1988 IMO Problems
Problems of the 1988 IMO.
Day I
Problem 1
Consider 2 concentric circle radii 
and 
(
) with centre 
Fix 
on the small circle and consider the variable chord 
of the small circle. Points 
and 
lie on the large circle; 
are collinear and 
is perpendicular to 
i.) For which values of 
is the sum 
extremal?
ii.) What are the possible positions of the midpoints 
of 
and 
of 
as varies?
Problem 2
Let 
be an even positive integer. Let 
be sets having elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly 
zeros?
Problem 3
A function 
defined on the positive integers (and taking positive integers values) is given by:
for all positive integers 
Determine with proof the number of positive integers 
for which 
Day II
Problem 4
Show that the solution set of the inequality
![\[\sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4}\]](http://latex.artofproblemsolving.com/a/c/7/ac7df0a9a7a36456ad31615aced9a17c3d870953.png)
is a union of disjoint intervals, the sum of whose length is 1988.
Problem 5
In a right-angled triangle 
let 
be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles 
intersect the sides 
at the points 
respectively. If 
and 
denote the areas of triangles and 
respectively, show that
![\[\frac {E}{E_1} \geq 2.\]](http://latex.artofproblemsolving.com/6/7/8/67878b24ad731c2b6823a11a88ba4f7a9348aa49.png)
Problem 6
Let 
and 
be two positive integers such that 
divides 
. Show that 
is a perfect square.
- 1988 IMO
- IMO 1988 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
| 1988 IMO (Problems) • Resources | ||
| Preceded by 1987 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1989 IMO |
| All IMO Problems and Solutions | ||
