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2014年南非数学奥林匹克(第三轮)(pdf版,无答案)


The South African Mathematical Olympiad Third Round 2014 Senior Division (Grades 10 to 12) Time : 4 hours (No calculating devices are allowed) 1. Determine the last two digits of the product of the squares of all positive odd integers less than 2014. 2. Given that a?c a?b =2 and =3 c?d b?d for certain real numbers a, b, c, d, determine the value of a?d . b?c 3. In obtuse triangle ABC, with the obtuse angle at A, let D, E, F be the feet of the altitudes through A, B, C respectively. DE is parallel to CF, and DF is parallel to the angle bisector of ∠BAC. Find the angles of the triangle. 4. (a) Let a, x, y be positive integers. Prove: if x ?= y, then also ax + gcd(a, x) + lcm(a, x) ?= ay + gcd(a, y) + lcm(a, y). (b) Show that there are no two positive integers a and b such that ab + gcd(a, b) + lcm(a, b) = 2014. 5. Let n > 1 be an integer. An n × n-square is divided into n2 unit squares. Of these unit squares, n are coloured green and n are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row? 6. Let O be the centre of a two-dimensional coordinate system, and let A1 , A2 , . . . , An be points in the ?rst quadrant and B1 , B2 , . . . , Bm points in the second quadrant. We associate numbers a1 , a2 , . . . , an to the points A1 , A2 , . . . , An and

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