MathDB

1

Part of 2009 Mexico National Olympiad

Problems(2)

Similar Triangles

Source: OMM 2009 1

7/15/2014
In △ABC\triangle ABC, let DD be the foot of the altitude from AA to BCBC. A circle centered at DD with radius ADAD intersects lines ABAB and ACAC at PP and QQ, respectively. Show that △AQP∼△ABC\triangle AQP\sim\triangle ABC.
geometry unsolvedgeometry
Plus or Minuses

Source: OMM 2009 4

7/15/2014
Let n>1n>1 be an odd integer, and let a1a_1, a2a_2, …\dots, ana_n be distinct real numbers. Let MM be the maximum of these numbers and mm the minimum. Show that it is possible to choose the signs of the expression s=±a1±a2±⋯±ans=\pm a_1\pm a_2\pm\dots\pm a_n so that
m<s<Mm<s<M
combinatorics unsolvedcombinatorics