MathDB

1

Part of 2003 China National Olympiad

Problems(2)

A, I and T are collinear iff [BKS]=[CKR]

Source: Chinese Mathematical Olympiad 2003 Problem 1

2/18/2012
Let II and HH be the incentre and orthocentre of triangle ABCABC respectively. Let P,QP,Q be the midpoints of AB,ACAB,AC. The rays PI,QIPI,QI intersect AC,ABAC,AB at R,SR,S respectively. Suppose that TT is the circumcentre of triangle BHCBHC. Let RSRS intersect BCBC at KK. Prove that A,IA,I and TT are collinear if and only if [BKS]=[CKR][BKS]=[CKR].
Shen Wunxuan
geometryincentercircumcirclegeometry proposed
a^m+1 divides a^n + 203

Source: Chinese Mathematical Olympiad 2003 Problem 4

2/18/2012
Find all integer triples (a,m,n)(a,m,n) such that am+1∣an+203a^m+1|a^n+203 where a,m>1a,m>1.
Chen Yonggao
modular arithmeticnumber theory proposednumber theory