The top 1000 students had 100 minutes to solve these five problems.

1. Suppose A, B, and C are the angles of a triangle. Prove that
   
   1 – 8 cos A cos B cos C = sin²(B – C) + (cos(B – C) – 2 cos A)².
   
   
2. Let x₁, x₂, … , x₁₀₀ be integers whose values are either 0 or 1.
   
   
   1. Show that
      
      x₁ + x₂ + ··· + x₁₀₀ – (x₁ x₂ + x₂ x₃ + ··· + x₉₉ x₁₀₀ + x₁₀₀ x₁) ≤ 50.
      
      
   2. Give specific values for x₁, x₂, … , x₁₀₀ that give equality.
      
      
3. Let ABCD be a trapezoid whose area is 32 square meters. Suppose the lengths of the parallel segments AB and DC are 2 meters and 6 meters, respectively, and P is the intersection of the diagonals AC and BD. If a line through P intersects AD and BC at E and F , respectively, determine, with a proof, the minimum possible area for quadrilateral ABFE.
   
   
4. Let n be a positive integer and x be a real number. Show that
   
   fl(nx) = fl(x) + fl(x + 1/n) + fl(x + 2/n) + ··· + fl(x + (n – 1)/n)
   
   where fl(a) is the greatest integer less than or equal to a. (For example, fl(4.5) = 4 and fl(–4.5) = –5.)
   
   
5. A 3n-digit positive integer (in base 10) containing no zero is said to be quad-perfect if the number is a perfect square and each of the three numbers obtained by viewing the first n digits, the middle n digits and the last n digits as three n-digit numbers is in itself a perfect square. (For example, when n = 1, the only quad-perfect numbers are 144 and 441.) Find all 9-digit quad-perfect numbers.

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