Tomi's Challenges - #13

Simplify $L$ with the help of the second binomial formula and $N:=2021\:2021\:2021\:2021$ : $\begin{aligned} -\sqrt{L}=&(N+3)^2 - 2N(N+3) + (N+8)^2 - 2(N+1)(N+8) + 2(N+1)^2 - (2N+65)\\[.5em] & +(N+8)^2- 2N(N+8) + (N+3)^2-2(N+1)(N+3) + 2(N+1)^2- (2N+65)\\[.5em] =&(N+3-N)^2 + (N+8-N-1)^2+ (N+8-N)^2+ (N+3 - N - 1)^2\pm4N+2-130\\[.5em] =&3^2+7^2+8^2+2^2+2-130=-2\qquad\Rightarrow \qquad L=4,\qquad L^L=\boxed{256} \end{aligned}$

$L=(\textcolor{#D61F06}{2021202120212024^2-4042404240424042\cdot 2021202120212024+2021202120212029^2-4042404240424044\cdot 2021202120212029+2\cdot 2021202120212022^2-4042404240424106}+\textcolor{#3D99F6}{2021202120212029^2-4042404240424042\cdot 2021202120212029+2021202120212024^2-4042404240424044\cdot 2021202120212024+2\cdot 2021202120212022^2-4042404240424106})$

The red and blue expressions represent the same multivariable polynomial, but under different values. Let's call the polynomial $f(x, y)$ . The red expression represents $f(2021202120212024, 2021202120212029)$ , and the blue expression represents $f(2021202120212029, 2021202120212024)$ . For the sake of simplicity, let's view only the polynomial.

$f(x, y)=x^2-4...42\cdot x+y^2-4...44\cdot y+2\cdot 2...22^2-4...4107$

$f(x, y)=(x-2...21)^2+(y-2...22)^2-2...21^2-2...22^2+2\cdot 2...22^2-4...4107$

$f(x, y)=(x-2...21)^2+(y-2...22)^2+2...22^2-2...21^2-4...4107$

$f(x, y)=(x-2...21)^2+(y-2...22)^2+4...4043-4...4107$

$f(x, y)=(x-2...21)^2+(y-2...22)^2-64$ .

Plugging in $x$ and $y$ , we get the following:

$f(2...24, 2...29)=(2...24-2...21)^2+(2...29-2...22)^2-64=9+49-64=-6$

$f(2...29, 2...24)=(2...29-2...21)^2+(2...24-2...22)^2-64=64+4-64=4$

We see that $L=(f(2...29, 2...24)+f(2...24, 2...29))^2=(-2)^2=4$ , $L$ is 4, and $4^4=16^2=\boxed{256}$ .