MathDB

3

Part of 2004 IMC

Problems(2)

Problem 3 IMC 2004 Macedonia

Source:

7/25/2004
Let AnA_n be the set of all the sums ∑k=1narcsin⁡xk\displaystyle \sum_{k=1}^n \arcsin x_k , where n≥2n\geq 2, xk∈[0,1]x_k \in [0,1], and ∑k=1nxk=1\displaystyle \sum^n_{k=1} x_k = 1. a) Prove that AnA_n is an interval. b) Let ana_n be the length of the interval AnA_n. Compute lim⁡n→∞an\displaystyle \lim_{n\to \infty} a_n.
limitfunctioninequalitiesIMCcollege contests
Problem 9 IMC 2004 Macedonia

Source:

7/26/2004
Let DD be the closed unit disk in the plane, and let z1,z2,…,znz_1,z_2,\ldots,z_n be fixed points in DD. Prove that there exists a point zz in DD such that the sum of the distances from zz to each of the nn points is greater or equal than nn.
inequalitiestriangle inequalityIMCcollege contests