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3

Part of 2011 Spain Mathematical Olympiad

Problems(2)

Tangent points of a quadrilateral to a sphere are coplanar

Source: Spanish Mathematical Olympiad, National Stage, 2011

8/9/2011
Let AA, BB, CC, DD be four points in space not all lying on the same plane. The segments ABAB, BCBC, CDCD, and DADA are tangent to the same sphere. Prove that their four points of tangency are coplanar.
geometry3D geometryspheregeometry unsolved
Sequence where 2011 divides S_{2011a} - S_{a}

Source: Spanish Mathematical Olympiad, National Stage, 2011

8/9/2011
The sequence S0,S1,S2,…S_0,S_1,S_2,\ldots is defined by[*]Sn=1S_n=1 for 0≤n≤20110\le n\le 2011, and [*]Sn+2012=Sn+2011+SnS_{n+2012}=S_{n+2011}+S_n for n≥0n\ge 0.Prove that S2011a−SaS_{2011a}-S_a is a multiple of 20112011 for all nonnegative integers aa.
algebrapolynomialinductionfloor functionmodular arithmeticnumber theorynumber theory unsolved