The LHS Math Team is going to have a Secret Santa event! Nine members are going to participate, and each person must give exactly one gift to a specific recipient so that each person receives exactly one gift. But to make it less boring, no pairs of people can just swap gifts. The number of ways to assign who gives gifts to who in the Secret Santa Exchange with these constraints is $N$. Find the remainder when $N$ is divided by $1000$.

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$. Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$. Show that $BX$ bisects segment $AF$.

For a positive integer $n$ and any real number $c$, define $x_k$ recursively by : \[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$.

Each side of a convex quadrilateral $ ABCD $ is divided into three equal parts; a straight line is drawn through the dividing points of sides $ AB $ and $ AD $ that lie closer to vertex $ A $, and similarly for vertices $ B $, $ C $, $ D $. Prove that the center of gravity of the quadrilateral formed by the drawn lines coincides with the center of gravity of quadrilateral $ ABCD $.

Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that \[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]

Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ .

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

[b]p1.[/b] Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet? [b]p2.[/b] Let $n$ be a positive integer. There are $n$ non-overlapping circles in a plane with radii $1, 2, ... , n$. The total area that they enclose is at least $100$. Find the minimum possible value of $n$. [b]p3.[/b] How many integers between $1$ and $50$, inclusive, are divisible by $4$ but not $6$? [b]p4.[/b] Let $a \star b = 1 + \frac{b}{a}$. Evaluate $((((((1 \star 1) \star 1) \star 1) \star 1) \star 1) \star 1) \star 1$. [b]p5.[/b] In acute triangle $ABC$, $D$ and $E$ are points inside triangle $ABC$ such that $DE \parallel BC$, $B$ is closer to $D$ than it is to $E$, $\angle AED = 80^o$ , $\angle ABD = 10^o$ , and $\angle CBD = 40^o$. Find the measure of $\angle BAE$, in degrees. [b]p6. [/b]Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination? [b]p7. [/b]Point $A = (1, 211)$ and point $B = (b, 2011)$ for some integer $b$. For how many values of $b$ is the slope of $AB$ an integer? [b]p8.[/b] A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$ and $141$ are all palindromes. How many palindromes between $1$ and 1000 are divisible by $11$? [b]p9.[/b] Suppose $x, y, z$ are real numbers that satisfy: $$x + y - z = 5$$ $$y + z - x = 7$$ $$z + x - y = 9$$ Find $x^2 + y^2 + z^2$. [b]p10.[/b] In triangle $ABC$, $AB = 3$ and $AC = 4$. The bisector of angle $A$ meets $BC$ at $D$. The line through $D$ perpendicular to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. Compute $EC - FB$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/7/e26fbaeb7d1f39cb8d5611c6a466add881ba0d.png[/img] [b]p11.[/b] Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three remaining faces, given that all three are positive primes less than $200$. [b]p12.[/b] Let $d$ be a number chosen at random from the set $\{142, 143, ..., 198\}$. What is the probability that the area of a rectangle with perimeter $400$ and diagonal length $d$ is an integer? [b]p13.[/b] There are $3$ congruent circles such that each circle passes through the centers of the other two. Suppose that $A, B$, and $C$ are points on the circles such that each circle has exactly one of $A, B$, or $C$ on it and triangle $ABC$ is equilateral. Find the ratio of the maximum possible area of $ABC$ to the minimum possible area of $ABC$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/4/c/162554fcc6aa21ce3df3ce6a446357f0516f5d.png[/img] [b]p14.[/b] Let $k$ and $m$ be constants such that for all triples $(a, b, c)$ of positive real numbers, $$\sqrt{ \frac{4}{a^2}+\frac{36}{b^2}+\frac{9}{c^2}+\frac{k}{ab} }=\left| \frac{2}{a}+\frac{6}{b}+\frac{3}{c}\right|$$ if and only if $am^2 + bm + c = 0$. Find $k$. [b]p15.[/b] A bored student named Abraham is writing $n$ numbers $a_1, a_2, ..., a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples $$(a_1, a_2, a_3), (a_2, a_3, a_4), ..., (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, a_1, a_2)$$ are distinct from each other. What is the maximum possible value of $n$? Give the answer n, along with an example of such a sequence. Write your answer as an ordered pair. (For example, if the answer were $5$, you might write $(5, 12311)$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a non-isosceles triangle with circumcircle $\omega$ and let $H, M$ be orthocenter and midpoint of $AB$ respectively. Let $P,Q$ be points on the arc $AB$ of $\omega$ not containing $C$ such that $\angle ACP=\angle BCQ < \angle ACQ$.Let $R,S$ be the foot of altitudes from $H$ to $CQ,CP$ respectively. Prove that thé points $P,Q,R,S$ are concyclic and $M$ is the center of this circle.

What is the sum of real roots of the equation $x^4-8x^3+13x^2 -24x + 9 = 0$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 4 $

Let $D$ be a point on the inside of triangle $ABC$ such that $AD=CD$, $\angle DAB=70^{\circ}$, $\angle DBA=30^{\circ}$ and $\angle DBC=20^{\circ}$. Find the measure of angle $\angle DCB$.

The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

Let $a,b,c,d$ be positive real numbers. Prove that \[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]

Let $ABC$ be a triangle, with $AB\neq AC$. Let $O_1$ and $O_2$ denote the centers of circles $\omega_1$ and $\omega_2$ with diameters $AB$ and $BC$, respectively. A point $P$ on segment $BC$ is chosen such that $AP$ intersects $\omega_1$ in point $Q$, with $Q\neq A$. Prove that $O_1$, $O_2$, and $Q$ are collinear if and only if $AP$ is the angle bisector of $\angle BAC$.

In tetrahedron $SABC$, $\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})$. Let $\theta=A-SC-B$, prove that $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.

Find all triplets of primes in the form $ (p, 2p\plus{}1, 4p\plus{}1)$.

Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i]

The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.

$ABCD$ is a convex quadrilateral with \[\angle BAC = 30^\circ \]\[\angle CAD = 20^\circ\]\[\angle ABD = 50^\circ\]\[\angle DBC = 30^\circ\] If the diagonals intersect at $P$, show that $PC = PD$.

Find the interior angle between two sides of a regular octagon (degrees).

Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number.

Let $m$ be the answer to this question. What is the value of $2m - 5$?

If $a,b,c \neq 0$, prove that $\frac{a^2+b^2}{c^2} +\frac{b^2+c^2}{a^2} +\frac{c^2+a^2}{b^2} \geq 6$.

Find all positive integers $ k$ for which equation $ n^m\minus{}m^n\equal{}k$ has solution in positive integers.