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ΣP_iA^{2p}_i is an integer, where P_i projections of points A_i of an arc on OA

Source: Moldova 2002 TST 2 P4

August 25, 2018

IntegerSumdistancearcprojectiongeometrynumber theory

Problem Statement

Let

C

be the circle with center

O(0,0)

and radius

1

, and

A(1,0), B(0,1)

be points on the circle. Distinct points

A_1,A_2, ....,A_{n-1}

C

divide the smaller arc

AB

into

n

equal parts (

n \ge 2

). If

P_i

is the orthogonal projection of

A_i

OA

(

i =1, ... ,n-1

), find all values of

n

such that

P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}

is an integer for every positive integer

p