Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.

If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$

For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?

The sum of seven integers is $ \minus{}1$. What is the maximum number of the seven integers that can be larger than $ 13$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?

In a triangle $ABC$, a point $P$ in the interior of $ABC$ is such that $$ \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB.$$ Suppose $\angle BAC = 30^{\circ}$ and $AP = 12$. Let $D,E,F$ be the feet of perpendiculars from $P$ on to $BC,CA,AB$ respectively. If $m \sqrt{n}$ is the area of the triangle DEF where $m,n$ are integers with $n$ prime, then what is the value of the product $mn$?

There are three bins: one with 30 apples, one with 30 oranges, and one with 15 of each. Each is labeled "apples," "oranges," or "mixed." Given that all three labels are wrong, how many pieces of fruit must you look at to determine the correct labels?

Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?

Calculate all real numbers $r $ with the following properties: If real numbers $a, b, c$ satisfy the inequality$ | ax^2 + bx + c | \le 1$ for each $x \in [ - 1, 1]$, then they also satisfy the inequality $| cx^2 + bx + a | \le r$ for each $ x \in [- 1, 1]$.

Let $n$ be an integer such that $n > 3$. Suppose that we choose three numbers from the set $\{1, 2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than $\frac{n}{2}$, then the values of these combinations are all distinct. (b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p - 1$.

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if \[ \frac{n}{810}=0.d25d25d25\ldots \]

Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.

Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ\equal{}60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

Each side of a convex $2019$-gon polygon is dyed with red, yellow and blue, and there are exactly $673$ sides of each kind of color. Prove that there exists at least one way to draw $2016$ diagonals to divide the convex $2019$-gon polygon into $2017$ triangles, such that any two of the $2016$ diagonals don't have intersection inside the $2019$-gon polygon,and for any triangle in all the $2017$ triangles, the colors of the three sides of the triangle are all the same, either totally different.

The incircle of a triangle $ABC$ with $AB\ne BC$ touches $BC$ at $A_1$ and $AC$ at $B_1$. The segments $AA_1$ and $BB_1$ meet the incircle at $A_2$ and $B_2$, respectively. Prove that the lines $AB,A_1B_1,A_2B_2$ are concurrent.

Let $ABC$ be a non-degenerate triangle in the euclidean plane. Define a sequence $(C_n)_{n=0}^\infty$ of points as follows: $C_0:=C$, and $C_{n+1}$ is the incenter of the triangle $ABC_n$. Find $\lim_{n\to\infty}C_n$.

Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

Suppose we wish to pick a random integer between $1$ and $N$ inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between $0$ and $1$, then multiplying the result by $N$ and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping the flipping process after we have enough flips to determine the ceiling of the number. For instance, if $N=3$, we could conclude that the number is $2$ after flipping $.011_2$, but $.010_2$ is inconclusive. Suppose $N=2014$. The expected number of flips for such a process is $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $100m+n$. [i]Proposed by Lewis Chen[/i]

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees? [asy] size(100); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray); filldraw(circle((0,0),2),lightgray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy] $\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.