International Mathematical Olympiad · 1964

Problems

Problem 1(a) Find all positive integers $n$ for which $2^n-1$ is divisible by $7$ . (b) Prove that no positive integer $n$ makes $2^n+1$ divisible by $7$ .Solutions: 1
Problem 2Suppose $a,b,c$ are the sides of a triangle. Prove that $a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le3abc$ .Solutions: 1
Problem 3A circle is inscribed in a triangle $ABC$ with sides $a,b,c$ . Tangents to the circle parallel to the sides of the triangle are constructed. Each tangent cuts off a triangle from $\triangle ABC$ , and a circle is inscribed in each of these three triangles. Find the sum of the areas of all four inscribed circles in terms of $a,b,c$ .Solutions: 1
Problem 4Seventeen people correspond by mail with one another, each with all the rest. In their letters only three different topics are discussed, and each pair of correspondents deals with only one topic. Prove that at least three people write to each other about the same topic.Solutions: 1
Problem 5Suppose 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.Solutions: 1
Problem 6In tetrahedron $ABCD$ , vertex $D$ is joined to $D_0$ , the centroid of $\triangle ABC$ . Through $A,B,C$ draw lines parallel to $DD_0$ , meeting the planes $BCD,CAD,ABD$ at $A_1,B_1,C_1$ , respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$ . Is the result true if $D_0$ is selected anywhere within $\triangle ABC$ ?Solutions: 1