Subcontests
(5)Simple NT
Prove that there are no pairs of positive integers (a,b,c,d,n) such that
a2+b2+c2+d2−4abcd=7⋅22n−1. Z-Shaped Polyline
In the xy-plane, points with integer coordinates ranging from 1 to 2000 for both the x and y coordinates are called good points. Moreover, for any four points A(x1,y1),B(x2,y2),C(x3,y3),D(x4,y4) satisfying the following conditions, the polyline ABCD is called a Z-shaped polyline.[*] All A,B,C,D are good points.
[*] x1<x2, y1=y2
[*] x2>x3, y2−x2=y3−x3
[*] x3<x4, y3=y4Find the smallest possible positive integer n such that Z-shaped polylines Z1,Z2,…,Zn satisfy the following condition: for any good point P, there exists an integer i between 1 and n inclusive such that P lies on Zi. Magical Set
Let n≥2 be an integer. Determine all sets of real numbers (a1,a2,…,an) such that a1−2a2,a2−2a3,…,an−1−2an,an−2a1 are rearrangements of a1,a2,…,an.