1. Prove the Dirichlet box principle: If n objects are put into m boxes, in college when the book doesn't tell you how to pronounce `Dirichlet'. some box must contain  objects, and some box...

1. Prove the Dirichlet box principle: If n objects are put into m boxes, in college when the book doesn't tell you how to pronounce `Dirichlet'. some box must contain
 objects, and some box must contain

2. Egyptian mathematicians in 1800 b.c. represented rational numbers between 0 and 1 as sums of unit fractions 1/x₁
+ · · · + 1/x_k, where the x's were distinct positive integers. For example, they wrote
 +

instead of
. Prove that it is always possible to do this in a systematic way: If 0

(This is Fibonacci's algorithm, due to Leonardo Fibonacci, a.d. 1202.)