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permutation is quadratic if k+a_k is square, existence for every n

Source: Dutch BxMO TST 2010 p5

August 24, 2019

permutationQuadraticnumber theoryPerfect Square

Problem Statement

For any non-negative integer

n

, we say that a permutation

(a_0,a_1,...,a_n)

\{0,1,..., n\}

is quadratic if

k + a_k

is a square for

k = 0, 1,...,n

. Show that for any non-negative integer

n

, there exists a quadratic permutation of

\{0,1,..., n\}