MathDB

Problem 2

Part of 2014 VJIMC

Problems(2)

x+y=z in Fp * subgroup, 6|order

Source: VJIMC 2014 2.2

5/21/2021
Let pp be a prime number and let AA be a subgroup of the multiplicative group Fp∗\mathbb F^*_p of the finite field Fp\mathbb F_p with pp elements. Prove that if the order of AA is a multiple of 66, then there exist x,y,z∈Ax,y,z\in A satisfying x+y=zx+y=z.
group theoryabstract algebra
faro shuffle 8 times =Id (VJIMC 2014 1.2)

Source:

5/20/2021
We have a deck of 2n2n cards. Each shuffling changes the order from a1,a2,…,an,b1,b2,…,bna_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n to a1,b1,a2,b2,…,an,bna_1,b_1,a_2,b_2,\ldots,a_n,b_n. Determine all even numbers 2n2n such that after shuffling the deck 88 times the original order is restored.
combinatorics