1. Find the sum of the digits in the square of the number 111 111 111.

2. A train takes ¼ minute to pass a telegraph pole, and ¾ minute to pass through a tunnel 540 metres long. What is the length of the train?

3. In Δ

*ABC*, the length of*AB*is 13cm, the length of*AC*is 15cm, and the length of the perpendicular from*A*to*BC*(i.e. the "altitude" from*A*) is 12cm. Find the two possible lengths of*BC*.4. Circle

*K*has diameter*AB*. Circle*L*touches*K*internally and also touches the line*AB*at the centre of circle*K.*Circle*M*touches*L*externally and*K*internally and also has tangent*AB*. Find the ratio of the area of circle*K*to the area of circle*M*.5. How many odd numbers greater than 60000 can be made from the digits 5, 6, 7, 8, 9, 0 if no number contains any digit more than once?

6. The total area of all the faces of a cuboid is 22 cm

^{2}, and the total length of all its edges is 24 cm. Find the length of any one of its internal diagonals.7. Let

*n*be an integer greater than 6. Prove that if*n*– 1 and*n*+ 1 are both prime, then*n*^{2}(*n*^{2}+ 16) is divisible by 720. Is the converse true?8. Let

*G*be a convex quadrilateral. Show that there is a point*X*in the plane of*G*with the property that every straight line through*X*divides*G*into two regions of equal area if and only if*G*is a parallelogram.
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