19th Vietnamese Mathematical Olympiad 1981 Problems

19th Vietnamese Mathematical Olympiad 1981 Problems

A1.  Show that the triangle ABC is right-angled iff sin A + sin B + sin C = cos A + cos B + cos C + 1.
A2.  Find all integral values of m such that x³ + 2x + m divides x¹² - x¹¹ + 3x¹⁰ + 11x³ - x² + 23x + 30.

A3.  Given two points A, B not in the plane p, find the point X in the plane such that XA/XB has the smallest possible value.
B1.  Find all real solutions to:
w² + x² + y² + z² = 50
w² - x² + y² - z² = -24
wx = yz
w - x + y - z = 0.
B2.  x₁, x₂, x₃, ... , x_n are reals in the interval [a, b]. M = (x₁ + x₂ + ... + x_n)/n, V = (x₁² + x₂² + ... + x_n²)/n. Show that M² ≥ 4Vab/(a + b)².

B3.  Two circles touch externally at A. P is a point inside one of the circles, not on the line of centers. A variable line L through P meets one circle at B (and possibly another point) and the other circle at C (and possibly another point). Find L such that the circumcircle of ABC touches the line of centers at A.