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The 30th Vietnam Mathematical Olympiad
1992年第30届越南数学奥林匹克

A1. ABCD is a tetrahedron. The three face angles at A sum to 180o, and the three face angles at B sum to 180o. Two of the face angles at C, ∠ACD and ∠BCD, sum to 180o. Find the sum of the areas of the four faces in terms of AC + CB = k and ∠ACB = x.

A2. For any positive integer n, let f(n) be the number of positive divisors of n which equal ±1 mod 10, and let g(n) be the number of positive divisors of n which equal ±3 mod 10. Show that f(n) ≥ g(n).

A3. Given a > 0, b > 0, c > 0, define the sequences a, bn, cn by a0 = a, b0 = b, c0 = c, an+1 = an + 2/(bn + cn), bn+1 = 2/(cn + an), cn+1 = cn + 2/(an + bn). Show that an tends to infinity.

B1. Label the squares of a 1991 x 1992 rectangle (m, n) with 1 ≤ m ≤ 1991 and 1 ≤ n ≤ 1992. We wish to color all the squares red. The first move is to color red the squares (m, n), (m+1, n+1), (m+2, n+1) for some m < 1990, n < 1992. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way ?

B2. ABCD is a rectangle with center O and angle AOB ≤ 45o. Rotate the rectangle about O through an angle 0 < x < 360o. Find x such that the intersection of the old and new rectangles has the smallest possible area.

B3. Let p(x) be a polynomial with constant term 1 and every coefficient 0 or 1. Show that p(x) does not have any real roots > (1 - √5)/2.

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