37th IMO 1996 shortlist

37th IMO 1996 shortlisted problems


1. x, y, z are positive real numbers with product 1. Show that xy/(x⁵ + xy + y⁵) + yz/(y⁵ + yz + z⁵) + zx/(z⁵ + zx + x⁵) ≤ 1. When does equality occur?
2. x₁ ≥ x₂ ≥ ... ≥ x_n are real numbers such that x₁^k + x₂^k + ... + x_n^k ≥ 0 for all positive integers k. Let d = max{ \|x₁\|, ... , \|x_n\| }. Show that x₁ = d and that (x - x₁)(x - x₂) ... (x - x_n) ≥ xⁿ - dⁿ for all real x > d.
3. Given a > 2, define the sequence a₀, a₁, a₂, ... by a₀ = 1, a₁ = a, a_n+2 = a_n+1(a_n+1²/a_n² - 2). Show that 1/a₀ + 1/a₁ + 1/a₂ + ... + 1/a_n < 2 + a - (a² - 4)^1/2.
4. Show that the polynomial xⁿ - a₁x^n-1 - ... - a_n = 0, where a_i are non-negative reals, not all zero, has just one positive real root. Let this root be k. Put s = a₁ + a₂ + ... + a_n and s' = a₁ + 2a₂ + 3a₃ + ... + na_n. Show that s^s ≤ k^s'.
5. The real polynomial p(x) = ax³ + bx² + cx + d is such that \|p(x)\| ≤ 1 for all x such that \|x\| ≤ 1. Show that \|a\| + \|b\| + \|c\| + \|d\| ≤ 7.
6. Show that there are polynomials p(x), q(x) with integer coefficients such that p(x) (x + 1)²ⁿ + q(x) (x²ⁿ + 1) = k, for some positive integer k. Find the smallest such k (for each n).
7. f is a real-valued function on the reals such that \| f(x) \| <= 1 and f(x + 13/42) + f(x) = f(x + 1/6) + f(x + 1/7) for all x. Show that there is a real number c > 0 such that f(x + c) = f(x) for all x.
9. The sequence a₁, a₂, a₃, ... is defined by a₁ = 0 and a_4n = a_2n + 1, a_4n+1 = a_2n - 1, a_4n+2 = a_2n+1 - 1, a_4n+3 = a_2n+1 + 1. Find the maximum and minimum values of a_n for n = 1, 2, ... , 1996 and the values of n at which they are attained. How many terms a_n for n = 1, 2, ... , 1996 are 0?
10. ABC is a triangle with orthocenter H. P is a point on the circumcircle (distinct from the vertices). BE is an altitude. Q is the intersection of the line through A parallel to PB and the line through B parallel to AP. R is the intersection of the line through A parallel to PC and the line through C parallel to PA. The lines HR and AQ meet at X. Show that EX is parallel to AP.
12. ABC is an acute-angled triangle with BC longer than AC. O is the circumcenter and H is the orthocenter. CF is an altitude. The line through F perpendicular to OF meets the side AC at P. Show that ∠FHP = ∠A.
13. P is a point inside the equilateral triangle ABC. The lines AP, BP, CP meet the opposite sides at A', B', C' respectively. Show that A'B'·B'C'·C'A' ≥ A'B·B'C·C'A.
15. One rectangle has sides a, b and another has sides c, d, where a < c ≤ d < b and ab < cd. Show that the rectangle with smaller area can be placed inside the other rectangle iff (b² - a²)² <= (bc - ad)² + (bd - ac)².
16. ABC is an acute-angled triangle with circumcenter O and circumradius R. The line AO meets the circumcircle of BOC again at A'. B' and C' are defined similarly. Show that OA'.OB'.OC' ≥ 8R³. When does equality occur?
17. ABCD is a convex quadrilateral. R_A is the circumradius of the three vertices other than A. R_B, R_C and R_D are defined similarly. Show that R_A + R_C > R_B + R_D iff ∠A + ∠C > ∠B + ∠D.
18. On the plane are given a point O and a polygon U (not necessarily convex). Let P denote the perimeter of U, D the sum of the distances from O to the vertices of U, and H the sum of the distances from O to the lines containing the sides of U. Prove that D² - H² ≥ P²/4.
19. We start with the numbers a, b, c, d. We then replace them with a' = a - b, b' = b - c, c' = c - d, d' = d - a. We carry out this process 1996 times. Is it possible to end up with numbers A, B, C, D such that \|BC - AD\|, \|AC - BD\|, \|AB - CD\| are all primes?
21. A finite sequence of integers a₀, a₁, ... , a_n is called quadratic if \|a₁ - a₀\| = 1², \|a₂ - a₁\| = 2², ... , \|a_n - a_n-1\| = n². Show that any two integers h, k can be linked by a quadratic sequence (in other words for some n we can find a quadratic sequence a_i with a₀ = h, a_n = k). Find the shortest quadratic sequence linking 0 and 1996.
22. Find all positive integers m and n such that [m²/n] + [n²/m] = [m/n + n/m] + mn.
23. Let X be the set of non-negative integers. Find all bijections f on X such that f(3mn + m + n) = 4 f(m) f(n) + f(m) + f(n) for all m, n.
24. An (n-1) x (n-1) square is divided into (n-1)² unit squares. Each of the n² vertices is colored red or blue. Find the number of possible colorings such that every small square has two vertices of each color.
26. k, m, n are positive integers such that 1 < n < m < k+2. What is the largest subset of {1, 2, 3, ... , k} which has no n distinct elements with sum m.
27. Do there exist two disjoint infinite sets A and B of points in the plane such that: (1) no three points in the union of A and B are collinear; (2) the distance between any two points in the union of A and B is at least 1; (3) given any three points in B, there is a point of A in the triangle formed by the three points; (4) given any three points in A, there is a point of B in the triangle formed by the three points? A triangle means all points in or on the triangle (the vertices, all points on the sides of the triangle and all points in the interior of the triangle).
28. A finite number of beans are placed in an infinite row of baskets. A move is to select a basket with at least two beans and to remove two beans, putting one in the basket on the left and the other in the basket on the right. Moves are made until no basket contains more than one bean. Show that the number of moves and the final configuration depend only on the initial configuration.
30. X is a finite set and f, g are bijections on X such that for any point x in X either f( f(x) ) = g( g(x) ) or f( g(x) ) = g( f(x) ) or both. Show that for any x, f( f( f(x) ) ) = g( g( f(x) ) ) iff f( f( g(x) ) ) = g( g( g(x) ) ).
Note: problems 8, 11, 14, 20, 25, 29 were used in the Olympiad and are not shown here.