The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?

Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.

A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$. a) Show that the number of people in the group is a square. b) Show that the number of seats on each bus is a triangular number.

Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.