MathDB

2

Part of 2008 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Assuming every distance between points in n closed intervals

Source: XVII Olimpíada Matemática Rioplatense (2008)

7/25/2011
On a line, there are nn closed intervals (none of which is a single point) whose union we denote by SS. It's known that for every real number dd, 0<d≤10<d\le 1, there are two points in SS that are a distance dd from each other. (a) Show that the sum of the lengths of the nn closed intervals is larger than 1n\frac{1}{n}. (b) Prove that, for each positive integer nn, the 1n\frac{1}{n} in the statement of part (a) cannot be replaced with a larger number.
algebra unsolvedalgebra
Perpendicularity in incircle/circumcircle/arc midpt diagram

Source: XVII Olimpíada Matemática Rioplatense (2008)

7/25/2011
In triangle ABCABC, where AB<ACAB<AC, let XX, YY, ZZ denote the points where the incircle is tangent to BCBC, CACA, ABAB, respectively. On the circumcircle of ABCABC, let UU denote the midpoint of the arc BCBC that contains the point AA. The line UXUX meets the circumcircle again at the point KK. Let TT denote the point of intersection of AKAK and YZYZ. Prove that XTXT is perpendicular to YZYZ.
geometrycircumcircletrigonometrycyclic quadrilateralangle bisectorgeometry unsolved