4th Austrian-Polish Mathematics Competition 1981 Problems

4th Austrian-Polish Mathematics Competition 1981 Problems

1.  Find the smallest n for which we can find 15 distinct elements a₁, a₂, ... , a₁₅ of {16, 17, ... , n} such that a_k is a multiple of k.

2.  The rational sequence a₀, a₁, a₂, ... satisfies a_n+1 = 2a_n² - 2a_n + 1. Find all a₀ for which there are four distinct integers r, s, t, u such that a_r - a_s = a_t - a_u.

3.  The diagram shows the incircle of ABC and three other circles inside the triangle, each touching the incircle and two sides of the triangle. The radius of the incircle is r and the radius of the circle nearest to A, B, C is r_A, r_B, r_C respectively. Show that r_A + r_B + r_C ≥ r, with equality iff ABC is equilateral. 

4.  n symbols are arranged in a circle. Each symbol is 0 or 1. A valid move is to change any 1 to 0 and to change the two adjacent symbols. For example one could change ... 01101 ... to ... 10001 ... . The initial configuration has just one 1. For which n can one obtain all 0s by a sequence of valid moves? 

5.  A quartic with rational coefficients has just one real root. Show that the root must be rational. 

6.  The real sequences x₁, x₂, x₃, ... , y₁, y₂, y₃, ... , z₁, z₂, z₃, ... satisfy x_n+1 = y_n + 1/z_n, y_n+1 = z_n + 1/x_n, z_n+1 = x_n + 1/y_n. Show that the sequences are all unbounded. 

7.  If N = 2ⁿ > 1 and k > 3 is odd, show that k^N - 1 has at least n+1 distinct prime factors. 

8.  A is a set of r parallel lines, B is a set of s parallel lines, and C is a set of t parallel lines. What is the smallest value of r + s + t such that the r + s + t lines divide the plane into at least 1982 regions?

9.  Let X be the closed interval [0, 1]. Let f: X → X be a function. Define f¹ = f, fⁿ⁺¹(x) = f( fⁿ(x) ). For some n we have |fⁿ(x) - fⁿ(y)| < |x - y| for all distinct x, y. Show that f has a unique fixed point.