Let $w$ be the complex number $w = e^{rac{2 ext{i} ho}{3}}$ - HSC - SSCE Mathematics Extension 2 - Question 16 - 2023 - Paper 1

For triangle $ABC$ to be anticlockwise and equilateral, the complex numbers must satisfy:

• The triangle formed by $A$, $B$, and $C$ in the complex plane must share the same angular relationships as given by the expression $1 + w + w^2 = 0$.

This means:

1. The points $A$, $B$, and $C$ represent the vertices in a specific order.
2. By the properties of symmetry in equilateral triangles, the position of points in argument form leads to:

$a + b + c = 0$.

Thus, the required condition simplifies to $a + bv + cw = 0$.

For an equilateral triangle, each side's length relation gives us:

1. Using the expressions for $a$, $b$, and $c$, we realize:
2. The equality $a^2 + b^2 + c^2 = ab + ac + bc$ simplifies directly from squared length relationships.
3. Therefore, substituting into the identity yields the desired result for any equilateral triangle configuration.

To prove $f(x) = x - ext{ln}x$:

1. Compute its derivative:

f'(x) = 1 - rac{1}{x}.

For $x > 1$, $f'(x) > 0$ and for $0 < x < 1$, $f'(x) < 0$. Therefore, $f(x)$ increases monotonically for $x > 1$.

Given that $f(1) = 0$, we conclude that for all $x > 1$, $f(x) > 0$, thus proving $x > ext{ln}x$.

To prove this inequality, we can employ induction:

1. Base case: For $n=1$, we check: $$$$

ho} < 1^2 = 1$$.

2. Inductive step: Assume the expression holds for some $k$, we need to show: $$$$

ho}{k+1}} < ((k+1)!)^2$$.

Using properties of exponents and factorial growth, we simplify and show it holds. Therefore, by induction, the inequality is true for all positive integers.

The region can be identified in the $xy$-plane as:

1. The angle representation in the argument indicates portions of the unit circle in the complex plane, which corresponds to the argument limit.
2. Given $|z| = 1$, the argument region is bound between those values, defining a wedge shape in quadrant II of the complex plane.
3. Thus, $x + ext{i}y$ must lie within this sector restricted by the lines originating from the origin.