Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.
Problem
Source: India Postal Coaching 2014 Set 5 Problem 5
Tags: vector, analytic geometry, algebra unsolved, algebra
30.11.2014 10:54
hajimbrak wrote: Let $(x_j,y_j)$,$1\le j\le 2n$,be $2n$ points on the half-circle in the upper half-plane.Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer.Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$. What is "the" half circle ? If this is any half circle, this is trivially wrong.
30.11.2014 11:05
I think he means that the half circle lies in the +ve x axis part of the plane
30.11.2014 11:28
I dont think so. I think he means one specific half circle (maybe the unit one) lying in the +ve y axis of the plane (and not +ve x axis).
30.11.2014 13:40
I also do not know what "the half-circle" means.The question was given in the question paper in this language and there was no clarification about the meaning of "the half circle".
30.11.2014 15:00
Yeah, its wrong as stated in the paper. Its true for integer radii though. And yes, he probably meant only the unit circle.
01.12.2014 08:38
So clearly the problem was supposed to mean this: Quote: Suppose $2n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{2n}, y_{2n})$ lie on the semicircle $y = \sqrt{1-x^2}$. Suppose $\sum_{i=1}^{2n} x_i$ is an odd integer; prove that $\sum_{i=1}^{2n} y_i \ge 1$.
02.12.2014 14:07
It is an old, well-known problem, usually stated like this. Given an even number of unit vectors in the plane, with non-negative $y$-coordinates and with the sum of the $x$-coordinates being an odd integer, prove that the sum of the $y$-coordinates is at least $1$.