Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

Binomial

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

Remainder Hunt

What are the possible remainders when the 100-th power of an integer is divided by 125?

Summit

Stage: 5 Challenge Level:

Prove that the sum

$$\sum_{t=0}^m {(-1)^t\over t!(m-t)!} = 0$$