Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?

The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$. [asy] path seg1, seg2; seg1=(6,0)--(0,3); seg2=(2,0)--(4,2); dot((0,0)); dot((1,0)); fill(circle((2,0),0.1),black); dot((3,0)); dot((4,0)); dot((5,0)); fill(circle((6,0),0.1),black); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((5,1)); dot((6,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); fill(circle((4,2),0.1),black); dot((5,2)); dot((6,2)); fill(circle((0,3),0.1),black); dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3)); dot((5,3)); dot((6,3)); draw(seg1); draw(seg2); pair [] x=intersectionpoints(seg1,seg2); fill(circle(x[0],0.1),black); label("$A$",(0,3),NW); label("$B$",(6,0),SE); label("$C$",(4,2),NE); label("$D$",(2,0),S); label("$E$",x[0],N);[/asy] $\text{(A)}\ \frac{4\sqrt5}{3}\qquad\text{(B)}\ \frac{5\sqrt5}{3}\qquad\text{(C)}\ \frac{12\sqrt5}{7}\qquad\text{(D)}\ 2\sqrt5 \qquad\text{(E)}\ \frac{5\sqrt{65}}{9}$

The median $AM$ is drawn in the acute-angled triangle $ABC$ with different sides. Its extension intersects the circumscribed circle $w$ of this triangle at the point $P$. Let $A {{H} _ {1}}$ be the altitude $\Delta ABC$, $H$ be the point of intersection of its altitudes. The rays $MH$ and $P {{H} _ {1}}$ intersect the circle $w$ at the points $K$ and $T$, respectively. Prove that the circumscribed circle of $\Delta KT {{H} _ {1}}$ touches the segment $BC$. (Hilko Danilo)

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

Integers $a, b, c, d$ satisfy $a+b+c+d=0$. Show that $$n=(ab-cd)\cdot(bc-ad)\cdot(ca-bd)$$ is a perfect square.

Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

2. What’s the closest number to $169$ that’s divisible by $9$?

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ [i]Gheorghe Iurea[/i]

Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]

Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that: $$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$ [i]Cristinel Mortici[/i]

There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$

For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.

Let $D$ be a point inside $\triangle ABC$ such that $\angle CDA + \angle CBA = 180^{\circ}.$ The line $CD$ meets the circle $\odot ABC$ at the point $E$ for the second time. Let $G$ be the common point of the circle centered at $C$ with radius $CD$ and the arc $\overset{\LARGE \frown}{AC}$ of $\odot ABC$ which does not contain the point $B$. The circle centered at $A$ with radius $AD$ meets $\odot BCD$ for the second time at $F$. Prove that the lines $GE, FD, CB$ are concurrent or parallel.

Find the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5$'s. [i]Proposed by Ephram Chun and Taiki Aiba[/i]

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

If $S=\{s_1,s_2,\dots,s_n\}$ is a set of integers with $s_1<s_2<\dots<s_n$, define $$f(S)=\sum_{k=1}^n (-1)^k k^2 s_k.$$ (If $S$ is empty, $f(S)=0$.) Compute the average value of $f(S)$ as $S$ ranges over all subsets of $\{1^2,2^2,\dots,100^2\}$. [i]Proposed by Connor Gordon and Nairit Sarkar[/i]

For given positive integer $a$, find every $(x_1, x_2, …, x_{2002})$ that satisfies the following: (1) $x_1 \geq x_2 \geq … \geq x_{2002} \geq 0$ (2) $0< x_1+x_2+…+x_{2003}<a+1$ (3) $ x^2_1+x^2_2+…+x^2_{2003}+9=a^2$

Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$ Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)

Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$, $O(x)=|x|\log |x|\log\log |x|$, $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$, and $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$ for any real number $x$ such that the functions are defined. Determine $$C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))).$$ [i]2016 CCA Math Bonanza Lightning #5.3[/i]

There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$? $ \textbf{(A)}\ \minus{}16\qquad \textbf{(B)}\ \minus{}8\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 20$

Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$, $x_2$, $\cdots$, $x_n$ are distinct postive integers. Find the maximum value of $n$. [i]Proposed by Le Duc Minh[/i]

Let triangle $\vartriangle ABC$ have circumcenter $O$ and circumradius $r$, and let $\omega$ be the circumcircle of ntriangle $\vartriangle BOC$. Let $F$ be the intersection of $\overleftrightarrow{AO}$ and $\omega$ not equal to $O$. Let $E$ be on line $\overleftrightarrow{AB}$ such that $\overline{EF} \perp \overline{AE}$, and let $G$ be on line $\overleftrightarrow{AC}$ such that $\overline{GF} \perp \overline{AG}$. If $AC =\frac{65}{63}$ , $BC = \frac{24}{13}r$, and $AB = \frac{126}{65}r$, compute $AF \cdot EG$.

Let $\omega_1$ and $\omega_2$ be circles with radii $3$ and $12$ and externally tangent at point $P$. Let a common external tangent intersect $\omega_1$, $\omega_2$ at $S$, $T$ respectively and the common internal tangent at point $Q$. Define $X$ to be the point on $\overrightarrow{QP}$ such that $QX=10$. If $XS$, $XT$ intersect $\omega_1$, $\omega_2$ a second time at $A$ and $B$, determine $\tan\angle APB$. . [i]2015 CCA Math Bonanza Individual Round #15[/i]

When the World Wide Web first became popular in the $1990$s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$-megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.) $\textbf{(A)} ~0.6\qquad\textbf{(B)} ~10\qquad\textbf{(C)} ~1800\qquad\textbf{(D)} ~7200\qquad\textbf{(E)} ~36000\qquad$