In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.

Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?

In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)

Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

Tow circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.

Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer $n$, it prints exactly one of the numbers $n,2n,3n$. We know that the machine prints $2$. Prove that it doesn't print $13824$.