From the Tournament of the Towns:
In a school, more than 90% of the students study both English and French, and more than 90% of the students study English and German. Prove that more than 90% of the students who study both French and German also study English.
This problem took me a long time to solve. At first glance, it seems innocuous – surely you just play around with the numbers, possibly with a Venn diagram thrown in, and it just falls out.
After five minutes, you wonder why you haven’t solved it yet; maybe it’s too early or too late in the day, or you’ve expended your mental energy elsewhere.
After fifteen minutes, you think to yourself ‘What is going on?! This problem is simple, right? Why is it taking this long…?’
Finally, after 30 minutes or longer, you realise just how tough the problem is. The simplicity of the statement deceived you into believing that a solution would be easy to come by; oh how you were fooled.
It turns out there are several successful approaches, at least one of which is fairly short when polished, but certainly not trivial. A Venn diagram is highly recommended.
Let A be the proportion who study English and French but not German, let B be the proportion who study English and German but not French, let C be the proportion who study French and German but not English, and let D be the proportion who study all three languages.
We are told that A+D>0.9, and B+D>0.9. We are required to prove that D/(C+D)>0.9, which is equivalent to D>9C after some rearranging.
Adding the two initial inequalities yields
A+B+2D>1.8
Adding C-D to both sides of this leads to
A+B+C+D>1.8+C-D
But of course A+B+C+D is at most 1, so in fact we have
1.8+C-D<1, or
D>C+0.8
Finally, we observe that since A+D>0.9, it must be the case that C<0.1, so that
D>C+0.8>C+8C
At last we have the desired result D>9C, and the problem is solved.
This is yet another exquisite problem from the Tournament of the Towns, which really is the gift that keeps on giving as far as mathematics competitions are concerned.
But while many problems are attractive for having a simple solution following a seemingly impenetrable statement, this example is quite different. On the face of it, the problem seems straightforward; it is only after you start investigating that you realise how wrong you were.