h*****y
发帖数: 66 | 1 问这里的高手们一个问题。万一是个简单的问题，也请高手们不要笑话
find the smallest possible value of a+b+c where a, b, and c are different
positive integers that satisfy the following equation:
1/a + 1/b + 1/c = 7/10
请问用什么方法呢？ | h*****y
发帖数: 66 | 2 谢谢，终于有人指点了。
这个问题要求，a,b,c是三个不同的正整数。 这个条件如何加进去呢？ 再次谢谢 | n*******l
发帖数: 2911 | 3 If a,b,c can be any positive number, then
7/10 = 1/a +1/b+1/c >= 3 (abc)^{-1/3}.
(abc)^{1/3}>= 30/7.
and
a+b+c >= 3(abc)^{1/3} >= 90/7.
Equality holds when a=b=c=30/7.
Since you require a,b,c to be positive integers, I
would just stare at 1/a+1/b+1/c = 7/10 and
conclude that {a,b,c} = {10,10,2}. Then
a+b+c=22.
【在 h*****y 的大作中提到】
: 问这里的高手们一个问题。万一是个简单的问题，也请高手们不要笑话
: find the smallest possible value of a+b+c where a, b, and c are different
: positive integers that satisfy the following equation:
: 1/a + 1/b + 1/c = 7/10
: 请问用什么方法呢？
| n*******e
发帖数: 2213 | 4 1/2 + 1/5 = 0.7 ->2 and 5 do not appear simutaneously.
5 has to appear.
therefore, we are left with [3,4,5], [3,5,6], and [4,5,6], so and so on.
among them, [3,5,6] works.
【在 h*****y 的大作中提到】
: 问这里的高手们一个问题。万一是个简单的问题，也请高手们不要笑话
: find the smallest possible value of a+b+c where a, b, and c are different
: positive integers that satisfy the following equation:
: 1/a + 1/b + 1/c = 7/10
: 请问用什么方法呢？
| h*****y
发帖数: 66 | 5 多谢大伙的帮忙！
【在 n*******e 的大作中提到】
: 1/2 + 1/5 = 0.7 ->2 and 5 do not appear simutaneously.
: 5 has to appear.
: therefore, we are left with [3,4,5], [3,5,6], and [4,5,6], so and so on.
: among them, [3,5,6] works.
| n*******l
发帖数: 2911 | 6 有道理，应该从最小的开始一个个尝试。对于一般情形，看来应该编个程序穷举。
【在 n*******e 的大作中提到】
: 1/2 + 1/5 = 0.7 ->2 and 5 do not appear simutaneously.
: 5 has to appear.
: therefore, we are left with [3,4,5], [3,5,6], and [4,5,6], so and so on.
: among them, [3,5,6] works.
| N******t
发帖数: 11 | 7 1/10+1/10+1/2 The only one? | s*****u
发帖数: 164 | |
| N******t
发帖数: 11 | | k*r
发帖数: 19 | 10 By Cauchy's Inequality, a + b + c > 3^2 *(10/7) = 12.8xxx
a + b + c >= 13.
Therefore the smallest triples are:
a = 4
b = 4
c = 5
【在 h*****y 的大作中提到】
: 问这里的高手们一个问题。万一是个简单的问题，也请高手们不要笑话
: find the smallest possible value of a+b+c where a, b, and c are different
: positive integers that satisfy the following equation:
: 1/a + 1/b + 1/c = 7/10
: 请问用什么方法呢？
| z*********g
发帖数: 37 | 11 The problem requires distinct positive integers! 3, 5, 6 is the answer. |
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