Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ [i]Proposed by Şahin Emrah[/i]

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

In country some cities are connected by oneway flights( There are no more then one flight between two cities). City $A$ called "available" for city $B$, if there is flight from $B$ to $A$, maybe with some transfers. It is known, that for every 2 cities $P$ and $Q$ exist city $R$, such that $P$ and $Q$ are available from $R$. Prove, that exist city $A$, such that every city is available for $A$.

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to [asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75)); [/asy] $ \textbf{(A)}\ k \qquad\textbf{(B)}\ \frac{6k}{5} \qquad\textbf{(C)}\ \frac{9k}{8} \qquad\textbf{(D)}\ \frac{5k}{4} \qquad\textbf{(E)}\ 2k $

Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.

Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.

Let $ABC$ be a triangle with vertices at lattice points. Suppose one of its sides in $\sqrt{n}$, where $n$ is square-free. Prove that $\frac{R}{r}$ is irraational . The symbols have usual meanings.

Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$? $\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$

Let $ x$ and $ y$ be positive real numbers with $ x^3 \plus{} y^3 \equal{} x \minus{} y.$ Prove that \[ x^2 \plus{} 4y^2 < 1.\]

$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align} Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.

Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.) $\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$

Let $f(n)$ be the sum of the digits of $n$. Find $\displaystyle{\sum_{n=1}^{99}f(n)}$.

When simplified, the third term in the expansion of $\left(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2}\right)^6$ is: ${{ \textbf{(A)}\ \frac{15}{x}\qquad\textbf{(B)}\ -\frac{15}{x}\qquad\textbf{(C)}\ -\frac{6x^2}{a^9} \qquad\textbf{(D)}\ \frac{20}{a^3} }\qquad\textbf{(E)}\ -\frac{20}{a^3} } $

What is $4^0 - 3^1 - 2^2 - 1^3$? $\textbf{(A) }{-}8\qquad\textbf{(B) }{-}7\qquad\textbf{(C) }{-}5\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Two standard six sided dice labeled with the numbers $1$-$6$ are rolled, and the numbers that come up are multiplied. What is the probability that their product is a multiple of five? $\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac5{18}\qquad\textbf{(C) } \dfrac{11}{36}\qquad\textbf{(D) } \dfrac13\qquad\textbf{(E) } \dfrac49$

Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$? $ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $

Before going to vacation, each of the $ 7$ pupils decided to send to each of the $ 3$ classmates one postcard. Is it possible that each student receives postcards only from the classmates he has sent postcards?

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .