January 2002, Stage 3&4

Problems

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Eminit

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

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

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Get Cross

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

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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Matter of Scale

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove Pythagoras Theorem using enlargements and scale factors.

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Deep Roots

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

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Sixational

Stage: 4 and 5 Challenge Level: Challenge Level:1

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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