| A1. Find all integers a, b, c satisfying 1 < a < b < c such that (a - 1)(b -1)(c - 1) is a divisor of abc - 1. |
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| A2. Find all functions f defined on the set of all real numbers with real values, such that f(x2 + f(y)) = y + f(x)2 for all x, y. |
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| A3. Consider 9 points in space, no 4 coplanar. Each pair of points is joined by a line segment which is colored either blue or red or left uncolored. Find the smallest value of n such that whenever exactly n edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color. |
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| B1. L is a tangent to the circle C and M is a point on L. Find the locus of all points P such that there exist points Q and R on L equidistant from M with C the incircle of the triangle PQR. |
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B2. Let S be a finite set of points in three-dimensional space. Let Sx, Sy, Sz be the sets consisting of the orthogonal projections of the points of S onto the yz-plane, zx-plane, xy-plane respectively. Prove that:
|S|2 ≤ |Sx| |Sy| |Sz|, where |A| denotes the number of points in the set A. [The orthogonal projection of a point onto a plane is the foot of the perpendicular from the point to the plane.] |
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B3. For each positive integer n, S(n) is defined as the greatest integer such that for every positive integer k ≤ S(n), n2 can be written as the sum of k positive squares.
(a) Prove that S(n) ≤ n2 - 14 for each n ≥ 4.
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