33rd IMO shortlist 1992/8 solution

33rd IMO 1992 shortlist

Problem 8

Show that there is a convex 1992-gon with sides 1, 2, 3, ... , 1992 in some order, and an inscribed circle (touching every side).

Solution

Suppose the polygon has vertices A₁, A₂, ... , A₁₉₉₂. Let the length of the tangents from A_i to the inscribed circle be x_i. Then we must have:
x₁ + x₂ = a₁
x₂ + x₃ = a₂
...
x₁₉₉₂ + x₁ = a₁₉₉₂
for some permutation a₁, a₂, ... , a₁₉₉₂ of 1, 2, ... , 1992.

This has a solution x_4n+1 = ½, x_4n+2 = 4n+½, x_4n+3 = 1½, x_4n+4 = 4n+2½, and a_4n+1 = 4n+1, a_4n+2 = 4n+2, a_4n+3 = 4n+4, a_4n+4 = 4n+3.

Take a circle centre O of large radius r and a broken line A₁A₂A₃...A₁₉₉₃ with A_iA_i+1 = a_i and wrap it around the circle. Take the distance of A₁ from the first point of contact to be ½. Then the distance of A₁₉₉₃ to the last point of contact will also be ½. We have OA₁² = ½² + r² = OA₁₉₉₃². Now if we gradually reduce the radius A₁ and A₁₉₉₃ will approach and eventually coincide to give the required polygon.

33rd IMO shortlist 1992