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Part of 2001 Poland - Second Round

Problems(2)

Difference of polynomials is not squarefree for odd primes

Source: Polish Second Round 2001

k,n>1

be integers such that the number

p=2k-1

is prime. Prove that, if the number

\binom{n}{2}-\binom{k}{2}

is divisible by

p

, then it is divisible by

p^2

algebrapolynomialmodular arithmeticnumber theory proposednumber theory

Arithmetic progression contains at least one rational term

Source: Polish Second Round 2001

Find all integers

n\ge 3

for which the following statement is true: Any arithmetic progression

a_1,\ldots ,a_n

n

terms for which

a_1+2a_2+\ldots+na_n

is rational contains at least one rational term.

modular arithmeticarithmetic sequenceirrational numbernumber theory proposednumber theory