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Binomial

Age 16 to 18

Challenge Level Yellow star

Show that

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

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Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$ .

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?