 
Summary: Interesting Problems
Charles Martin
October 8, 2009
1. One hundred balls are placed into one hundred boxes. Suppose that for each k = 1, 2, . . . , 99 no set of k
boxes contains exactly k balls. Prove that all balls are in the same box.
2. Two linear operators P, Q on a Hilbert space satisfy [P, Q] = i I, where R\{0} and [P, Q] = PQQP.
Prove that the underlying space is infintedimensional and that at least one of the operators is unbounded.
Hint: consider [P, Qn
].
3. Let n N. Evaluate
n
j=0
2n
2j
(3)j
.
4. The numbers a and b are randomly chosen independently and uniformly from the interval [1, 1]. Find the
probability that a2/3
+ b2/3
1.
