MathDB

9

Part of 2011 JBMO Shortlist

Problems(2)

sums with minimums in R

Source: JBMO 2011 Shortlist A9

10/14/2017
Let x1,x2,...,xnx_1,x_2, ..., x_n be real numbers satisfying ∑k=1n−1min(xk;xk+1)=min(x1;xn)\sum_{k=1}^{n-1} min(x_k; x_{k+1}) = min(x_1; x_n). Prove that ∑k=2n−1xk≥0\sum_{k=2}^{n-1} x_k \ge 0.
JBMOalgebra
2011 in plane, distance <1, 1005 points outside unit circles

Source: JBMO 2011 Shortlist C9

10/14/2017
Decide if it is possible to consider 20112011 points in a plane such that the distance between every two of these points is different from 11 and each unit circle centered at one of these points leaves exactly 10051005 points outside the circle.
JBMOcombinatoricscombinatorial geometry