Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC \equal{} BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$ \textbf{(A)} \ \frac {1}{6} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {2}{3}$

Let $x_1=\sqrt{10}$ and $y_1=\sqrt3$. For all $n\geq 2$, let \begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*} Find $x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.$

$x$ and $y$ are positive and relatively prime and $z$ is an integer. They satisfy $(5z-4x)(5z-4y)=25xy$. Show that at least one of $10z+x+y$ or quotient of this number divided by $3$ is a square number (i.e. prove that $10z+x+y$ or integer part of $\frac{10z+x+y}{3}$ is a square number).

Let $k$ be a real number such that the system \begin{align*} &|25+20i-z|=5\\ &|z-4-k|=|z-3i-k| \\ \end{align*} has exactly one complex solution $z.$ The sum of all possible values of $k$ can be written as $\dfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Here $i=\sqrt{-1}.$

In parallelogram $ ABCD$, point $ M$ is on $ \overline{AB}$ so that $ \frac{AM}{AB} \equal{} \frac{17}{1000}$ and point $ N$ is on $ \overline{AD}$ so that $ \frac{AN}{AD} \equal{} \frac{17}{2009}$. Let $ P$ be the point of intersection of $ \overline{AC}$ and $ \overline{MN}$. Find $ \frac{AC}{AP}$.

Let $S$ be a subset with $673$ elements of the set $\{1,2,\ldots ,2010\}$. Prove that one can find two distinct elements of $S$, say $a$ and $b$, such that $6$ divides $a+b$.

Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2a+4\tan^2 b$.

Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points.

Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?

Prove that there exists a constant $K$ such that the following inequality holds for any sequence of positive numbers $a_1 , a_2 , a_3 , \ldots:$ $$\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.$$

Call a function $\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}$ $\emph{M-rugged}$ if it is unbounded and satisfies the following two conditions: $(1)$ If $f(n)|f(m)$ and $f(n)<f(m)$ then $n|m$. $(2)$ $|f(n+1)-f(n)|\leq M$. a. Find all $1-rugged$ functions. b. Determine if the number of $2-rugged$ functions is smaller than $2019$.

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.

$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively. The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral. Prove that $K$ lies on the diagonal $AC$. [i]Proposed by S. Berlov[/i]

[b]Q[/b] Let $p,q,r$ be non negative reals such that $pqr=1$. Find the maximum value for the expression $$\sum_{cyc} p[r^{4}+q^{4}-p^{4}-p]$$ [i]Proposed by Aritra12[/i]

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.

The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

Show that there is no infinite sequence of primes $p_1, p_2, p_3, . . .$ there any for each $ k$: $p_{k+1} = 2p_k - 1$ or $p_{k+1} = 2p_k + 1$ is fulfilled. Note that not the same formula for every $k$.

It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals: $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

On the sides $BC$ and $AC$ of the isosceles triangle $ABC$ ($AB = BC$), points $E$ and $D$ are marked, respectively, so that $DE \parallel AB$. On the extendsion of side $CB$ beyond the point $B$, point $K$ was arbitrarily marked. Let $P$ be the intersection point of the lines $AB$ and $KD$. Let $Q$ be the intersection point of the lines $AK$ and $DE$. Prove that $CA$ is the bisector of angle $\angle PCQ$.