Difference between revisions of "1983 IMO Problems"
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== Problem 6 == | == Problem 6 == | ||
Let <math> a</math>, <math> b</math> and <math> c</math> be the lengths of the sides of a triangle. Prove that | Let <math> a</math>, <math> b</math> and <math> c</math> be the lengths of the sides of a triangle. Prove that | ||
| − | + | <cmath>a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.</cmath> | |
| − | |||
Determine when equality occurs. | Determine when equality occurs. | ||
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== See Also == | == See Also == | ||
{{IMO box|year=1983|before=[[1982 IMO]]|after=[[1984 IMO]]}} | {{IMO box|year=1983|before=[[1982 IMO]]|after=[[1984 IMO]]}} | ||
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Latest revision as of 13:47, 14 October 2024
Problem 1
Find all functions 
defined on the set of positive reals which take positive real values and satisfy: 
for all 
; and 
as 
.
Problem 2
Let 
be one of the two distinct points of intersection of two unequal coplanar circles 
and 
with centers 
and 
respectively. One of the common tangents to the circles touches at 
and at 
, while the other touches at 
and at 
. Let 
be the midpoint of 
and 
the midpoint of 
. Prove that 
.
Problem 3
Let 
and 
be positive integers, no two of which have a common divisor greater than 
. Show that 
is the largest integer which cannot be expressed in the form 
, where 
are non-negative integers.
Problem 4
Let 
be an equilateral triangle and 
the set of all points contained in the three segments 
, 
, and 
(including , 
, and 
). Determine whether, for every partition of into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.
Problem 5
Is it possible to choose 
distinct positive integers, all less than or equal to 
, no three of which are consecutive terms of an arithmetic progression?
Problem 6
Let 
, 
and be the lengths of the sides of a triangle. Prove that
![\[a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.\]](http://latex.artofproblemsolving.com/5/f/c/5fcaedd680c4342fdcb088fbfe889483ad8250ce.png)
Determine when equality occurs.
See Also
| 1983 IMO (Problems) • Resources | ||
| Preceded by 1982 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1984 IMO |
| All IMO Problems and Solutions | ||
