2008 USAMO Problems
(Titu Andreescu) Prove that for each positive integer , there are pairwise relatively prime integers , all strictly greater than 1, such that is the product of two consecutive integers.
(Zuming Feng) Let be an acute, scalene triangle, and let , , and be the midpoints of , , and , respectively. Let the perpendicular bisectors of and intersect ray in points and respectively, and let lines and intersect in point , inside of triangle . Prove that points , , , and all lie on one circle.
(Gabriel Carroll) Let be a positive integer. Denote by the set of points with integer coordinates such that A path is a sequence of distinct points in such that, for , the distance between and is (in other words, the points and are neighbors in the lattice of points with integer coordinates). Prove that the points in cannot be partitioned into fewer than paths (a partition of into paths is a set of nonempty paths such that each point in appears in exactly one of the paths in ).
(Gregory Galparin) Let be a convex polygon with sides, . Any set of diagonals of that do not intersect in the interior of the polygon determine a triangulation of into triangles. If is regular and there is a triangulation of consisting of only isosceles triangles, find all the possible values of .
(Kiran Kedlaya) Three nonnegative real numbers , , are written on a blackboard. These numbers have the property that there exist integers , , , not all zero, satisfying . We are permitted to perform the following operation: find two numbers , on the blackboard with , then erase and write in its place. Prove that after a finite number of such operations, we can end up with at least one on the blackboard.
(Sam Vandervelde) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form for some positive integer ).
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