News: IMO Paper 2 (13 July 2006)

Problem 4. Determine all pairs (x,y) of integers such that

1 + 2^x + 2^2x+1 = y².

(USA)

Problem 5. Let P(x) be a polynomial of degree n > 1 with integer coefficients and let k be a positive integer. Consider the polynomial Q(x) = P(P(…P(P(x))…)), where P occurs k times. Prove that there are at most n integers t such that Q(t) = t.

(Romania)

Problem 6. Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.

(Serbia and Montenegro)

Time allowed: 4 hours 30 minutes
Each problem is worth 7 points

Previous news item: IMO Paper 1 (12 July 2006).

Next news item: IMO 2006 results (14–15 July 2006).