Difference between revisions of "1990 APMO Problems"
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== Problem 3 == | == Problem 3 == | ||
| − | Consider all the triangles <math>ABC</math> which have a fixed base <math>AB</math> and whose altitude from <math>C</math> is a constant <math>h</math>. For which triangles is the product of its altitudes a | + | Consider all the triangles <math>ABC</math> which have a fixed base <math>AB</math> and whose altitude from <math>C</math> is a constant <math>h</math>. For which triangles is the product of its altitudes a maximum? |
[[1990 APMO Problems/Problem 3|Solution]] | [[1990 APMO Problems/Problem 3|Solution]] | ||
| Line 40: | Line 40: | ||
[[1990 APMO Problems/Problem 5|Solution]] | [[1990 APMO Problems/Problem 5|Solution]] | ||
| − | == See | + | == See Also == |
* [[Asian Pacific Mathematics Olympiad]] | * [[Asian Pacific Mathematics Olympiad]] | ||
* [[APMO Problems and Solutions]] | * [[APMO Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
Latest revision as of 22:07, 12 September 2016
Problem 1
Given triangle 
, let 
, 
, 
be the midpoints of 
, 
, 
respective and let 
be the centroid of the triangle. For each value of 
, how many non-similar triangles are there in which 
is a cyclic quadrilateral?
Problem 2
Let 
be positive real numbers, and let 
be the sum of the products of taken 
at a time. Show that
![\[S_kS_{n-k}\geq \binom{n}{k}^2 a_1a_2\cdots a_n\]](http://latex.artofproblemsolving.com/d/6/7/d673d6e323d4a70c89036829e7fab9b227278b49.png)
for 
Problem 3
Consider all the triangles which have a fixed base and whose altitude from 
is a constant 
. For which triangles is the product of its altitudes a maximum?
Problem 4
A set of 
persons is divided into non-intersecting subsets in such a way that
1. No one in a subset knows all the others in the subset,
2. Among any three persons in a subset, there are always at least two who do now know each other, and
3. For any two persons in a subset who do now know each other, there is exactly one person in the same subset knowing both of them.
(a) Prove that within each subset, every person has the same number of acquaintances.
(b) Determine the maximum possible number of subsets.
Note: It is understood that if a person 
knows person 
, then person will know person ; an acquaintance is someone who is known. Every person is assumed to know one's self.
Problem 5
Show that for every integer 
, there exists a convex hexagon which can be dissected into exactly 
congruent triangles.
