MathDB

11.1

Part of VMEO IV 2015

Problems(2)

Integral points

Source: VMEO IV - Junior Division

8/6/2016
On Cartesian plane, given a line defined by y=x+12y=x+\frac{1}{\sqrt{2}}. a) Prove that every circle has center I∈dI\in d and radius is 18\frac{1}{8} has no integral point inside. b) Find the greatest k>0k>0 such that the distance of every integral points to dd is greater or equal than kk.
number theoryalgebraanalytic geometry
a^2 + b^2 + c^2 <= (k^2 + 1)(ab + bc + ca) if a/b+b/c+ c/a= (k + 1)^2 + 2/(k+1)

Source: 2015 VMEO IV Seniors 11.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/19/2021
Let k≥0k \ge 0 and a,b,ca, b, c be three positive real numbers such that ab+bc+ca=(k+1)2+2k+1.\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}. Prove that a2+b2+c2≤(k2+1)(ab+bc+ca).a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).
algebrainequalities