

1. A triangle area T is divided into six regions by lines drawn through a point inside the triangle parallel to the sides. The three triangular regions have areas T_{1}, T_{2}, T_{3}. Show that √T =√T_{1} + √T_{2} + √T_{3}.


2. Find n > 1 so that with stamp denominations n and n+2 it is possible to obtain any value ≥ 2n+2.


3. For x ≥ 1, define p_{n}(x) = ½(x + √(x^{2}1) )^{n} + ½(x  √(x^{2}1) )^{n}. Show that p_{n}(x) ≥ 1 and p_{mn}(x) = p_{m}(p_{n}(x)).


4. The pentagon ABCDE is incribed in a circle. ∠A ≤ ∠B ≤ ∠C ≤ ∠D ≤ ∠E. Show that ∠C > π/2 and that this is the best possible lower bound.


5. Show that we can divide {1, 2, 3, ... , 2^{n}} into two disjoint parts S, T such that ∑_{k∈S} k^{m} = ∑_{k∈T} k^{m} for m = 0, 1, 2, ... , n1.


6. A rectangle is constructed from 1 x 6 rectangles. Show that one of its sides is a multiple of 6.

