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.

I think he means that the half circle lies in the +ve x axis part of the plane

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).

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".

Yeah, its wrong as stated in the paper. Its true for integer radii though. And yes, he probably meant only the unit circle.

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 $\sum_{i=1}^{2n} y_i \ge 1$.

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 $1$.