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ASU 157 All Soviet Union MO 1971 f_a(x,y) = x^2 + axy + y^2

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July 3, 2019
algebrapolynomial

Problem Statement

a) Consider the function f(x,y)=x2+xy+y2f(x,y) = x^2 + xy + y^2 Prove that for the every point (x,y)(x,y) there exist such integers (m,n)(m,n), that f((x−m),(y−n))≤1/2f((x-m),(y-n)) \le 1/2
b) Let us denote with g(x,y)g(x,y) the least possible value of the f((x−m),(y−n))f((x-m),(y-n)) for all the integers m,nm,n. The statement a) was equal to the fact g(x,y)≤1/2g(x,y) \le 1/2. Prove that in fact, g(x,y)≤1/3g(x,y) \le 1/3 Find all the points (x,y)(x,y), where g(x,y)=1/3g(x,y)=1/3.
c) Consider function fa(x,y)=x2+axy+y2   (0≤a≤2)f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2) Find any cc such that ga(x,y)≤cg_a(x,y) \le c. Try to obtain the closest estimation.
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