Regular hexagon $ABCDEF$ with side length $2$ is inscribed within a sphere of radius $4$. Let point $X$ be on the sphere. Find the maximum value of the volume of the pyramid $ABCDEFX$.

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space. Determine the smallest number $g(n)$ of a points of a set in $\mathbb{R}^n$ such that every point in $\mathbb{R}^n$ is an irrational distance from at least one point in that set.

Will the number $1/1996$ decrease or increase and by how many times if in the decimal notation of this number the first non-zero digit after the decimal point is crossed out?

Find all positive integers $N$ with the following properties: (i) $N$ has at least two distinct prime factors, and (ii) if $d_1 < d_2 < d_3 < d_4$ are the four smallest divisors of $N$ then $N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2$

The real factors of $ x^2\plus{}4$ are: $\textbf{(A)}\ (x^2+2)(x^2+2) \qquad \textbf{(B)}\ (x^2+2)(x^2-2) \qquad \textbf{(C)}\ x^2(x^2+4) \qquad\\ \textbf{(D)}\ (x^2-2x+2)(x^2+2x+2) \qquad \textbf{(E)}\ \text{Non-existent}$

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner: \[12345678 \rightarrow 14352678 \rightarrow 43152678\] Compute the number of possible final orderings of the eight numbers.

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).

Let $\omega$ be a circle centered at $O$ with radius $R=2018$. For any $0 < r < 1009$, let $\gamma$ be a circle of radius $r$ centered at a point $I$ satisfying $OI =\sqrt{R(R-2r)}$. Choose any $A,B,C\in \omega$ with $AC, AB$ tangent to $\gamma$ at $E,F$, respectively. Suppose a circle of radius $r_A$ is tangent to $AB,AC$, and internally tangent to $\omega$ at a point $D$ with $r_A=5r$. Let line $EF$ meet $\omega$ at $P_1,Q_1$. Suppose $P_2,P_3,Q_2,Q_3$ lie on $\omega$ such that $P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3$ are tangent to $\gamma$. Let $P_2P_3,Q_2Q_3$ meet at $K$, and suppose $KI$ meets $AD$ at a point $X$. Then as $r$ varies from $0$ to $1009$, the maximum possible value of $OX$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd (a,c)=1$. Compute $10a+b+c$. [i]Proposed by Vincent Huang

Fix a point $A$, a circle $\Omega$ centered at $O$, and reals $r$ and $\theta$. Let $X$ and $Y$ be variable points on $\Omega$ so that $\measuredangle XOY = \theta$. The tangents to $\Omega$ at $X$ and $Y$ meet at $T$, and a dilation at $T$ with scale factor $r$ sends $A$ to $A'$. Let $P$ be the foot from $A'$ to $TX$. $ $ $ $ $ $ $ $ $ $ Suppose that some point $P^*$ is the same for two different $X$. Show that $\measuredangle TXY = \measuredangle AP^\ast O$. (All angles are directed.) Proposed by [i]Karn Chutinan[/i]

Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which $$\sin(x+y)\leq \sin(x)+\sin(y)$$ for every $x$ between $0$ and $\pi$, inclusive? $\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi $

$P$ is a point on the plane which square $ABCD$ belongs to, satisfying that $\triangle PAB,\triangle PBC,\triangle PCD,\triangle PDA$ are isosceles triangles. What's the number of such points? $\text{(A)}9\qquad\text{(B)}17\qquad\text{(C)}1\qquad\text{(D)}5$

Define $M_n = \{1,2,...,n\}$, for any $n\in N$. A collection of $3$-element subsets of $M_n$ is said to be [i]fine [/i] if for any coloring of elements of $M_n$ in two colors there is a subset of the collection all three elements of which are of the same color. For any $n\ge 5$ find the minimal possible number of the $3$-element subsets of $M_n$ in the fine collection. (E. Barabanov, S. Mazanik, I. Voronovich)

Suppose $a,b,c\in[0,2]$ and $a+b+c=3$. Find the maximal and minimal value of the expression $$\sqrt{a(b+1)}+\sqrt{b(c+1)}+\sqrt{c(a+1)}.$$

For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.

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 $a_1a_2a_3 ... a_{200}$.

We are given a non-obtuse $\Delta ABC$ $(BC>AC)$ with an altitude $CD$ $(D\in AB)$, center $O$ of its circumscribed circle, and a middle point $M$ of its side $AB$. Point $E$ lies on the ray $\overrightarrow{BA}$ in such way that $AE.BE=DE.ME$. If the line $OE$ bisects the area of $\Delta ABC$ and $CO=CD.cos\angle ACB$, determine the angles of $\Delta ABC$.