Hong Kong Maths Olympiad Heat Group (90-91)

1.Find the unit-digit of 1357⁷⁸⁹⁰.

2.If (1/2)+(1/6)+(1/12)+(1/20)+(1/30)+(1/42)+”K+(1/2450)=(x/100), find x.

3.(a/3)+(b/4) and (c/6) are three proper fractions in their simplest form, where a, b and c are positive integers. If c is added to the numerator of each fraction, then the sum of the fractions formed will be equal to 6. Find the value of a+b+c.

4.Study the Pascal's triangle shown below:

5.In the multiplication”¼”¼”¼×”¼”¼=”¼”¼×”¼”¼=5568, each of the above boxes represents an integer from 1 to 9. If the integers for the nine boxes above are all different, find the number represented by ”¼”¼”¼.

6.Find the remainder when 1997¹⁹⁹⁰-1991 is divided by 1996.

7.Find the least positive integral value of n such that

8.One of the solutions of the equation 32x+59y=3259 in positive integers is given by (x,y) = (100, 1). It is known that there is exactly one more pair of positive integers a, b(a is not equal to 100 and b is not equal to 1) such that 32a +59b= 3259. Find a.

Figure1

10.In figure 2, two chords AOB, COD cut at O. If the tangents at A and C meet at X, the tangents at B and D meet at Y and angle AXC=130^o, angle AOD=120^o, angle BYD=k^o, find k.

Figure2