MathDB

1

Part of 2012 China Western Mathematical Olympiad

Problems(2)

Find smallest positive integer

Source: China Western Mathematical Olympiad 2012 P1

9/29/2012
Find the smallest positive integer mm satisfying the following condition: for all prime numbers pp such that p>3p>3,have 105∣9p2−29p+m.105|9^{ p^2}-29^p+m. (September 28, 2012, Hohhot)
modular arithmeticnumber theoryprime numbersnumber theory proposed
through the midpoint of $OH$

Source: 2012 CWMO P5

9/30/2012
OO is the circumcenter of acute ΔABC\Delta ABC, HH is the Orthocenter. AD⊥BCAD \bot BC, EFEF is the perpendicular bisector of AOAO,D,ED,E on the BCBC. Prove that the circumcircle of ΔADE\Delta ADE through the midpoint of OHOH.
geometrycircumcircletrapezoidratioparallelogram