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 ak is a multiple of k.
2. The rational sequence a0, a1, a2, ... satisfies an+1 = 2an2 - 2an + 1. Find all a0 for which there are four distinct integers r, s, t, u such that ar - as = at - au.
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 rA, rB, rC respectively. Show that rA + rB + rC ≥ 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 xn+1 = yn + 1/zn, yn+1 = zn + 1/xn, zn+1 = xn + 1/yn. Show that the sequences are all unbounded.
7. If N = 2n > 1 and k > 3 is odd, show that kN - 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.
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