The top 1,119 students had 100 minutes to solve these five problems.

1. Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are 3 and 4. The lengths of the segments of the other chord are 6 and 2. Find the diameter of the circle.
   
   
2. Determine the greatest integer that will divide 13,511, 13,903 and 14,589 and leave the same remainder.
   
   
3. Suppose A, B and C are the angles of a triangle. Show that
   
   cos²A + cos²B + cos²C + 2 cos A cos B cos C = 1.
   
   
4. Given the linear fractional transformation
   
   f₁(x) = (2x − 1)/(x + 1),
   
   define f_n+1(x) = f₁(f_n(x)) for n = 1, 2, 3, … .
   It can be shown that f₃₅ = f₅.
   
   (a) Find a function g such that f₁(g(x)) = g(f₁(x)) = x.
   (b) Find f₂₈.
   
   
5. Suppose a is a complex number such that a¹⁰ + a⁵ + 1 = 0.
   
   Determine the value of a²⁰⁰⁵ + 1/a²⁰⁰⁵.

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