Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .

Given a chess-board $99\times 99$ with a set $F$ of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set $F$. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement: [i]"There is a beetle, that either stayed on the same field or moved to the neighbouring one".[/i] Is it always valid if the figure $F$ is: a) A central cross, i.e. the union of the $50$-th row and the $50$-th column? b) A window frame, i.e. the union of the $1$-st, $50$-th and $99$-th rows and the $1$-st, $50$-th and $99$-th columns? c) All the chess-board?

Let $ABC$ be a triangle such that $AB<AC<BC$. Let $D,E$ be points on the segment $BC$ such that $BD=BA$ and $CE=CA$. If $K$ is the circumcenter of triangle $ADE$, $F$ is the intersection of lines $AD,KC$ and $G$ is the intersection of lines $AE,KB$, then prove that the circumcircle of triangle $KDE$ (let it be $c_1$), the circle with center the point $F$ and radius $FE$ (let it be $c_2$) and the circle with center $G$ and radius $GD$ (let it be $c_3$) concur on a point which lies on the line $AK$.

Prove that for every real number $x>0$ and each integer $n>0$ we have \[x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)\]

Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$. Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.

Let be three positive real numbers $ x,y,z, $ whose product is $ 1. $ Prove that: $$ \sum_{\text{cyc}} \frac{3}{\sqrt{1+x+xy}} \le \sqrt 3<3\sqrt 3\le \sum_{\text{cyc}} \sqrt{1+x+xy} $$

Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$ , where $p, q$ are relatively prime, find $p + q$.

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$? $\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$

Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ?

Find all positive integers $n$ that satisfy the following inequalities: $$-46 \leq \frac{2023}{46-n} \leq 46-n$$

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Triangle $ABC$ is right angled at $C$. Lines $AM$ and $BN$ are internal angle bisectors. $AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively. Prove that the line which passes through the midpoints of segments $QN$ and $PM$ is parallel to $AB$.

Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.

Given is a convex quadrilateral $ABCD$ with $\angle BAD = \angle BCD$ and $\angle ABC < \angle ADC$. Point $M$ is the midpoint of segment $AC$. Prove that there exist points $X$ and $Y$ on the segments $AB$ and $BC$, respectively, such that $XY \perp BD, MX = MY$ and $\angle XMY = \angle ADC - \angle ABC$. [i]Proposed by Mykhailo Shtandenko[/i]

Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

Prove that if $a,b,c, d \in [1,2]$, then $$\frac{a + b}{b + c}+\frac{c + d}{d + a}\le 4 \frac{a + c}{b + d}$$ When does the equality hold?

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

Compute all prime numbers $p$ such that $8p+1$ is a perfect square.

Let $p$ and $q$ be integers. If $k^2+pk+q>0$ for every integer $k$, show that $x^2+px+q>0$ for every real number $x$.

In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.

A man has $ \$10,000$ to invest. He invests $ \$4000$ at $ 5\%$ and $ \$3500$ at $ 4\%$. In order to have a yearly income of $ \$500$, he must invest the remainder at: $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 6.1\% \qquad\textbf{(C)}\ 6.2\% \qquad\textbf{(D)}\ 6.3\% \qquad\textbf{(E)}\ 6.4\%$

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number? (D. Fomin, Leningrad)