35th Polish Mathematical Olympiad Problems 1984

35th Polish Mathematical Olympiad Problems 1984

A1.  X is a set with n > 2 elements. Is there a function f : X → X such that the composition f ^n-1 is constant, but f ^n-2 is not constant?

A2.  Given n we define a_i,j as follows. For i, j = 1, 2, ... , n, a_i,j = 1 for j = i, and 0 for j ≠ i. For i = 1, 2, ... , n, j = n+1, ... , 2n, a_i,j = -1/n. Show that for any permutation p of (1, 2, ... , 2n) we have ∑_i=1ⁿ |∑_k=1ⁿ a_i,p(k) | ≥ n/2.

A3.  W is a regular octahedron with center O. P is a plane through the center O. K(O, r₁) and K(O, r₂) are circles center O and radii r₁, r₂ such that K(O, r₁) ⊆ P∩W ⊆ K(O, r₂). Show that r₁/r₂ ≤ (√3)/2.

B1.  We throw a coin n times and record the results as the sequence α₁, α₂, ... , α_n, using 1 for head, 2 for tail. Let β_j = α₁ + α₂ + ... + α_j and let p(n) be the probability that the sequence β₁, β₂, ... , β_n includes the value n. Find p(n) in terms of p(n-1) and p(n-2).

B2.  Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon with side 1. Show that no vertex of the hexagon is covered by two or more disks.

B3.  There are 1025 cities, P₁, ... , P₁₀₂₅ and ten airlines A₁, ... , A₁₀, which connect some of the cities. Given any two cities there is at least one airline which has a direct flight between them. Show that there is an airline which can offer a round trip with an odd number of flights.