Analysis, PDE and Mathematical Physics at MSU

Humor Problems


In a dark, narrow alley, a function and a differential operator meet:
"Get out of my way - or I'll differentiate you till you're zero!"
"Try it - I'm e^x..."
Same alley, same function, but a different operator:
"Get out of my way - or I'll differentiate you till you're zero!"
"Try it - I'm e^x..."
"Too bad... I'm d/dy."
Q: What is the value of the contour integral around Western Europe?
A: Zero.
Q: Why?
A: Because all poles are in Eastern Europe!
Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four...
A newlywed husband is discouraged by his wife's obsession with mathematics. Afraid of being second fiddle to her profession, he finally confronts her: "Do you love math more than me?"
"Of course not, dear - I love you much more!"
Happy, although sceptical, he challenges her: "Well, then prove it!"
Pondering a bit, she responds: "Ok... Let epsilon be greater than zero..."

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Analysis is full of elementary formulated problems which have a large significance for quite non-elementary subjects. We suggest to your attention some of them.

Let D1,…, DN are discs on the plane. Their centers are c1,…, cN. For each center count two numbers: m(ci)= the number of our discs conataining ci, i=1,…,N; and M(ci) =the number of discs in the sequence 2D1,…, 2DN conatining ci, i=1,…,N. Here 2D stands for the disc with the same center but the doubled radius.
Obviously 1 ≤ mi≤ Mi ≤ N. Give an elementary prove that there exists an absolute constant A such that independently of the number N and independently of collection D1,…, DN there exists 1≤j≤ N such that Mj ≤ A≤ mj.
One is given the collection of N different positive integers: a1, a2,…, aN. Prove that whatever the collection of N-1 positive integers
0<b1<b2<...<bN-1< a1+a2+...+aN is given one can permute her/his given ai in such a way that
ai_1≠ b1,
ai_1+ai_2≠ b2,

ai_1+ai_2+…+ai_(N-1) ≠ bN-1.
Give an elementary prove of the following fact: let f, g, h be real valued, mesurable functions on the real line. Also they are 1-periodic. Assume also that |f|< 1, |g|< 1, |h|< 1. We are interested in the following integral:

Obviously |L| is less or equal to 1, and 1 can be obviously attained. We add the property:

Prove that |L| then is at most ½.

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