Good question!

Find all solutions in integers, of the equation 1m+n+k=1k+1m+1n\frac{1}{m+n+k} = \frac{1}{k} + \frac{1}{m} + \frac{1}{n}

Note by Kiran Patel
7 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

1m+n+k1k=1m+1n\large \frac{1}{m+n+k}-\frac{1}{k}=\frac{1}{m}+\frac{1}{n}

(m+n)k(m+n+k)=m+nmn\large \frac{-(m+n)}{k(m+n+k)}=\frac{m+n}{mn}

mn=k(m+n+k)-mn=k(m+n+k)

k2+k(m+n)+mn=0k^2+k(m+n)+mn=0

Solving the quadratic gives us:

k=nk=-n or k=mk=-m

An analogous argument gives us that m=nm=-n is also a solution.

My doubt is that whether we can have m,n,k=0m,n,k=0 as solutions or not?

Although the equation actually satisfies, I am not sure.

Aditya Parson - 7 years, 11 months ago

Log in to reply

If any of the variables were zero, then the fractions in the problem would be undefined. This the solutions are in the form (a, a, -a), where a is a real number not equal to zero.

Bob Krueger - 7 years, 11 months ago

Log in to reply

But, what if we simplify the expression such that the variables are only in the numerator?

Aditya Parson - 7 years, 11 months ago

Log in to reply

@Aditya Parson We cannot clear out any factor or expression unless we know about the values it takes. Since we are given the expression, we will have to keep in mind that it is the base expression and solve accordingly.

Nishant Sharma - 7 years, 11 months ago

@Aditya Parson It doesn't change the fact that the original expression would be undefined if the variables were 0.

Michael Tang - 7 years, 11 months ago
×

Problem Loading...

Note Loading...

Set Loading...