1st Brasil 1979


1. Show that if a < b are in the interval [0, π/2] then a - sin a < b - sin b. Is this true for a < b in the interval [π, 3π/2]?
2. The remainder on dividing the polynomial p(x) by x² - (a+b)x + ab (where a and b are unequal) is mx + n. Find the coefficients m, n in terms of a, b. Find m, n for the case p(x) = x²⁰⁰ divided by x² - x - 2 and show that they are integral.
3. The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter.
4. Show that the number of positive integer solutions to x₁ + 2³x₂ + 3³x₃ + ... + 10³x₁₀ = 3025 () equals the number of non-negative integer solutions to the equation y₁ + 2³y₂ + 3³y₃ + ... + 10³y₁₀ = 0. Hence show that () has a unique solution in positive integers and find it.
5.(i) ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN. (ii) The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is 2/5 the area of the parallelogram. (iii) ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is 1/3 of the area of ABC. (iv) Is (ii) true for all convex quadrilaterals ABCD?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.