $BC,$ and $K$ be the other intersection of the reflection of $AM$ over the angle bisector of $\angle BAC$ with $(ABC).$ Let $E$ the intersection of $DG$ and $KM.$ Find the area of $ABCE.$ [i]Team #10[/i]

In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear. [i]Ray Li.[/i]

We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contains $111$. What is the length of the shortest string $s$ which contains the binary representations of all the positive integers less than or equal to $2048$?

Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2A_1=10$, $A_4=A_3A_2=101$, $A_5=A_4A_3=10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.

$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$.

Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_0=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n$. Define \[ q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j}, \] and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n$. Evaluate $M(2020)$.

Given a regular $n$-sided polygon with $n \ge 3$. Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way. Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.

Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.

Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.