1977 IMO Problems
Problems of the 19th IMO 1977 in Yugoslavia.
Problem 1
In the interior of a square 
we construct the equilateral triangles 
Prove that the midpoints of the four segments 
and the midpoints of the eight segments 
are the 12 vertices of a regular dodecagon.
Problem 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Problem 3
Let 
be a given number greater than 2. We consider the set 
of all the integers of the form 
with 
A number 
from is called indecomposable in if there are not two numbers 
and 
from so that 
Prove that there exist a number 
that can be expressed as the product of elements indecomposable in in more than one way. (Expressions which differ only in order of the elements of will be considered the same.)
Problem 4
Let 
be given reals. We consider the function defined by![\[f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x).\]](http://latex.artofproblemsolving.com/7/9/2/792a9f8a71c29da6bd80b3305bc44e6b8ce0bdca.png)
Prove that if for any real number 
we have 
then 
and 
Problem 5
Let 
be two natural numbers. When we divide 
by 
, we the the remainder 
and the quotient 
Determine all pairs 
for which 
Problem 6
Let 
be the set of positive integers. Let 
be a function defined on , which satisfies the inequality 
for all 
. Prove that for any we have 
- 1977 IMO
- IMO 1977 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
| 1977 IMO (Problems) • Resources | ||
| Preceded by 1976 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1978 IMO |
| All IMO Problems and Solutions | ||
