# Your Math Problems for the Night: Coffin Problems

A friend of mine successfully nerd sniped me today.  He sent me the following list of problems:

http://arxiv.org/pdf/1110.1556v2.pdf

Now, before I start talking about math, this collection is fascinating in part because of where it comes from:

The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, was at that time actively trying to keep Jewish students (and other “undesirables”) from enrolling in the department. One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam. I was told that these problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could easily be rejected, so this system was an effective method of controlling admissions. These kinds of math problems were informally referred to as “Jewish” problems or “coffins”. “Coffins” is the literal translation from Russian; they have also been called “killer” problems in English.

The author goes on to elaborate on this.  As a top math student, she was recruited by people fighting against this situation to help solve these problems so that students, particularly Jewish ones, could be trained in the types of tricks needed to solve them.  The problems were so difficult that in a month her team only solved half of them.  She was so shocked and shaken by the blatant behind-the-scenes discrimination that she kept the problems and explored them online with other mathematicians much later, after coming to the United States.  “It turned out that not all of the coffins even had elementary solutions,” she writes.  “Some were intentionally ambiguous questions, some were just plain hard, some had impossible premises.”

And these were being given to high school seniors!

The awfulness of their origins aside, the paper completely sidetracked me for the whole morning.  A handful I was able to see a solution for right away, but the lion’s share I am still stumped on.  (The paper includes the solutions, but that’s no fun!)  The first one, for instance, looks easy but has stymied me:

$x(8\sqrt{1-x}+\sqrt{1+x}) \leq 11\sqrt{1+x}-16\sqrt{1-x}$

to be solved for positive x.

I’m guessing that it involves some very clever substitution, but I haven’t figured out what yet.

The second one, however, was one of the ones I was able to get pretty quickly:

$F(x_1) - F(x_2) \leq (x_1 - x_2)^2$

to be solved for all real-valued functions of real variables F such that the above holds for any x1 and x2.

Here was my solution: