Difference between revisions of "2010 OIM Problems/Problem 4"
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The arithmetic, geometric and harmonic means of two different positive integers are integer numbers. Find the smallest possible value for the arithmetic mean. | The arithmetic, geometric and harmonic means of two different positive integers are integer numbers. Find the smallest possible value for the arithmetic mean. | ||
| − | '''Note:''' If <math>a</math> and <math>b</math> are positive numbers, their arithmetic, geometric and harmonic means are, respectively: <math>\frac{a+b}{2}</math>, <math>\sqrt{ab}</math>, and | + | '''Note:''' If <math>a</math> and <math>b</math> are positive numbers, their arithmetic, geometric and harmonic means are, respectively: <math>\frac{a+b}{2}</math>, <math>\sqrt{ab}</math>, and <math>\frac{2ab}{a+b}</math>. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 16:12, 14 December 2023
Problem
The arithmetic, geometric and harmonic means of two different positive integers are integer numbers. Find the smallest possible value for the arithmetic mean.
Note: If 
and 
are positive numbers, their arithmetic, geometric and harmonic means are, respectively: 
, 
, and 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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