14th Austrian-Polish 1991


1. Show that there are infinitely many integers m > 1 such that mC2 = 3(nC4) for some integer n > 3, where aCb denotes the binomial coefficient a!/(b! (a-b)!). Find all such m.
2. Find all real solutions to: (x² - 6x + 13)y = 20 (y² - 6y + 13)z = 20 (z² - 6z + 13)x = 20.
3. A₁, A₂ are distinct points in the plane. Find all points A₃ for which we can find n > 2 and points P₁, P₂, ... , P_n (not necessarily distinct) such that the midpoints of P₁P₂, P₂P₃, P₃P₄, ... , P_n-1P_n, P_nP₁ are A₁, A₂, A₃, A₁, A₂, A₃, A₁, ... respectively.
4. The polynomial p(x) is non-negative for all 0 ≤ x ≤ 1. Show that there are polynomials q(x), r(x), s(x) which are non-negative for all x such that p(x) = q(x) + x r(x) + (1 - x) s(x).
5. Show that if the real numbers x, y, z satisfy xyz = 1, then x² + y² + z² + xy + yz + zx ≥ 2(√x + √y + √z).
6. ABCD is a convex quadrilateral. P is a point inside ABCD such that PAB, PBC, PCD, PDA have equal area. Show that area ABC = area ADC, or area BCD = area BAD.
7. Find the maximum value of (x + x² + x³ + ... + x^2n-1)/(1 + xⁿ)² for positive real x and the values of x at which the maximum is achieved.
8. Find all solutions to ab = -1 mod c, bc = 1 mod a, ca = 1 mod b, such that a, b, c are all distinct positive integers and one of them is 19.
9. Let X be the set {1, 2, 3, ... , 2n}. g is a function X → X such that g(k) ≠ k and g(g(k)) = k for all k. How many functions f: X → X are there such that f(k) ≠ k and f(f(f(k))) = g(k) for all k?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.