MathDB

3

Part of 1995 IberoAmerican

Problems(2)

10th ibmo - chile 1995/q3.

Source: September 23rd - 30th

5/7/2006
Let r r and s s two orthogonal lines that does not lay on the same plane. Let AB AB be their common perpendicular, where A∈r A\in{}r and B∈s B\in{}s(*).Consider the sphere of diameter AB AB. The points M∈r M\in{r} and N∈s N\in{s} varies with the condition that MN MN is tangent to the sphere on the point T T. Find the locus of T T. Note: The plane that contains B B and r r is perpendicular to s s.
geometry3D geometrysphereanalytic geometryfunctionsimilar trianglesangle bisector
10th ibmo - chile 1995/q6.

Source: Spanish Communities

5/7/2006
A function f:N→Nf: \N\rightarrow\N is circular if for every p∈Np\in\N there exists n∈N, n≤pn\in\N,\ n\leq{p} such that fn(p)=pf^n(p)=p (ff composed with itself nn times) The function ff has repulsion degree k>0k>0 if for every p∈Np\in\N fi(p)≠pf^i(p)\neq{p} for every i=1,2,…,⌊kp⌋i=1,2,\dots,\lfloor{kp}\rfloor. Determine the maximum repulsion degree can have a circular function.
Note: Here ⌊x⌋\lfloor{x}\rfloor is the integer part of xx.
functionfloor functioninductioncalculusintegrationalgebra unsolvedalgebra