Find all pairs $(x,y)$ of nonnegative integers that satisfy \[x^3y+x+y=xy+2xy^2.\]

Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ where $D$, $E$, $F$ lie on $BC$, $AC$, $AB$, respectively. Let $M$ be the midpoint of $BC$. The circumcircle of triangle $AEF$ cuts the line $AM$ at $A$ and $X$. The line $AM$ cuts the line $CF$ at $Y$. Let $Z$ be the point of intersection of $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.

Some cells of a $2007\times 2007$ table are colored. The table is charrua if none of the rows and none of the columns are completely colored. (a) What is the maximum number $k$ of colored cells that a charrua table can have? (b) For such $k$, calculate the number of distinct charrua tables that exist.

Let $ABCDE$ be a convex pentagon that satisfies all of the following: There is a circle $\Gamma$ tangent to each of the sides. The lengths of the sides are all positive integers. At least one of the sides of the pentagon has length $1$. The side $AB$ has length $2$. Let $P$ be the point of tangency of $\Gamma$ with $AB$. (a) Determine the lengths of the segments $AP$ and $BP$. (b) Give an example of a pentagon satisfying the given conditions.

Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.