problem

Favourite

### Binomial

By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn

problem
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Binomial

Favourite

By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn

problem
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Partitioning revisited

Favourite

We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4

problem
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Cubes within Cubes revisited

Favourite

Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?

problem
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Particularly general

By proving these particular identities, prove the existence of general cases.

problem
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Powerful Factors

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³
= (x+y) (x²-xy+y²) to find the highest power of 2 and the
highest power of 3 which divide 5^{36}-1.