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Let $k$ be an odd positive integer. Can every positive rational number $n$ be written as $\frac{a^k+b^k}{c^k+d^k}$ where $a,b,c,d$ are positive rational numbers/ rational numbers?

The answer is true for $k=1$, and probably also true for $k=3$ by explicit construction (see http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html). How about the case $k=5$ and $k=7$? For general $k$?

Motivation: By general philosophy of arithemetic for surfaces, there will be little rational points on a surface of general type. And for $k=3$ and fixed rational number $n$, we are considering rational points on cubic surfaces which are del Pezzo so shall contain rational curves.

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    $\begingroup$ The k=3 case has an affirmative solution. This appeared on the shortlist of the 1999 IMO. $\endgroup$
    – Peter McNamara
    Commented Jan 11, 2020 at 7:49
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    $\begingroup$ Note that if 3 can be written in this form with $k=7$ then the map $(x,y)\mapsto x^7+3y^7$ from $\mathbb{Q}^2$ to $\mathbb{Q}$ is not injective. However, this map actually believed to be injective, see mathoverflow.net/questions/21003/… $\endgroup$
    – Dmitry Krachun
    Commented Jan 11, 2020 at 22:50

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