Shortlisted Problems for the 37th IMO

Bombay (India), 1996

Original HTML-version by Enrique Valeriano Cuba

Problem 1.
Let a, b and c be positive real numbers such that abc = 1. Prove that:

When does equality hold?

Problem 2.
Let a1 ³ a2 ³ ... an ³ be a real numbers such that for all integers k > 0,
ak1 + ak2 + ... + akn ³ 0.
Let p = max {|a1| , ... , |an|}. Prove that p = a1 and that
(x - a1) (x - a2) ... (x - an) ³ xn - an1
for all x > a1

Problem 3.
Let a > 2 be given, and define recursively:
a0 = 1, a1 = a,
Show that for all integers k > 0, we have
Problem 4.
Let a1, a2, ... , an be non-negative numbers, not all zero.

Problem 5.
Let P be the real polynomial, P(x) = ax3 + bx3 + cx + d. Prove that if |P(x)| £ 1 for all x such that |x| £ 1, then
|a| + |b| + |c| + |d| £ 7.

Problem 6.
Let n be an even integer. Prove that there exist a positive integer k such that
k = f(x)(x + 1)n + g(x)(xn + 1)
for some polynomials f, determine k0 as a function of n.

Problem 7.
Let f be a function from the set of real numbers  into itself such that for all x Î Â, we have |f(x)| £ 1 and
Prove that f is a periodic function (that is, there exist a non-zero real number c, such that f(x + c) = f(x) for all x Î Â).

Problem 8.
Let the sequence a(n), n = 1, 2, 3, ... , be generated as follows: a(1) = 0, and for n > 1,
a(n) = a([n/2]) + (-1)n(n + 1)/2.
(Here [t] is the greatest integer less than or equal to t.)

Problem 9.
Let triangle ABC have orthocentre H, and let P be a point on its circumcircle, distinct from A, B, C. Let E be the foot of the altitude BH, let PAQB and PARC be parallelograms, and let AQ meet HR in X. Prove that EX is paralle to AP.

Problem 10.
Let ABC be an acute-angled triangle with |BC| > |CA|, and let O be the circumcentre, H its orthocentre, and F the foot of its altitude CH. Let the perpendicular to OF at F meet the side CA at P. Prove that ÐFHP = ÐBAC.

Problem 11.
Let ABC be equilateral, and let P be a point in its interior. Let the lines AP, BP, CP meet the sides BC, CA, AB in the points A1, B1, C1 respectively. Prove that
A1B1 . B1C1 . C1A1 ³ A1B . B1C . C1A

Problem 12.
Let the sides of two rectangles be {a, b} and {c, d} respectively, with a < c £ d < b and ab < cd. Prove that the firt rectangle can be placed within the second one if and only if
(b2 - a2)2 £ (bc - ad)2 + (bd + ac) 2.
Problem 13.
Let ABC be an acute-angled triangle with circumcentre O and circumradius R. Let AO meet the circle BOC again in A', let BO meet the circle COA again in B' and let CO meet the circle AOB again in C'. Prove that
OA' . OB' . OC' ³ 8R3.
When does equality hold?

Problem 14.
Let ABCD be a convex quadrilateral, and let RA, RB, RC, RD denote the circumradio of the triangles DAB, ABC, BCD, CDA respectively. Prove that RA + RC > RB + RD if and only if ÐA + ÐC > ÐB + ÐD.

Problem 15.
On the plane are given a point O and a polygon Á (not necessarily convex). Let P denote the perimeter of Á, D the sum of the distances from O to the lines containing the sides of Á. Prove that D2 - H2 ³ P2/4.

Problem 16.
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and the next number on the circle, moving in clockwise direction; that is, the numbers a, b, c, d are replaced by a - b, b - c, c - d, d - a. Is it posible after 1996 such steps to have numbers a, b, c, d such that the numbers |bc - ad|, |ac - bd|, |ab - cd| are primes?

Problem 17.
A finite sequence of integers a0, a1, ... , an is called quadratic if for each i in the set {1, 2, ... , n} we have the equality |ai - ai-1| = i2.

Problem 18.
Find all positive integers a and b for which
where, as usual, [t] refers to greatest integer which is less than or equal to t.

Problem 19.
Let N0 refer to set of non-negative integers. Fiind a bijective function f from N0 into N0 such that for all m, n + N0,
f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).

Problem 20.
A square (n - 1) X (n - 1) is divided into (n - 1)2 unit squares in the usual manner. Each of the n2 vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemes are regarded as diferent if at least one vertex is coloured differently in the two schemes.)

Problem 21.
Let k, m, n be integers such that 1 < n £ m - 1 £ k. Determine the maximun size of a subset S of the set {1, 2, 3, ... , k - 1, k} such that no n distinct elements of S add up to m.

Problem 22.
Determine whether or not there exits two disjoint infinite sets A and B of points in the plane satisfying the following conditions:

Problem 23.
A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one bean is chosen. Two beans are taken from ths square; one of them is placed on the square inmediately to the left while the other is placed on the square inmediately to the right of the chosen square. The sequence terminates if at some point there is at most one bean on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

Problem 24.
Let U be a finite set and f, g be bijective function from U onto itself. Let
S = {w Î U : f(f(w)) = g(g(w))}
T = {w Î U : f(g(w)) = g(f(w))},
and suppose that U = S È T. Prove that, for m ÎU, f(w)ÎS if and only if g(w) ÎS.