## Shortlisted Problems for the 37^{th} 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 a_{1} ³ a_{2} ³ ...
a_{n} ³ be a real numbers such that for all integers *k* > 0,

a^{k}_{1} + a^{k}_{2} + ... + a^{k}_{n}
³ 0.
Let *p* = max {|a_{1}| , ... , |a_{n}|}. Prove that *p* = *a*_{1} and that

*(x - a*_{1}) (x - a_{2}) ... (x - a_{n}) ³ x^{n} -
a^{n}_{1}
for all *x* > *a*_{1}

**Problem 3.**

Let a > 2 be given, and define recursively:
a_{0} = 1, a_{1} = a,
Show that for all integers *k* > 0, we have
**Problem 4. **

Let a_{1}, a_{2}, ... , a_{n} be non-negative numbers, not all zero.

- (a)Prove that
*x*^{n} - a_{1}x^{n-1} - ... - a_{n-1}x - a_{n} = 0
has precisely one positive real root.
- (b)Let A = S
^{n}_{j=1} a_{j}, and
S^{n}_{j=1} ja_{j} and let R be the positive real
root of the equation in (a). Prove that *A*^{A} £ *R*^{B}.

**Problem 5.**

Let **P** be the real polynomial, **P**(x) = *ax*^{3} + bx^{3} + 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)(x^{n} + 1)
for some polynomials *f*, determine *k*_{0} as a function of *n*.

**Problem 7.**

Let *f* be a function from the set of real numbers Â 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*.)

- (a) Determine the maximun and the minimun value of
*a(n)* over *n* £ 1996
and find *n* £ 1996 for which these extreme values are attained.
- (b) How many terms
*a(n), n * £ 1996, are equal to 0?

**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 Ð*FHP* =
Ð*BAC*.

**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 *A*_{1}, B_{1}, C_{1} respectively. Prove that

*A*_{1}B_{1} . B_{1}C_{1} . C_{1}A_{1}
³
A_{1}B . B_{1}C . C_{1}A

**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

*(b*^{2} - a^{2})^{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 8R*^{3}.
When does equality hold?

**Problem 14.**

Let *ABCD* be a convex quadrilateral, and let *R*_{A}, R_{B}, R_{C}, R_{D}
denote the circumradio of the triangles *DAB, ABC, BCD, CDA* respectively. Prove that *R*_{A} +
R_{C} > R_{B} + R_{D} 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 *D*^{2} - H^{2}
³ P^{2}/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 *a*_{0}, a_{1}, ... , a_{n} is called quadratic if for
each *i* in the set {1, 2, ... , n} we have the equality *|a*_{i} - a_{i-1}| = i^{2}.

- (a) Prove that for any two integers
*b* and *c*, there exist anatural number *n* and a quadratic
sequence with *a*_{0} = b and *a*_{n} = c.
- (b) Find the smallest natural number
*n* for which there exists a quadratic sequence with
*a*_{0} = 0 and *a*_{n} = 1996.

**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 *N*_{0} refer to set of non-negative integers. Fiind a bijective function *f* from
*N*_{0} into *N*_{0} such that for all *m, n *+
*N*_{0},

*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 *n*^{2} 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:

- (a) No three points in A È B
are collinear, and the distance brtween any two point in
A È B is at least 1.
- (b) There is a point ofA in any triangle whose vertices are in B,
and there is a point of B in any triangle whose vertice are in A.

**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))}*
and

*T = {w Î U : f(g(w)) = g(f(w))}*,
and suppose that *U* = *S* È *T*. Prove that, for *m*
Î*U, f(w)ÎS* if and only if *g(w)
Î**S*.