

1. The positive integers are grouped as follows: 1, 2+3, 4+5+6, 7+8+9+10, ... Find the value of the nth sum.


2. Show that cos x^{2} + cos y^{2}  cos xy < 3 for reals x, y.


3. The equations 2x_{1}  x_{2} = 1, x_{1} + 2x_{2}  x_{3} = 1, x_{2} + 2x_{3}  x_{4} = 1, x_{3} + 3x_{4}  x_{5} = 1, ... , x_{n2} + 2x_{n1}  x_{n} = 1, x_{n1} + 2x_{n} = 1 have a solution in positive integers x_{i}. Show that n must be even.


4. C, C' are concentric circles with radii R, R'. A rectangle has two adjacent vertices on C and the other two vertices on C'. Find its sides if its area is as large as possible.


5. Show that a unit square can be covered with three equal disks with radius < 1/√2. What is the smallest possible radius?


6. Show that the only real solution to x(x+y)^{2} = 9, x(y^{3}  x^{3}) = 7 is x = 1, y = 2.

