There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square? [asy] size(130); defaultpen(fontsize(11)); int i, j; for(i=0; i<9; i=i+1) { for(j=0; j<9; j=j+1) if((i==4) && (j==4)) { dot((i,j),linewidth(5)); } else { dot((i,j),linewidth(3)); } } dot("$P$",(4,4),NE); draw((0,0)--(0,8)--(8,8)--(8,0)--cycle); [/asy] $\textbf{(A) } \frac{1}{5} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{2}{5} \qquad\textbf{(D) } \frac{9}{20} \qquad\textbf{(E) } \frac{1}{2}$

Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX} \equal{} \angle{DAH}$. [i]Zuming Feng.[/i]

Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$? $\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$

Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.

Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$. Let $K$ be the area of rhombus $\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$. [asy] size(150);defaultpen(linewidth(0.7)+fontsize(10)); draw(rotate(45)*polygon(4)); pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180); draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$\mathcal{P}$", intersectionpoint( A--(-1,0), F--(0,1) )); label("$\mathcal{S}$", intersectionpoint( E--(-1,0), F--(0,-1) )); label("$\mathcal{R}$", intersectionpoint( D--(1,0), C--(0,-1) )); label("$\mathcal{Q}$", intersectionpoint( B--(1,0), C--(0,1) )); label("$\mathcal{T}$", point); dot(A^^B^^C^^D^^E^^F);[/asy]

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.

Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B$. The probability that team $A$ finishes with more points than team $B$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

Let the triangle $ABC$ with $BC$ the smallest side. Let $P$ on ($AB$) such that angle $PCB$ equals angle $BAC$. and $Q$ on side ($AC$) such that angle $QBC$ equals angle $BAC$. Show that the line passing through the circumenters of triangles $ABC$ and $APQ$ is perpendicular on $BC$.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.

If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $