I wrote how the written entrance exam was used to keep Jewish students from studying at Moscow State University, but the real brutality happened at the oral exam. Undesirable students were given very difficult problems. Here is a sample “Jewish” problem:
Solve the following equation for real y:
Here is how my compatriots who studied algebra in Soviet high schools would have approached this problem. First, cube it and get a 9th degree equation. Then, try to use the Rational Root Theorem and find that y = 1 is a root. Factoring out y − 1 gives an 8th degree equation too messy to deal with.
The most advanced students would have checked if the polynomial in question had multiple roots by GCDing it with its derivative, but in vain.
We didn’t study any other methods. So the students given that problem would have failed it and the exam.
Unfortunately, this problem is impossible to appeal, because it has an elementary solution that any applicant could have understood. It goes like this:
Let us introduce a new variable: x = (y3 + 1)/2. Now we need to solve a system of equations:
This system has a symmetry which we can exploit. The graphs of the functions x = (y3 + 1)/2 and y = (x3 + 1)/2 are reflections of each other across the line x = y. As both functions are increasing, the solution to the system of equations should lie on the line x = y. Hence, we need to solve the cubic y = (y3 + 1)/2, one of whose roots we already know.
Now I offer you another problem without telling you the solution:
Four points on a plane used to belong to four different sides of a square. Reconstruct the square by compass and straightedge.