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In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

### Mediant

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Ice Cream Tangent

##### Stage: 4 Short Challenge Level:

Let the radius of the circle be r and let the perpendicular height of the triangle be h.

$\tan x^{\circ}= h/r$

Now, the area of the semicircle = ${1\over2}\pi r^2$ and the area of the triangle = ${1\over2}\times 2r \times h$

Which gives $r h$ = ${1\over2}\pi r^2$, so ${h\over r} = {\pi \over 2}$

This problem is taken from the UKMT Mathematical Challenges.