British Mathematical Olympiad

News: IMO Paper 2 (13 July 2006)

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

1 + 2x + 22x+1 = y2.


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.


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

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