Difference between revisions of "1978 IMO Problems"
| Line 23: | Line 23: | ||
[[1978 IMO Problems/Problem 4|Solution]] | [[1978 IMO Problems/Problem 4|Solution]] | ||
| − | ==Problem 5== | + | ===Problem 5=== |
Let <math>f</math> be an injective function from <math>{1,2,3,\ldots}</math> in itself. Prove that for any <math>n</math> we have: <math>\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.</math> | Let <math>f</math> be an injective function from <math>{1,2,3,\ldots}</math> in itself. Prove that for any <math>n</math> we have: <math>\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.</math> | ||
[[1978 IMO Problems/Problem 5|Solution]] | [[1978 IMO Problems/Problem 5|Solution]] | ||
| − | ==Problem 6== | + | ===Problem 6=== |
An international society has its members from six different countries. The list of members contain <math>1978</math> names, numbered <math>1, 2, \dots, 1978</math>. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country. | An international society has its members from six different countries. The list of members contain <math>1978</math> names, numbered <math>1, 2, \dots, 1978</math>. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country. | ||
Latest revision as of 15:56, 29 January 2021
Problems of the 20th IMO 1978 in Romania.
Contents
Day I
Problem 1
Let 
and 
be positive integers such that 
. In their decimal representations, the last three digits of 
are equal, respectively, to the last three digits of 
. Find and such that 
has its least value.
Problem 2
We consider a fixed point 
in the interior of a fixed sphere
We construct three segments 
, perpendicular two by two
with the vertexes 
on the sphere We consider the vertex 
which is opposite to in the parallelepiped (with right angles) with 
as edges Find the locus of the point when take all the positions compatible with our problem.
Problem 3
Let 
a sequence with all its terms positive The 
positive integer which doesn't belong to the sequence is 
Find 
Day II
Problem 4
In a triangle 
we have 
A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides 
in the points 
respectively 
Prove that the midpoint of 
is the center of the inscribed circle of the triangle 
Problem 5
Let 
be an injective function from 
in itself. Prove that for any we have: 
Problem 6
An international society has its members from six different countries. The list of members contain 
names, numbered 
. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.
- 1978 IMO
- IMO 1978 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
| 1978 IMO (Problems) • Resources | ||
| Preceded by 1977 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1979 IMO |
| All IMO Problems and Solutions | ||
