Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$

Each cell of the 4x4 board has a grasshopper. When a grasshopper jumps, it moves to one of the adjacent cells (down, up, right, or left). The grasshopper cannot move diagonally or go off the board. At most how many cells can remain empty after each grasshopper jumps once?

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

Let $\omega$ be the circumcircle of an acute angled tirangle $ABC.$ The line tangent to $\omega$ at $A$ intersects the line $BC$ at the point $T.$ Let the midpoint of segment $AT$ be $N,$ and the centroid of $\triangle ABC$ be the point $G.$ The other tangent line drawn from $N$ to $\omega$ intersects $\omega$ at the point $L.$ The line $LG$ meets $\omega$ at $S\neq L.$ Prove that $AS\parallel BC.$