50th Polish Mathematical Olympiad Problems 1999

50th Polish Mathematical Olympiad Problems 1999

A1.  D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the side AC such that AE/EC = BD/(AD-BC). Show that AD > BE.

A2.  Given 101 distinct non-negative integers less than 5050 show that one choose four a, b, c, d such that a + b - c - d is a multiple of 5050.

A3.  Show that one can find 50 distinct positive integers such that the sum of each number and its digits is the same.

B1.  For which n do the equations have a solution in integers:

x₁² + x₂² + 50 = 16x₁ + 12x₂

x₂² + x₃² + 50 = 16x₂ + 12x₃

...

x_n-1² + x_n² + 50 = 16x_n-1 + 12x_n

x_n² + x₁² + 50 = 16x_n + 12x₁

B2.  Show that ∑_1≤i<j≤n (|a_i-a_j| + |b_i-b_j|) ≤ ∑_1≤i<j≤n |a_i-b_j| for all integers a_i, b_i.

B3.  The convex hexagon ABCDEF satisfies ∠A + ∠C + ∠E = 360^o and AB·CD·EF = BC·DE·FA. Show that AB·FD·EC = BF·DE·CA.