Section A (answers only)
A1 Evaluate (1 + 998 ) / 1998
A2 9 = 32 = 3 x 3 is a square ( or "square number" ). How many ( positive ) square numbers are there less than 1000?
A3 ABCD is a square with each side of length 2 units. M is the midpoint of AB and N is the midpoint of BC. What is the area of triangle DMN?
A4 I choose a number, multiply it by 2, then subtract 3, then divide by 4, then add 5, and get back to my original number. What was my number?
A5 Two touching circles have the same radius. One of them is fixed. The other has an upward pointing arrow designed on it. The circle with the arrow rolls without slipping three times around the fixed circle. How many complete turns does the arrow make?
A6 Meg ate one quarter of her packet of sweets on the way to the cinema, one third of what was left while queueing for tickets, and one half of the rest before the film started. What fraction of the packet was left?
A7 A cube has twelve edges: that is, twelve pairs of adjacent vertices. How many pairs of non-adjacent vertices are there in a cube?
A8 A 7/8 cm by 5/6 cm by 3/4 cm plasticene cuboid (i.e. brick shape) is made into two cubes. One cube has edge length 1/2 cm. What is the edge length of the other?
A9 Notice that "24" is divisible both by 2 and by 4. How many two-digit numbers are divisible by both the tens digit and by the units digit?
A10 Matt the mad mathematician was late for work again! "What's your excuse this time?" asked his boss. "While getting dressed I got carried away," said Matt. "Just suppose you had a pair of socks and a pair of shoes ( and no other clothes! ) In how many different orders can you put them on?" ( You can't put on your left shoe until your left foot has a sock on it ! )
Section B (full written solutions)
B1 The year 1995 was unusual: the last two digits are an exact multiple of the first two digits. How many such years have there been since 1000AD?
B2 Four wounded soldiers must cross a damaged bridge at night. The bridge takes at most two at a time. They have only one torch. Each pair moves at the speed of the slowest, and one person must return with the torch before the next pair can cross. Find ( with proof ) the shortest time needed for all to cross if the four soldiers take 1, 2, 4 and 6 minutes respectively.
B3 You are challenged to place the numbers 1 to 8 at the vertices of a cube so that the sums of the two numbers at the end of each edge are all different. Show how to do this, or prove that it is impossible.
B4 Two identical touching circles just fit inside a square with both their centres on one of the diagonals of the square. What fraction of the square is occupied by the two circles?
B5 A wire framework forms an OCTOHEDRON (two square-based pyramids placed base-to-base). It has six vertices, with four triangles meeting at each vertex.
(a) Draw a "circuit" which uses no edge more than once and which visits each vertex exactly once before returning to the start.
(b) How many such circuits are there in all? How can you be sure?
B6 ABCD is a square of side 2 units; M is the midpoint of AD. The square is folded so that C folds exactly on top of M. How long is the crease?
|