Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.

One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Michael explains a bit of strategy. "You have the first move and you occupy three of the four territories in the Australian continent. You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn." "Don't tell her $\textit{that!}$" complains Joshua. Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia. Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack. Finally, Wendy is left with one chance. Wendy and Joshua each roll just one six-sided die. Wendy wins if her roll is $\textit{higher}$ than Joshua's roll. Let $a$ and $b$ be relatively prime positive integers so that $a/b$ is the probability that Wendy rolls higher, giving her control over the continent of Australia. Find the value of $a+b$.

Consider the set $M=\{1,2,...,n\},n\in\mathbb N$. Find the smallest positive integer $k$ with the following property: In every $k$-element subset $S$ of $M$ there exist two elements, one of which divides the other one.

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

It is known that in the triangle $ABC$ the smallest side is $BC$. Let $X, Y, K$ and $L$ - points on the sides $AB, AC$ and on the rays $CB, BC$, respectively, are such that $BX = BK = BC =CY =CL$. The line $KX$ intersects the line $LY$ at the point $M$. Prove that the intersection point of the medians $\vartriangle KLM$ coincides with the center of the inscribed circle $\vartriangle ABC$.

Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that a) lines $AA_{1}, BB_{1}, CC_{1}$ concur; b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

In triangle $ABC$, we consider the concurrent lines $AA_1$, $BB_1$ and $CC_1$, with $A_1$, $B_1$ and $C_1$ lying on the segments $BC$, $CA$ and respectively $AB$. If the point of intersection of the lines is the incenter of $\triangle A_1B_1C_1$, prove that it is also the orthocenter of $\triangle ABC$.

Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities \[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad\text{and}\quad c < z. \] [i]Proposed by Evan Chen[/i]

A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second. At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A. B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins. [list=a] [*] Which player has a winning strategy? [*] Suppose now both players has $k$ chips numbered 1 to $k$. Which player has a winning strategy? [*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? [/list]

A sequence $(a_n)_{-\infty}^{-\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} +...+ a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0,a_1,...,a_{97}$.

202 participants arrived at a mathematical conference from three countries: Israel, Greece, and Japan. On the first day of the conference, every pair of participants from the same country shook hands. On the second day, every pair of participants exactly one of whom was Israeli shook hands. On the third day, every pair of participants one of whom was Israeli and the other Greek shook hands. In total 20200 handshakes occurred. How many Israelis participated in the conference?

Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

Construct a triangle $ABC$ given the side $AB$ and the distance $OH$ from the circumcenter $O$ to the orthocenter $H$, assuming that $OH$ and $AB$ are parallel.

If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio $y:x$ is: $\textbf{(A) }23:17\qquad \textbf{(B) }17y:23\qquad \textbf{(C) }23x:17\qquad$ $\textbf{(D) }\text{dependent upon the cost}\qquad \textbf{(E) }\text{none of these.}$

Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ dene $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$

For real numbers $a$ and $b$, define $$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$ Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these? $ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $

For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum^{\lfloor \log_{23} n\rfloor}_{i=1} s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\, \sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115$$ Compute $s_{20}(n) - s_{23}(n)$.

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

Let $\mathcal{P}$ be a regular $10$-gon in the coordinate plane. Mark computes the number of distinct $x$-coordinates that vertices of $\mathcal{P}$ take. Across all possible placements of $\mathcal{P}$ in the plane, compute the sum of all possible answers Mark could get.

How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.) [i]Proposed by Miles Yamner and Andrew Wu[/i] (Note: wording changed from original to clarify)

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Given a cube $ABCD-A'B'C'D'$, in the $12$ lines:$AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'$, how many sets of lines are skew lines? $\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48$