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The 19th Vietnam Mathematical Olympiad
1981年第19届越南奥林匹克数学竞赛

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, ... , xn are reals in the interval [a, b]. M = (x1 + x2 + ... + xn)/n, V = (x12 + x22 + ... + xn2)/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.

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