# Problem Of The Week # 305 - Apr 10, 2018

• MHB
• Ackbach
In summary, the topic of the conversation was about the benefits of meditation and mindfulness. The benefits discussed included reduced stress, improved focus, and increased self-awareness. One person shared their personal experience with meditation, while another mentioned the scientific research supporting its effectiveness. They also talked about different techniques and resources for incorporating meditation into daily life. Overall, the conversation highlighted the many positive effects of practicing meditation and mindfulness.
Ackbach
Gold Member
MHB
Here is this week's POTW:

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Solve for $x, y, z$ (in terms of $a, r, s, t$):
\begin{align*}
yz&=a(y+z)+r\\
zx&=a(z+x)+s\\
xy&=a(x+y)+t.
\end{align*}
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Congratulations to castor28 for his correct solution, and an honorable mention to kiwi for a mostly correct solution, to this week's POTW, which was Problem 150 in the MAA Challenges. castor28's solution follows:

[sp]Each of the equations represents a hyperbola with center $(a,a)$. If we write $x=u+a$, $y=v+a$, $z=w+a$, the equations become:
\begin{align*} uv &= a^2 + t\\ uw &= a^2 + s\\ vw &= a^2 + r \end{align*}
Multiplying the equations together, we get:
$$(uvw)^2 = (a^2+r)(a^2+s)(a^2+t)$$
and we obtain:
\begin{align*} u &= \pm\sqrt{\frac{(a^2+s)(a^2+t)}{a^2+r}}\\ v &= \pm\sqrt{\frac{(a^2+r)(a^2+t)}{a^2+s}}\\ w &= \pm\sqrt{\frac{(a^2+r)(a^2+s)}{a^2+t}}\\ \end{align*}
Note that the signs are not independent. Once a sign is chosen for $u$, the signs of $v$ and $w$ are determined by the signs of $uv$ and $uw$; in general, the system has two solutions. This is also true if some unknowns are imaginary (if $(a^2+r)(a^2+s)(a^2+t)<0$).

We can then recover $x$, $y$, $z$ as $u+a$, $v+a$ and $w+a$.

If exactly one of $(a^2+r)$, $(a^2+s)$, $(a^2+t)$ is $0$, there is no solution. Indeed, if $a^2 + r = vw = 0$, then at least one of $uv$ or $uw$ must be $0$.

If at least two of $(a^2+r)$, $(a^2+s)$, $(a^2+t)$ are $0$, there are infinitely many solutions (at least two of the equations are the same). For example, if $a^2+r=a^2+s=0$, we can take $w=0$ and we are left with the equation $uv=a^2+t$, which has infinitely many solutions.[/sp]

## 1. What is the "Problem Of The Week #305 - Apr 10, 2018"?

The "Problem Of The Week #305 - Apr 10, 2018" is a weekly challenge or puzzle presented by a scientific organization or publication. It is usually designed to test critical thinking and problem-solving skills in a specific field of science.

## 2. Who can participate in the "Problem Of The Week #305 - Apr 10, 2018"?

Anyone with an interest in science and the ability to solve complex problems can participate in the "Problem Of The Week #305 - Apr 10, 2018". It is open to students, researchers, and professionals from all over the world.

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## 4. What is the purpose of the "Problem Of The Week #305 - Apr 10, 2018"?

The purpose of the "Problem Of The Week #305 - Apr 10, 2018" is to promote critical thinking and problem-solving skills in the scientific community. It also serves as an opportunity for scientists to engage and collaborate with each other in finding solutions to complex problems.

## 5. Are there any rewards for solving the "Problem Of The Week #305 - Apr 10, 2018"?

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