Olympiad problems

1979 Brazil National Olympiad, 2

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

1979 Brazil National Olympiad, 5

Tags: geometry , Brazilian Math Olympiad , Brazilian Math Olympiad 1979

[list=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. [*] 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 $\frac{2}{5}$ the area of the parallelogram. [*] ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC. [*] Is (ii) true for all convex quadrilaterals ABCD? [/list]

1979 Brazil National Olympiad, 4

Tags: Brazilian Math Olympiad 1979 , number theory , Diophantine equation , Brazilian Math Olympiad

Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.

1979 Brazil National Olympiad, 1

Tags: Brazilian Math Olympiad , trigonometry , function , inequalities , algebra , Brazilian Math Olympiad 1979

Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?

1979 Brazil National Olympiad, 3

Tags: geometry , Brazilian Math Olympiad , Brazilian Math Olympiad 1979 , orthocenter

The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter.