Subcontests
(3)F,C,H are collinear
The circle Γ and the line ℓ have no common points. Let AB be the diameter of Γ perpendicular to ℓ, with B closer to ℓ than A. An arbitrary point C=A, B is chosen on Γ. The line AC intersects ℓ at D. The line DE is tangent to Γ at E, with B and E on the same side of AC. Let BE intersect ℓ at F, and let AF intersect Γ at G=A. Let H be the reflection of G in AB. Show that F,C, and H are collinear. Two fixed points of a function
Suppose that f:{1,2,…,1600}→{1,2,…,1600} satisfies f(1)=1 and
f^{2005}(x)=x \text{for}\ x=1,2,\ldots ,1600.
(a) Prove that f has a fixed point different from 1.
(b) Find all n>1600 such that any f:{1,…,n}→{1,…,n} satisfying the above condition has at least two fixed points.