Hong Kong Maths Olympiad Heat Group (94-95)

1.Find the number of positive integral solutions of the equation x³+(x+1)³+(x+2)³=(x+3)³.

2.In figure1, ABCD is a quadrilateral whose diagonals intersect at O. If ¡çAOB=30^o, AC=24 and BD=22, find the area of the quadrilateral ABCD.

Figure1

3.Given that (1/n)+(2/n)+(3/n)+...+(n-1)/n=(n-1)/2, find the value of (1/2)+(1/3)+(2/3)+(1/4)+(2/4)+(3/4)+(1/5)+...+(1/10+...+9/10).

4.Suppose x and y are positive integers such that x²=y²+2000, find the lease value of x.

5.Given that 37¹⁰⁰ is a 157-digit number, and 37¹⁵ is an n-digit number. Find n.

6.Given that 1²+2²+3²+...+n²=(n/6)(n+1)(2n+1), find the value of 19×21+18×22+17×23+...+1×39.

7.In figure 2, ABCD is a square where AB=1 and CPQ is an equilateral triangle. Find the area of triangle.

Figure2

8.The number of ways to pay a sum of $17 by using $1 coins, $2 coins and $5 coins is n. Find n. (Assume that all types of coins must be used each time.)

9.In figure 3, find the total number of triangles in the 3×3 square.

Figure3

10.In figure 4, the radius of the quadrant and the diameter of the large semi-circle is 2. Find the radius of the small semi-circle.

Figure4