Powers and roots

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Ab Surd Ity

Age

16 to 18

Challenge level

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).
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How Many Solutions?

Age

16 to 18

Challenge level

Find all the solutions to the this equation.
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Em'power'ed

Age

16 to 18

Challenge level

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?
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Surds

Age

14 to 16

Challenge level

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay
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Sept 03

Age

11 to 14

Challenge level

What is the last digit of the number 1 / 5^903 ?
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Root to Poly

Age

14 to 16

Challenge level

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.
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What an Odd Fact(or)

Age

11 to 14

Challenge level

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?
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Equal Temperament

Age

14 to 16

Challenge level

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
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Cubes Within Cubes

Age

7 to 14

Challenge level

We start with one yellow cube and build around it to make a 3×3×3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
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Pocket Money

Age

11 to 14

Challenge level

Which of these pocket money systems would you rather have?