this month, Stage 3 & 4

October 1999

 

Problems

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Like Powers

Stage:3 Challenge Level:Challenge Level:1

Investigate 1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n and 2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n for different values of n.

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Thirty Six Exactly

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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All Square

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

Solve the system of equations xy = 1 yz = 4 zx = 9

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Three Four Five

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

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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Rachel's Problem

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

Is it true that 99^n has 2n digits and 999^n has 3n digits? Investigate!