Thread: Putnam Problems ~~

Can anybody explain these two problems?
i know the answser for #1 is 6, but how do we solve it? Thank u ^^

1. S = { (x, y) : |x| - |y| ≤ 1 and |y| ≤ 1 }, sketch the region S and find its area. (Putnam Test, A1, 1988)
2. Consider a polynomial f(x) with real coefficients having the property f(g(x))=g(f(x)) for every polynomial g(x) with real coefficients. Determines and prove the nature of f(x). (Putnam competition)

For problem 1, just sketch the region S and think about triangles.

For problem 2, f commutes with all g under functional composition. That's quite a strong condition. I think the angle of attack would be to consider really simple cases. For example. if f(x) = c for all x, then f(g(x)) = c but g(f(x)) = g(c), which is not true for all polynomials g. Pick g(x) = c+1 and you have g(f(x)) = g(c) = c+1 which doesn't equal c. So f(x) is not a constant. What if you try linear polynomials?