June 2002, Stage 4&5

Problems

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Prime Magic

Stage: 2, 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

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Look Before You Leap

Stage: 4 Challenge Level: Challenge Level:1

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

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Are You Kidding

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

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CD Heaven

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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Thank Your Lucky Stars

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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Basic Rhythms

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Explore a number pattern which has the same symmetries in different bases.

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Amida

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

To draw lots each player chooses a different upright, the paper is then unrolled, the paths charted and the results declared. Prove that no two paths ever end up at the foot of the same upright?

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Target Six

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.