

A1. (a) Find all natural numbers n for which 7 divides 2^{n}  1.
(b) Prove that there is no natural number n for which 7 divides 2^{n} + 1.


A2. Suppose that a, b, c are the sides of a triangle. Prove that:
a^{2}(b + c  a) + b^{2}(c + a  b) + c^{2}(a + b  c) ≤ 3abc.


A3. Triangle ABC has sides a, b, c. Tangents to the inscribed circle are constructed parallel to the sides. Each tangent forms a triangle with the other two sides of the triangle and a circle is inscribed in each of these three triangles. Find the total area of all four inscribed circles.


B1. Each pair from 17 people exchange letters on one of three topics. Prove that there are at least 3 people who write to each other on the same topic. [In other words, if we color the edges of the complete graph K_{17} with three colors, then we can find a triangle all the same color.]


B2. 5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel or perpendicular. Through each point lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersections these perpendiculars can have.


B3. ABCD is a tetrahedron and D_{0} is the centroid of ABC. Lines parallel to DD_{0} are drawn through A, B and C and meet the planes BCD, CAD and ABD in A_{0}, B_{0}, and C_{0} respectively. Prove that the volume of ABCD is onethird of the volume of A_{0}B_{0}C_{0}D_{0}. Is the result true if D_{0} is an arbitrary point inside ABC?

