|
|
|
|
A1. Given a triangle ABC, let I be the incenter. The internal bisectors of angles A, B, C meet the opposite sides in A', B', C' respectively. Prove that:
1/4 < AI·BI·CI/(AA'·BB'·CC') ≤ 8/27.
|
|
|
A2. Let n > 6 be an integer and let a1, a2, ... , ak be all the positive integers less than n and relatively prime to n. If
a2 - a1 = a3 - a2 = ... = ak - ak-1 > 0,
prove that n must be either a prime number or a power of 2.
|
|
|
A3. Let S = {1, 2, 3, ... 280}. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime.
|
|
|
B1. Suppose G is a connected graph with k edges. Prove that it is possible to label the edges 1, 2, ... , k in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is 1.
[A graph is a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of edges belongs to at most one edge. The graph is connected if for each pair of distinct vertices x, y there is some sequence of vertices x = v0, v1, ... , vm = y, such that each pair vi, vi+1 (0 ≤ i < m) is joined by an edge.]
|
|
|
B2. Let ABC be a triangle and X an interior point of ABC. Show that at least one of the angles XAB, XBC, XCA is less than or equal to 30o.
|
|
|
B3. Given any real number a > 1 construct a bounded infinite sequence x0, x1, x2, ... such that |xi - xj| |i - j|a ≥ 1 for every pair of distinct i, j.
[An infinite sequence x0, x1, x2, ... of real numbers is bounded if there is a constant C such that |xi| < C for all i.]
|
|
