MathDB

1

Part of 1975 Bundeswettbewerb Mathematik

Problems(2)

sequence of lattice points with non-negative integer coordinates

Source: 1975 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

11/20/2022
In a planar coordinate system, the points have non-negative integer coordinates numbered according to the figure. E.g. the point (3,1)(3,1) has the number 1212. https://cdn.artofproblemsolving.com/attachments/a/a/28725d75f281ac4618129067037d751c8d8f83.png What is the number of the point(x,y)(x,y)?
analytic geometrycombinatorics
Bundeswettbewerb Mathematik 1975 Problem 2.1

Source: Bundeswettbewerb Mathematik 1975 Round 2

10/22/2022
Let a,b,c,da, b, c, d be distinct positive real numbers. Prove that if one of the numbers c,dc, d lies between aa and bb, or one of a,ba, b lies between cc and dd, then (a+b)(c+d)>ab+cd\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd} and that otherwise, one can choose a,b,c,da, b, c, d so that this inequality is false.
inequalitiesRealalgebra