From 64th Putnam 2003

     1.  Do there exist polynomials a(x), b(x), c(y), d(y) such that 
 
                  {1+xy+x^2+y^2=a(x)c(y)+b(x)d(y)} ?
                  
     2. Show that

           {\int_0^1\int_0^1 |f(x)+f(y)|dx dy \ge \int_0^1|f(x)|dx}

           for any continuous real-valued function on [0,1].

     3. Is it possible to partition {0, 1, 2, 3, ... } into two parts such that n = x + y
         with x ≠ y has the same number of solutions in each part for each n?