Solutions in math book titled Narrative Approaches to the International Mathematical Problems

Page 1

Narrative approaches to the international mathematical problems

The following math problems have solutions in the math book with title Narrative Approaches to the International Mathematical Problems. The book is available for purchase at Amazon.com and other online outlets. Problem 1 of the United States Mathematical Olympiad 1973 Two points, P and Q, lie in the interior of a regular tetrahedron ABCD. Prove that angle PAQ < 60°. Problem 1 of the United States Mathematical Olympiad 2010 Let AXYZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P, Q, R, S the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines PQ and RS is half the size of ∠XOZ, where O is the midpoint of segment AB. Problem 1 of the International Mathematical Olympiad 2006 Let ABC be a triangle with incenter I. A point P in the interior of the triangle satisfies ∠PBA + ∠PCA = ∠PBC + ∠PCB. Show that AP > AI, and that equality holds if and only if P = I. Problem 4 of the United States Mathematical Olympiad 1975 Two given circles intersect in two points P and Q. Show how to construct a segment AB passing through P and terminating on the two circles such that AP×PB is a maximum. Problem 4 of the United States Mathematical Olympiad 1979 Show how to construct a chord FPE of a given angle A through a fixed point P within the angle A such that + is a maximum. Problem 4 of the United States Mathematical Olympiad 2010

1


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.