Growing

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

Remainder Hunt

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

Summit

Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.

Binomial

Stage: 5 Challenge Level:

Show that

$\sum_{k=0}^n {n\choose k}^2 = {2n \choose n}.$