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The 26th Vietnam Mathematical Olympiad
1988年第26届越南奥林匹克数学竞赛

A1. 994 cages each contain 2 chickens. Each day we rearrange the chickens so that the same pair of chickens are never together twice. What is the maximum number of days we can do this ?

A2. The real polynomial p(x) = xn - nxn-1 + (n2-n)/2 xn-2 + an-3xn-3 + ... + a1x + a0 (where n > 2) has n real roots. Find the values of a0, a1, ... , an-3.

A3. The plane is dissected into equilateral triangles of side 1 by three sets of equally spaced parallel lines. Does there exist a circle such that just 1988 vertices lie inside it ?

B1. The sequence of reals x1, x2, x3, ... satisfies xn+2 <= (xn + xn+1)/2. Show that it converges to a finite limit.

B2. ABC is an acute-angled triangle. Tan A, tan B, tan C are the roots of the equation x3 + px2 + qx + p = 0, where q is not 1. Show that p ≤ √27 and q > 1.

B3. For a line L in space let R(L) be the operation of rotation through 180 deg about L. Show that three skew lines L, M, N have a common perpendicular iff R(L) R(M) R(N) has the form R(K) for some line K.

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