Problem 1

Part of 2000 VJIMC

Problems(2)

product (k^2+1)=4 mod p from k=1 to k=p if p=4n-1

Source: VJIMC 2000 2.1

p

be a prime of the form

p=4n-1

n

is a positive integer. Prove that

\prod_{k=1}^p(k^2+1)\equiv4\pmod p.

uncountable subset family, pairwise intersections are finite

Source: VJIMC 2000 1.1

Is there a countable set

Y

and an uncountable family

\mathcal F

of its subsets such that for every two distinct

A,B\in\mathcal F

, their intersection

A\cap B

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