06Q

1.

(a) Find all positive integers n for which 2n1 is divisible by 7.

(b) Prove that there is no positive integer n such that 2n1 is divisible by 7. 2. 3.

A circle is inscribed in triangle ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from ABC. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c).

Let a, b, c be the sides of a triangle. Prove that a2(bca)b2(cab)c2(abc)3abc.

4.

Seventeen people correspond by mail with one another — each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

5.

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

6.

In tetrahedron ABCD, vertex D is connected with D0 the centroid of ABC. Lines parallel to DD0 are drawn through A, B and C. These lines intersect the planes BCD, CAD and ABD in points A1, B1 and C1, respectively. Prove that the volume of ABCD is one third the volume of A1BC11D0. Is the result true if point D0 is selected anywhere within ABC?