MathDB

1

Part of 2006 Iran MO (2nd round)

Problems(2)

Prove that A'B' = radius of C_1 - Iran NMO 2006 - Problem1

Source:

9/23/2010
Let C1,C2C_1,C_2 be two circles such that the center of C1C_1 is on the circumference of C2C_2. Let C1,C2C_1,C_2 intersect each other at points M,NM,N. Let A,BA,B be two points on the circumference of C1C_1 such that ABAB is the diameter of it. Let lines AM,BNAM,BN meet C2C_2 for the second time at A′,B′A',B', respectively. Prove that A′B′=r1A'B'=r_1 where r1r_1 is the radius of C1C_1.
geometry proposedgeometry
Finite n such that m+n|mn+1 - Iran NMO 2006 - Problem4

Source:

9/23/2010
a.) Let m>1m>1 be a positive integer. Prove there exist finite number of positive integers nn such that m+n∣mn+1m+n|mn+1.
b.) For positive integers m,n>2m,n>2, prove that there exists a sequence a0,a1,⋯ ,aka_0,a_1,\cdots,a_k from positive integers greater than 22 that a0=ma_0=m, ak=na_k=n and ai+ai+1∣aiai+1+1a_i+a_{i+1}|a_ia_{i+1}+1 for i=0,1,⋯ ,k−1i=0,1,\cdots,k-1.
number theory proposednumber theory