MathDB

1

Part of 2000 Cono Sur Olympiad

Problems(2)

Line through vertices of two squares tangent to circumcircle

Source: XI Olimpíada Matemática del Cono Sur (2000)

7/26/2011
In square ABCDABCD (labeled clockwise), let PP be any point on BCBC and construct square APRSAPRS (labeled clockwise). Prove that line CRCR is tangent to the circumcircle of triangle ABCABC.
geometrycircumcircleanalytic geometrygraphing linesslopegeometry unsolved
Can the digits of 16^n be nonincreasing

Source: XI Olimpíada Matemática del Cono Sur (2000)

7/26/2011
Call a positive integer descending if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example, 42214221 and 751751 are descending while 476476 and 455455 are not descending. Determine whether there exists a positive integer nn for which 16n16^n is descending.
number theory unsolvednumber theory