Jewish Problems

I already gave an example of the kinds of problems that were given to Jewish people at the oral entrance exam to the math department of Moscow State University. In fact, I have a whole page with a collection of such problems, called Jewish problems or Coffins. That page was one of the first pages I created when I started my website more than ten years ago.

When my son Alexey was in high school, I asked him to help me type these problems into a file and to recover their solutions from my more than laconic notes, and solve the problems that I didn’t have notes for. He did the job, but the file was lying dormant on my computer. Recently I resurrected the file and we prepared some of the solutions for a publication.

The problems that were given during these exams were very different in flavor: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises. In our joint paper “Jewish Problems” we presented problems with a special flavor. These are problems that have a short and “simple” solution, that is nonetheless very difficult to find. This way the math department of MSU was better protected from appeals and complaints.

Try the following problem from our paper:

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.

I will give a talk on the subject for UMA at MIT on October 18, at 5pm.

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10 Comments

  1. Andrei Zelevinsky:

    Tanya: 1) I thought using derivatives was not allowed on entrance exams. 2) Problem 4: this is an equation, not an inequality. 3) Solution of Problem 18: I don’t think the part “We see that they are getting smaller very fast” would be accepted on the exam …

  2. Jason Dyer:

    I might be confused, but #2 does seem to have a non-derivative answer. The answer must be symmetrical, that is, x_1 and x_2 must be switchable, so that

    F(x_1) <= F(x_2) + (x_1-x_2)^2

    but also

    F(x_2) <= F(x_1) + (x_1-x_2)^2

    which can only occur if F is the constant function.

  3. Count Iblis:

    In physics, there is no discrimination against Jews, so everyone is given difficult problems:

    http://arxiv.org/abs/physics/0605057

    :)

  4. Joseph:

    No need for derivatives on this one.

    Since x and y are “switchable”, we are basically given that |F(x)-F(y)|<=(x-y)^2 for all x and y. Pick any two distinct real numbers x and y (assume x<y, without loss of generality), and assume that the distance between them is m.

    Now choose any natural number n, and partition the interval [x,y] into n equal subintervals. Let x_0=x, x_1=x+(m/n), x_2=x+2(m/n), and so on, up to x_n=y. We then have:

    |F(x)-F(y)|
    =|[F(x_0)-F(x_1)]+[F(x_1)-F(x_2)]+…+[F(x_(n-1)-F(x_n)]|
    <=|F(x_0-F(x_1)|+|F(x_1)-F(x_2)|+…+|F(x_(n-1)-F(x_n)|
    <=(m/n)^2+(m/n)^2+…+(m/n)^2
    =m^2/n

    But if |F(x)-F(y)|<=m^2/n for ALL natural numbers n, we must have |F(x)-F(y)|=0. So the function values at any two points are equal; the function must be constant.

    Oh, and I’m a Jew, by the way. :-D

  5. buzz:

    “Εβραϊκά προβλήματα” ή “Φέρετρα” της ΕΣΣΔ…

    Μια Σοβιετική Εβραία απόφοιτος των Μαθηματικών Ολυμπιάδων, (αργυρό και χρυσό μετάλιο το 1975/1976), περιγράφει τα αδύνατα μαθηματικά προβλήμα…

  6. TMA:

    You are probably aware of the [ridiculously expensive, but also available for kindle] book http://www.amazon.com/Failed-Your-Math-Comrade-Einstein/dp/981256358X
    google books: http://books.google.com/books?id=ho6fMF8ehogC&lpg=PP1&pg=PP1#v=onepage&q&f=false

    which is in part about these exams, but also includes essays about Bella Abramovna/Jewish People’s University.

  7. TMA:

    oh, I see your paper references that book!

  8. Konstantin:

    Вот ведь хотел я дать тоже ссылку на Comrade Einstein, да опоздал…
    Ну тогда вот в чистом виде.
    http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/mekh-mat.html

  9. Konstantin:

    http://kerosinka.livejournal.com/1633.html (full collection of links, Russian)

  10. dani:

    F(x) is constant…but this means that (x-y)^2=0, for every x,y real…
    To be honest I am a bit disappointed…it looked nice…

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