MathDB

2

Part of 2008 JBMO Shortlist

Problems(3)

3 positive reals w product 1,sum their pairwise product sums

Source: JBMO 2008 Shortlist C2

10/14/2017
Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product 11 and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to 11. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).
JBMOcombinatorics
2008 JBMO Shortlist G2

Source: 2008 JBMO Shortlist G2

10/10/2017
For a fixed triangle ABCABC we choose a point MM on the ray CACA (after AA), a point NN on the ray ABAB (after BB) and a point PP on the ray BCBC (after CC) in a way such that AM−BC=BN−AC=CP–ABAM -BC = BN- AC = CP – AB. Prove that the angles of triangle MNPMNP do not depend on the choice of M,N,PM, N, P .
JBMOgeometry
set of rationals not multiple of an $n$-th power of a prime

Source: JBMO 2008 Shortlist N2

10/14/2017
Let n≥2n \ge 2 be a fixed positive integer. An integer will be called "nn-free" if it is not a multiple of an nn-th power of a prime. Let MM be an infi nite set of rational numbers, such that the product of every nn elements of MM is an nn-free integer. Prove that MM contains only integers.
JBMOnumber theory