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Đề thi APMOPS 2011 đầy đủ ở đây: Thichtoanhoc-downloaded-APMOPS2011-problems
Q1. If the following figure is folded into the shape of a cube, what is the number opposite to the face marked P?
Q2. Find the value of
Q3. A circle of radius 1 m has some points lying on its circumference. Find the minimum number of points such that at least two points are less than 1 m apart.
Q4. Three sides of a four-sided figure are of lengths 4 cm, 9 cm and 14 cm respectively. If the largest possible length of the fourth side is x cm where x is a whole number, find the value of x.
Q5. Find the largest prime number that divides the number
(1 x 2 x 3 x · · · x 97 x 98) + (1 x 2 x 3 x · · · x 99 x 100)
Q6. ABCD is a square and BCE is an equilateral triangle. If BC is 8 cm, find the radius of the circle passing through A,E and D in cm.
Q7. Find the value of
Q8. The following diagram shows a circle of radius 8 cm with the centre R. Two smaller circles with centre P and Q touch the circle with centre R and each other as shown in the diagram. Find the perimeter of the triangle PQR in cm.
Q9. Peter and Jane competed in a 5000 m race. Peter’s speed was 4 times that of Jane’s. Jane ran from the beginning to the end, whereas Peter stopped running every now and then. When Jane crossed the finish line, Peter was 100 m behind. Jane ran a total of x m during the time Peter was not running. Find the value of x.