Problem of the week:

Week #1.

When I was un undergraduate student I devised a problem for the Grade 9 Slovene Math Olympiad that starts (in English translation) as follows:
"Let a,b,c,d be prime numbers such that b-a=c-b=ac-d=sqrt(d-a-b-c) ..."
(sqrt denotes the square-root function.) I will not tell you what the students were asked to do with these numbers. I will instead ask you the following (possibly open ended) question:
What year did the problem appear? Justify your answer. For bonus marks also show that neither "b-a=c-b=ac-d" nor "b-a=c-b=sqrt(d-a-b-c)" would uniquely determine the primes in question.

Most interesting solutions will be rewarded and published on this website.

The first prize for the problem of Week 1 goes to Robert Dawson who submitted a complete solution within seconds of the posting. An honorable mention goes to Owen Sharpe who even LaTeX-ed his solution .

Week #2

Let be the positive real number satisfying a^(2020)=0.99 and let f be a real-valued function defined for every real number satisfying
f(f(x))=2af(x)-(a^2) x
for every x. Prove that f(0)=0. For part marks find an explicit formula for a nonzero function satifying the above condition.

Now for the interesting part (i.e., speculative and open-ended): Find a nice condition for f so that the function equation above only has one nonzero solution. Justify your answer.

In general: does the function equation above have more then one nonzero solution? I am guessing (but I am not sure) that without any extra assumptions (such as continuity or surjectivity) on f there are infinitely many posibilities. Can you prove this?

The first prize for the problem of Week 2 goes to Owen Sharpe. Congratualations! You can see the solution here The problem regarding what additional contitions to assume for f to force f(x)=ax to be the only nonzero solution still remains open. Please submit your observations if you can. The only nice condition I can come up is for f to be surjective and continuous (or, slightly more generally surjective and maps positive numbers to positive numbers and negative numbers to negative numbers or vice versa).

Week #3

The problem of the Week #3 is a modification of an old IMO problem. The problem asks us to prove that for positive integers m,n we have that the number n!m!(n+m)! divides the number (2n)!(2m)!. There is a rather easy solution that does not shed much light on ther problem (and uses the famous formula for computing the largest power of a prime p that divides n!). So I will exclude this solution as follows: "I will only accept solutions that give a nice combinatorial interpretation of the quotient of the two numbers involved."

Thank you to John Irving and Owen Sharpe, who independently pointed out to me that I was asking about a solution to an open problem. That being said, I will gladly accept solutions if you email them to me.

Week #4 and $5

Due to lack of response, we are extending the deadline for this problem for an extra week. We are dealing with a problem I confronted (unsuccessfully) at some international undergraduate math competition:

Let A,B be nxn real-entried matrices such that AB=A^2+B^2 and AB-BA is invertible. Prove that n is divisible by 3. Below are some additional questions for extra marks:

1. When I was sharing this problem a few years ago with Bamdad Yahaghi, he observed that the result holds for any field of characteristic 0. Please prove this. Part marks will be given for proving the result for complex-entried matrices.
2. Can you give a counter-example for a field of prime characteristic?
3. Can you give an example of 3x3 (or (3k)x(3k)) real-entried matrices that satisfy the above condition?

The prize goes to Owen Sharpe. Congratulations! You can see the solution here . The parts of finding (3k)x(3k) real examples and 2x2 example over a field of characteristic 3 remain outstanding - solutions will be rewarded.

Week # 6, #7

Since there will be no new problem next week (we have a reading week) this problem is for two weeks. I was told about this problem by Heydar Radjavi:

Let A,B be real nxn matrices that are similar as complex nxn matrices, i.e., there is an invertible complex nxn matrix P such that PAP^{-1}=B. Prove that A,B are also similar as real matrices. For additional marks generalize this problem to more general field extensions and give examples where for some field extensions the concusion fails.

Submit your solutions by email by Monday, November 16. Please put "Problem of Week #6" into the subject line.