Problem 8 - Complex numbers from A to...Z

Question

Let \(P\) be a point on the circumcircle of square \(A B C D .\) Find all integers \(n>0\) such that the sum $$ S_{n}(P)=P A^{n}+P B^{n}+P C^{n}+P D^{n} $$

Step-by-Step Solution

Answer

Answer: The sum Sn(P) is constant only for n = 1.

Step 1: Representing the points with complex numbers

Let's represent the points \(A\), \(B\), \(C\), and \(D\) as complex numbers \(a\), \(b\), \(c\), and \(d\) respectively. Also, let's represent the point \(P\) as the complex number \(z\). This allows us to treat distances and angles as operations with complex numbers.

Step 2: Identifying center and representing polar coordinates

Consider the center \(O\) of the circumcircle represented by the complex number \(\omega\). Since \(A\), \(B\), \(C\), and \(D\) are vertices of a square, and O is the center of the square and the circumcircle, we can see that: $$OA=OB=OC=OD=r$$ and $$\angle AOB=\angle BOC=\angle COD=\angle DOA=90^\circ$$ Now, let's represent these complex numbers using polar coordinates. We have: $$a=\omega + re^{i0}, \quad b=\omega+ re^{i\frac{\pi}{2}}, \quad c=\omega + re^{i\pi}, \quad d=\omega + re^{i\frac{3\pi}{2}}$$

Step 3: Calculating the distances

Now, we are ready to calculate the distances of \(P\) from the points \(A\), \(B\), \(C\), and \(D\). Since \(|p-a|=PA, |p-b|=PB, |p-c|=PC, |p-d|=PD\), we can write: $$S_n(P)=|p-a|^n+|p - b|^n + |p - c|^n + |p - d|^n$$

Step 4: Simplifying the distances formula

Using complex conjugates, we can simplify the formula for \(S_n(P)\) as follows: $$S_n(P) = \left| z-a \right|^n + \left| z-b \right|^n + \left| z-c \right|^n + \left| z-d \right|^n = \left| \left( z - \omega \right) - re^{i0} \right|^n + \left| \left( z - \omega \right) - re^{i\frac{\pi}{2}} \right|^n + \left| \left( z - \omega \right) - re^{i\pi} \right|^n + \left| \left( z - \omega \right) - re^{i\frac{3\pi}{2}} \right|^n$$ Now, let \(z - \omega = w\) and simplify: $$S_n(P) = \left| w - re^{i0} \right|^n + \left| w - re^{i\frac{\pi}{2}} \right|^n + \left| w - re^{i\pi} \right|^n + \left| w - re^{i\frac{3\pi}{2}} \right|^n$$

Step 5: Exploring for which n the sum is constant

Using geometric interpretation, we can conclude that: For \(n = 1\) the sum \(S_1(P)\) is constant. This is because P will always lie on a circle of constant circumference, and so the sum of its distances to A, B, C, and D will always be constant. However, for \(n > 1\), we must show that the sum is not constant. To do this, let's consider two cases: \((1)\) when \(P\) coincides with one of the vertices, and \((2)\) when \(P\) is at the midpoint of one of the sides of the square. \((1)\) When \(P\) coincides with one of the vertices (e.g. at point \(A\)), we have \(S_n(P)=0+|AB|^n+|AC|^n+|AD|^n\). \((2)\) When \(P\) is at the midpoint of one of the sides (e.g. midpoint \(E\) of \(AB\)), the sum is \(S_n(P)=|AE|^n+|BE|^n+|CE|^n+|DE|^n\). For \(n>1\), \(S_n(P)\) is not constant, as the terms will grow or shrink disproportionately when the distance changes. Therefore, the only integer value for which \(S_n(P)\) remains constant is \(n=1\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing complex numbers in terms of a magnitude and an angle. This is different from the usual Cartesian coordinates, which use an x and y axis. In polar coordinates, a complex number is given by its distance from the origin, known as the modulus, and the angle that it forms with the positive x-axis, known as the argument.
For example, suppose we have a point represented by the complex number \( z = x + yi \). This point has Cartesian coordinates \((x, y)\). To convert this to polar coordinates, we calculate:

• The modulus \( r = \sqrt{x^2 + y^2} \), which is the distance from the origin.
• The argument \( \theta = \tan^{-1}(\frac{y}{x}) \), which is the angle the line makes with the positive x-axis.

So the polar representation of \( z \) is \( re^{i\theta} \), where \( r \) is the modulus and \( \theta \) is the argument. In this exercise, points like \( A, B, C, \) and \( D \) were represented using such polar forms, with the center of the square as the origin.

Square Geometry

Square geometry involves understanding the properties and symmetries of a square. A square is a regular quadrilateral, meaning it has four equal sides and every angle is a right angle (90 degrees).
In geometry, this symmetry provides useful properties such as each diagonal of a square being equal and bisecting the opposite angles.

• The diagonals of a square are always equal in length, and they intersect each other at the center of the square, forming four right triangles.
• This intersection point is also the center of the square's circumcircle, which perfectly circumscribes the square.
• Each angle between consecutive vertices (such as \( \angle AOB \) or \( \angle BOC \)) is 90 degrees, a fact utilized when representing the vertices in the polar coordinate step.

Understanding these basic properties is crucial when dealing with more complex problems, such as the given exercise involving a point \( P \) on the square's circumcircle.

Circumcircle

A circumcircle of a polygon is a circle that passes through all the vertices of the polygon. For a square, the circumcircle is particularly simple to manage because of the square's inherent symmetry.
All squares have a circumcircle, and the center of this circumcircle is the same as the geometric center of the square. Its radius, known as the circumradius, can be computed as half the diagonal of the square.

• For a square with side length \( s \), the diagonal \( d = s\sqrt{2} \), so the circumradius is \( r = \frac{s\sqrt{2}}{2} \).
• This circumradius is equal to the distance from the center of the square to any of its vertices.

In the problem, \( P \) lies on this circumcircle, meaning the distance from \( P \) to \( A, B, C, \) or \( D \) is influenced by the symmetrical properties of both the square and its circumcircle.

Complex Conjugates

Complex conjugates help in simplifying calculations involving complex numbers. The complex conjugate of a complex number \( z = x + yi \) is \( \bar{z} = x - yi \).
This operation flips the sign of the imaginary component, which offers various benefits in computation and analysis.

• When a complex number and its conjugate are multiplied, the result is a real number: \( z \cdot \bar{z} = x^2 + y^2 \), effectively giving the squared modulus.
• The modulus of a complex number \( z \) is \( \sqrt{z \cdot \bar{z}} \).

In the context of this exercise, using complex conjugates helps in expressing the distances, such as \( |z-a| \), in a simplified form. It makes calculating powers of distances more manageable by converting complex absolute values into expressions involving real numbers.