The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? $\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$

A $8\times 8$ board is divided in dominoes (rectangles with dimensions $1 \times 2$ or $2 \times 1$). [list=a] [*] Prove that the total length of the border between horizontal and vertical dominoes is at most $52$. [*] Determine the maximum possible total length of the border between horizontal and vertical dominoes. [/list] [i]Proposed by B. Frenkin, A. Zaslavsky, E. Arzhantseva[/i]

Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

In triangle $ ABC$,$ \angle BAC \equal{} 120^o$. Let the angle bisectors of angles $ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively. Prove that the circle on diameter $ EF$ passes through $ D.$

Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.

In $\triangle ABC$ points $D$, $E$ lie on segment $BC$ where $BD = DE = EC.$ Points $X$, $Y$ lie on the perpendicular bisectors of $AD$, $AE$ respectively. If $XE$, $YD$ are tangent to the circumcircles of $\triangle AEC$, $\triangle ADB$ respectively, prove $X,A,Y$ are collinear.

Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.

Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.

In coordinate plane, we call a point $(x, y)$ "lattice point" if both $x$ and $y$ are integers. Knowing that the vertices of triangle $ABC$ are all lattice points, and there exists exactly one lattice point interior to triangle $ABC$ (there might exist lattice points on the sides of $ABC$). Prove that the area of triangle $ABC$ is no larger than $\frac 92.$

Let $n$ be a positive odd integer. $C$ is a set consists of integral points on a plane, which is defined by \[ C = \{(i, j): i, j = 0, 1, \dots, 2n-1\} \] and forms a $2n \times 2n$ array. On every point there is a Guinea pig, which is facing toward one of the following directions: [i]positive/negative $x$-axis[/i], or [i]positive/negative $y$-axis[/i]. Jeff wants to keep $n^2+1$ of the Guinea pigs on the plane and remove all the others. After that, the Guinea pigs on the plane will move as the following: 1. In every round, the Guinea pigs move toward by an unit, and keep facing the same direction. 2. If a Guinea pig move to a point $(i, j)$ which is [i]not[/i] in $C$, it will further move to another point $(p, q)$ in $C$, such that $p \equiv i \pmod {2n}$ and $q \equiv j \pmod {2n}$. [i](For example, if a Guinea pig move from $(2, 0)$ to $(2, -1)$, it will then further move to $(2, 2n-1)$.)[/i] The next round begins after all the Guinea pigs settle up. Jeff's goal is to keep the appropriate Guinea pigs on the plane, so that in every single round, any two Guinea pigs will never move to the same endpoint, and will never move to the startpoints[i](in that round)[/i] of each other simultaneously. Prove that Jeff can always succeed wherever the Guinea pigs initially face. [i]Proposed by Weijiun Kao[/i] Edit: By the way, it can be proven that the number $n^2+1$ is optimal, i.e. if the Guinea pigs face appropriately, Jeff can only keep at most $n^2+1$ of them on the plane to avoid any collision.

In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to like $DK$ meets the circle with diameter $BC$ at points $X,Y$. Prove that $AX \cdot DY = DX \cdot AY$

Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors. [i]Proposed by Evan Chang (squareman), USA[/i]

Let the altitude from $A$ and $B$ of triangle $ABC$ meet the circumcircle of $ABC$ again at $D$ and $E$ respectively. Let $DE$ meet $AC$ and $BC$ at $P$ and $Q$ respectively. Show that $ABQP$ is cyclic

We say that the string $d_kd_{k-1} \cdots d_1d_0$ represents a number $n$ in base $-2$ if each $d_i$ is either $0$ or $1$, and $n = d_k(-2)^k + d_{k-1}(-2)^{k-1} + \cdots + d_1(-2) + d_0$. For example, $110_{-2}$ represents the number $2$. What string represents $2016$ in base $-2$?

Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers. [i]Proposed by Akshaj[/i]

Let $ b$ be an even positive integer for which there exists a natural number n such that $ n>1$ and $ \frac{b^n\minus{}1}{b\minus{}1}$ is a perfect square. Prove that $ b$ is divisible by 8.