# New Year Challenge: Find Real Number Triples

• MHB
• anemone
In summary, to find all triples $(a,\,b,\,c)$ of real numbers that satisfy the given system, one approach would be to simplify the equations by multiplying both sides by the common denominator $abc$ and then rearranging them to isolate one variable in terms of the other two. This will result in a quadratic equation, which can be solved to find the possible values for that variable. These values can then be substituted back into the original equations to find the solutions to the system, which can be checked for validity.
anemone
Gold Member
MHB
POTW Director
Find all triples $(a,\,b,\,c)$ of real numbers such that the following system holds:

$a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\\a^2+b^2+c^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}$

To the original poster,

Thank you for your interesting question. my approach to solving this system would be to first simplify the equations by multiplying both sides by the common denominator $abc$. This would give us the following system:

$a^2bc+ab^2c+abc^2=1+bc+ac\\a^2b^2c^2+a^2bc^2+a^2b^2c=1+a^2c^2+b^2c^2$

Next, I would rearrange the equations to isolate one variable in terms of the other two. For example, in the first equation, we can isolate $c$ as follows:

$c=\dfrac{1+ab-a^2b}{ab+1-a^2}$

We can do the same for the other two variables in the second equation. This will give us three equations in terms of $a$, $b$, and $c$:

$c=\dfrac{1+ab-a^2b}{ab+1-a^2}\\a=\dfrac{1+bc-b^2c}{bc+1-b^2}\\b=\dfrac{1+ac-a^2c}{ac+1-a^2}$

From here, we can substitute these equations into each other and solve for one variable in terms of the other two. This will give us a quadratic equation, which can be solved to find the possible values for that variable. We can then substitute these values back into the original equations to find the corresponding values for the other two variables.

After solving for all three variables, we can check if the solutions satisfy the given equations. If they do, then they are valid solutions to the system. If not, then they are extraneous solutions and can be discarded.

I hope this helps in solving the system. Please let me know if you have any further questions. Good luck!

## 1. What is the purpose of the "New Year Challenge: Find Real Number Triples"?

The purpose of the challenge is to find three real numbers that satisfy a given equation. This challenge aims to test problem-solving skills and mathematical reasoning.

## 2. How can I participate in the "New Year Challenge: Find Real Number Triples"?

To participate in the challenge, you can visit the designated website or platform where it is being hosted. The challenge may require you to submit your solution or code, depending on the specific instructions provided.

## 3. What are the criteria for determining the winners of the "New Year Challenge: Find Real Number Triples"?

The criteria for determining the winners may vary depending on the challenge. It is usually based on the accuracy and efficiency of the solutions submitted. Some challenges may also consider creativity and originality.

## 4. Do I need any specific knowledge or skills to participate in the "New Year Challenge: Find Real Number Triples"?

Basic knowledge of mathematics and problem-solving skills are essential to participate in the challenge. Some challenges may also require coding skills, depending on the given instructions.

## 5. Are there any prizes or rewards for participating in the "New Year Challenge: Find Real Number Triples"?

Prizes and rewards may vary depending on the specific challenge. Some challenges may offer cash prizes, certificates, or other forms of recognition to the winners. It is best to check the details of the challenge for more information about the prizes.

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