tailieunhanh - ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 27

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 27', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1991 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Let G be the centroid of triangle ABC and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X Y and G are collinear and XY and BC are parallel. Suppose that XC and GB intersect at Q and YB and GC intersect at P. Show that triangle MPQ is similar to triangle ABC. Question 2 Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red show that there are at least 1991 red points in the plane. Can you find a special case with exactly 1991 red points Question 3 Let a-1 a2 . an bl b2 . bn be positive real numbers such that ai a2 an bl b2 bn. Show that al a2 an ai a2 ---- an ai bi a2 b2 an bn 2 Question 4 During a break n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy then he skips the next child and gives a candy to the next one then he skips 2 and gives a candy to the next one then he skips 3 and so on. Determine the values of n for which eventually perhaps after many rounds all children will have at least one candy each. Question 5 Given are two tangent circles and a point P on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point .