It is well known that each $n\in\mathbb N$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Motivated by this, here I pose the following novel question.

**Question 1.** Can each $n\in\mathbb N$ be written as $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?

I guess that the answer is positive, which has been verified for $n\le 10^6$. 

Similarly, I have the following two questions.

**Question 2.** Are $7$ and $487$ the only natural numbers which cannot be written as $w^2+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$?

**Question 3.** Is it true that eac $n\in\mathbb N$ with $n\not\equiv 3\pmod4$ can be written as 
$4w^2+x^2+y^2+z^2+xyz$ with $w,x,y,z\in\mathbb N$?

I also believe that the answers to Questions 2 and 3 should be positive, which have been verified for $n\le 4\times10^5$. It is easy to see that $$x^2+y^2+z^2+xyz\not\equiv3\pmod4$$ for any $x,y,z\in\mathbb Z$.

Any ideas to the above new questions? Your comments are welcome!