How to solve "Jewish" math problems

Back in the seventies, the mathematics department at the Soviet Union's Moscow State University (one of the most prestigious departments in the USSR at the time) used a special collection of math problems, informally referred to as "Jewish" problems or "coffins" ("coffin" problems translating roughly to "killer" problems in English), to keep Jewish students and other so-called "undesirables" out of the department.

The problems were notoriously difficult to solve, but had seemingly simple solutions, allowing the department to avoid criticism of the wait-a-minute-what-you're-doing-is-completely-unethical variety.

Here's an example of one of the problems:

Find all real functions of real variable F(x) such that for any x and y the following inequality holds: F(x) − F(y) ≤ (x − y)^2.

In the summer of 1975, math prodigy Tanya Khovanova was approached by a teacher named Valera Senderov, who had managed to ascertain a list of these problems, and asked to help solve them.

At the time, Khovanova was attending a Soviet math camp, preparing — along with 7 other young, brilliant mathematicians — to compete in the International Math Olympiad on behalf of the Soviet Union. Khovanova writes about the experience in a paper the recently posted to arXiv:

These problems and their solutions were, of course, kept secret, but Valera Senderov and his friends had managed to collect a list. In 1975, they approached us to solve these problems, so that they could train the Jewish and other students in these mathematical ideas. Our team of the best eight Soviet students, during the month we had the problems, solved only half of them. True, that we had other priorities, but this fact speaks to the dificulty of these problems.

Being young and impressionable, I was shaken by this whole situation. I had had no idea that such blatant discrimination had been going on. In addition to trying to solve them at the time, I kept these problems as my most valuable possession — I still have that teal notebook.

Later, I emigrated to the United States. When I started my own web page, one of the first things I did was to post some of the problems. People sent me more problems, and solutions to the ones I had. It turned out that not all of the coffins even had elementary solutions: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises.

This account is taken from a paper, recently posted to arXiv by Khovanova and her son, that collects problems with short and "simple" solutions that are nonetheless very difficult to solve (the example problem given up top is sampled from the paper, where you'll also find the solution).

Those interested in Khovanova's collection of deceptively difficult problems, and more on the backstory of discrimination against Jews in 1970's USSR, will find them here, available free of charge.

arXiv via Tanya Khovanova's Math Blog