Algebra - January 2004, Stage 4&5

Problems

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Chocolate 2010

Age 14 to 16 Challenge Level:

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Integer and Integer

Age 14 to 16 Short Challenge Level:

Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?

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Fibonacci Factors

Age 16 to 18 Challenge Level:

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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More Beads

Age 16 to 18 Challenge Level:

With red and blue beads on a circular wire; 'put a red bead between any two of the same colour and a blue between different colours then remove the original beads'. Keep repeating this. What happens?

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Poly Fibs

Age 16 to 18 Challenge Level:

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.