A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a,b$, $a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.
Let $n$ be a positive integer and $a_1,a_2,\ldots,a_{2n}$ be $2n$ distinct integers. Given that the equation $$|x-a_1||x-a_2|\cdots|x-a_{2n}|=(n!)^2$$ has an integer solution $x=m$, find $m$ in terms of $a_1,\ldots,a_{2n}$.
In $\triangle ABC$, $AB=AC$, $D$ is a point on the side $BC$ and $E$ is a point on the segment $AD$. Given that $\angle BED=\angle BAC=2\angle CED$, prove that $BD=2CD$.
Consider a polygon with $m+n$ sides where $m,n$ are positive integers. Colour $m$ of its vertices red and the remaining $n$ vertices blue. A side is given the number $2$ if both its end vertices are red, the number $1/2$ if both its end vertices are blue and the number $1$ otherwise. Let the product of these numbers be $P$. Find the largest possible value of $P$.
Let $a,b,c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac1a+\frac1b+\frac1c=0$. Find the number of such triples $(a,b,c)$ with $50\ge|a|\ge|b|\ge|c|\ge1$.