MathDB

2

Part of 2003 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Concurrent lines in circles tangent to circumcircle diagram

Source: XII Rioplatense Mathematical Olympiad (2003), Level 3

8/8/2011
Triangle ABCABC is inscribed in the circle Γ\Gamma. Let Γa\Gamma_a denote the circle internally tangent to Γ\Gamma and also tangent to sides ABAB and ACAC. Let A′A' denote the point of tangency of Γ\Gamma and Γa\Gamma_a. Define B′B' and C′C' similarly. Prove that AA′AA', BB′BB' and CC′CC' are concurrent.
geometrycircumcirclegeometric transformationvideosprojective geometrygeometry unsolvedmixtilinear incircle
Arithmetic progressions that cover 1 out of k integers

Source: XII Rioplatense Mathematical Olympaid (2003), Level 3

8/9/2011
Let nn and kk be positive integers. Consider nn infinite arithmetic progressions of nonnegative integers with the property that among any kk consecutive nonnegative integers, at least one of kk integers belongs to one of the nn arithmetic progressions. Let d1,d2,…,dnd_1,d_2,\ldots,d_n denote the differences of the arithmetic progressions, and let d=min⁡{d1,d2,…,dn}d=\min\{d_1,d_2,\ldots,d_n\}. In terms of nn and kk, what is the maximum possible value of dd?
arithmetic sequencealgebra unsolvedalgebra