Difference between revisions of "1976 IMO Problems"
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<cmath>3\log_2[u_n]=2^n-(-1)^n,</cmath> | <cmath>3\log_2[u_n]=2^n-(-1)^n,</cmath> | ||
where <math>[x]</math> is the integral part of <math>x</math>. | where <math>[x]</math> is the integral part of <math>x</math>. | ||
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+ | * [[1976 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1976 IMO 1976 Problems on the Resources page] * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1976|before=[[1975 IMO]]|after=[[1977 IMO]]}} |
Revision as of 15:22, 29 January 2021
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.
Problem 2
Let and for Prove that for any positive integer n the roots of the equation are all real and distinct.
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.
Day 2
Problem 4
Find the largest number obtainable as the product of positive integers whose sum is .
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 .
Problem 6
For all positive integral , , , and . Prove that where is the integral part of .
- 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 |