42nd Polish Mathematical Olympiad Problems 1991

42nd Polish Mathematical Olympiad Problems 1991

A1.  Do there exist tetrahedra T₁, T₂ such that (1) vol T₁ > vol T₂, and (2) every face of T₂ has larger area than any face of T₁?


A2.  Let F(n) be the number of paths P₀, P₁, ... , P_n of length n that go from P₀ = (0,0) to a lattice point P_n on the line y = 0, such that each P_i is a lattice point and for each i < n, P_i and P_i+1 are adjacent lattice points a distance 1 apart. Show that F(n) = (2n)Cn.

A3.  N is a number of the form ∑_k=1⁶⁰ a_k k^kk, where each a_k = 1 or -1. Show that N cannot be a 5th power.

B1.  Let V be the set of all vectors (x,y) with integral coordinates. Find all real-valued functions f on V such that (a) f(v) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all perpendicular v, w ∈ V. (The vector (0,0) is considered to be perpendicular to any vector.)

B2.  k₁, k₂ are circles with different radii and centers K₁, K₂. Neither lies inside the other, and they do not touch or intersect. One pair of common tangents meet at A on K₁K₂, the other pair meet at B on K₁K₂. P is any point on k₁. Show that there is a diameter of K₂ with one endpoint on the line PA and the other on the line PB.

B3.  The real numbers x, y, z satisfy x² + y² + z² = 2. Show that x + y + z ≤ 2 + xyz. When do we have equality?