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Part of 2010 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Min/max{a,b,c,d} if a^b divides b^c divides c^d divides d^a

Source: XIX Olimpíada Matemática Rioplatense (2010)

a

b

c

d

are distinct positive integers such that

a^b

b^c

b^c

c^d

c^d

d^a

. (a) Is it possible to determine which of the numbers

a

b

c

d

is the smallest? (b) Is it possible to determine which of the numbers

a

b

c

d

is the largest?

inequalitiesnumber theorynumber theory unsolved

999 distinct remainders with one of them zero

Source: XIX Olimpíada Matemática Rioplatense (2010)

r_2, r_3,\ldots, r_{1000}

denote the remainders when a positive odd integer is divided by

2,3,\ldots,1000

, respectively. It is known that the remainders are pairwise distinct and one of them is

0

. Find all values of

k

for which it is possible that

r_k = 0

modular arithmeticnumber theory unsolvednumber theory