### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

### Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

### Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

# Rarity

##### Age 16 to 18 Challenge Level:

Show that there are no integer solutions $m, n$ of the equation $$\left({5\over 4}\right)^m = \left({2\over 1}\right)^n$$ which gives the number of major thirds in an octave on a musical scale.

Given integers $a, b, c$ and $d$ where $a$ and $b$ are coprime and $c$ and $d$ are coprime, find necessary and sufficient conditions for there to exist positive integers $m$ and $n$ such that $$\left({a\over b}\right)^m = \left({c\over d}\right)^n.$$

[The set of three problems Tuning and Ratio , Euclid's Algorithm and Musical Intervals and this problem Rarity were devised by Benjamin Wardaugh who used to be a member of the NRICH team. Benjamin is now doing research on the history of music and mathematics at Oxford University. Read Benjamin's article Music and Euclid's Algorithm for more on this subject.]