This question comes from a Chinese high school olympiad training program. It seems remarkably more difficult (and indeed, interesting!) than all other problems arising in the same program, especially since an elementary (high-school level) solution is probably available.

> 
> Show that there exists integers $a,b,c,d,e>2018$ so that the equation $$a^2+b^2+c^2+d^2+e^2+65=abcde$$ is satisfied.
> 
> 
> 

For what it's worth, here's what I have tried. Rewriting the equation as a quadratic polynomial in $a$,
$$a^2-(bcde)a+b^2+c^2+d^2+e^2+65=0.$$
For there to be integer solutions, the discriminant must be a perfect square. Hence
$$ b^2c^2d^2e^2-4b^2-4c^2-4d^2-4e^2-4\cdot5\cdot13=n^2.$$
I don't however see how I can solve this equation, especially due to the large number of unknowns. Any ideas?

---

**Edit:** Ivan Neretin presents an excellent answer by Vieta Jumping, which I'm sure will yield results. However, the training program I mentioned has not discussed such advanced tactics as Vieta Jumping yet, and only covered $\gcd$, $\operatorname{lcm}$, factorisation of polynomials, discriminants, modular arithmetic, divisibility and quadratic residues. Hence despite Ivan's excellent solution, I would still be extremely appreciative of a more elementary solution.
['https://math.stackexchange.com/questions/3015046', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/496634/']
You are supposed to bruteforce or guess $(a,b,c,d,e)=(1,2,3,4,5)$ or any other small solution, and then go up by [Vieta jumping](https://en.wikipedia.org/wiki/Vieta_jumping). That is, once you have a solution, you rewrite it as a quadratic polynomial in $a$ (just like you did), and since one root is integer, so is the other. Then you do the same to $b,\;c...$ and repeat until the roots are big enough.
Here is the output of the [Maxima](http://maxima.sourceforge.net/) commands I used to calculate a solution according to [Ivan Neretin's answer](https://math.stackexchange.com/a/3015056/11206).

```
(%i2) ev(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x = 1,b = 2,c = 3,d = 4,e = 5)

(%o2)                                  0

(%i3) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),b = 2,c = 3,d = 4,e = 5)

(%o3)                          [x = 1, x = 119]

(%i4) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),b = 119,c = 3,d = 4,e = 5)

(%o4)                          [x = 7138, x = 2]

(%i5) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),b = 119,c = 7138,d = 4,e

                                                                            = 5)

(%o5)                        [x = 3, x = 16988437]

(%i6) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),b = 119,c = 7138,

         d = 16988437,e = 5)

(%o6)                     [x = 72151760667066, x = 4]

(%i7) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),b = 119,c = 7138,

         d = 16988437,e = 72151760667066)

(%o7)              [x = 1041175313471572184867943319, x = 5]

(%i8) ev(solve(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,x),

         b = 1041175313471572184867943319,c = 7138,d = 16988437,

         e = 72151760667066)

(%o8) [x = 9109630532627114315851511163018235051842553960810405, x = 119]

(%i9) ev(x^2-b*c*d*e*x+b^2+c^2+d^2+e^2+65,

         x = 9109630532627114315851511163018235051842553960810405,

         b = 1041175313471572184867943319,c = 7138,d = 16988437,

         e = 72151760667066)

(%o9)                                  0

```