1976 IMO Problems
Problems of the 18th IMO 1976 in Austria.
Contents
Day 1
Problem 1
In a convex quadrilateral (in the plane) with the area of the sum of two opposite sides and a diagonal is
. Determine all the possible values that the other diagonal can have.
[1976 IMO Problems/Problem 1|Solution]
Problem 2
Let and
for
Prove that for any positive integer n the roots of the equation
are all real and distinct.
[1976 IMO Problems/Problem 2|Solution]
Problem 3
A box whose shape is a parallelepiped can be completely filled with cubes of side If we put in it the maximum possible number of cubes, each of volume
, with the sides parallel to those of the box, then exactly
percent from the volume of the box is occupied. Determine the possible dimensions of the box.
[1976 IMO Problems/Problem 3|Solution]
Day 2
Problem 4
Find the largest number obtainable as the product of positive integers whose sum is .
[1976 IMO Problems/Problem 4|Solution]
Problem 5
Let a set of equations be given,
with coefficients
satisfying
,
, or
for all
, and
. Prove that if
, there exists a solution
of this system such that all
(
) are integers satisfying
and
for at least one value of
.
[1976 IMO Problems/Problem 5|Solution]
Problem 6
For all positive integral ,
,
, and
. Prove that
where
is the integral part of
.
[1976 IMO Problems/Problem 6|Solution]
- 1976 IMO
- IMO 1976 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1976 IMO (Problems) • Resources | ||
Preceded by 1975 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1977 IMO |
All IMO Problems and Solutions |