MathDB

2

Part of 2000 France Team Selection Test

Problems(2)

France TST 2000 D2Q2

Source:

3/2/2013
A,B,C,DA,B,C,D are points on a circle in that order. Prove that ∣AB−CD∣+∣AD−BC∣≥2∣AC−BD∣|AB-CD|+|AD-BC| \ge 2|AC-BD|.
geometry unsolvedgeometry
France TST 2000 D1Q2

Source:

3/2/2013
A function from the positive integers to the positive integers satisfies these properties 1. f(ab)=f(a)f(b)f(ab)=f(a)f(b) for any two coprime positive integers a,ba,b. 2. f(p+q)=f(p)+f(q)f(p+q)=f(p)+f(q) for any two primes p,qp,q. Prove that f(2)=2,f(3)=3,f(1999)=1999f(2)=2, f(3)=3, f(1999)=1999.
functionnumber theory unsolvednumber theory