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a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2, a_1 < a_2 < ...< a_{n-1} < a_n

Source: 2018 Romania District VII p4

September 1, 2024
number theoryinequalities

Problem Statement

a) Consider the positive integers a,b,ca, b, c so that a<b<ca < b < c and a2+b2=c2a^2+b^2 = c^2. If a1=a2a_1 = a^2, a2=aba_2 = ab, a3=bca_3 = bc, a4=c2a_4 = c^2, prove that a12+a22+a32=a42a_1^2+a_2^2+a_3^2=a_4^2 and a1<a2<a3<a4a_1 < a_2 < a_3 < a_4.
b) Show that for any n∈Nn \in N, n≥3n\ge 3, there exist the positive integers a1,a2,...,ana_1, a_2,..., a_n so that a12+a22+...+an−12=an2a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2 and a1<a2<...<an−1<ana_1 < a_2 < ...< a_{n-1} < a_n
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