answer:1. Let ( f(K) ) be the value assigned to point ( K ). We need to show that ( f(K) = 0 ) for all points ( K ) in the plane. 2. Consider any point ( A ) in the plane. Construct a polygon ( AB_{21}B_{31}ldots B_{n1} ) that is similar to the given convex polygon ( P ). 3. Now, draw another polygon ( B_{21}B_{22}ldots B_{2n} ) that is also similar to ( P ). 4. Consider a spiral symmetry about point ( A ) that maps the polygon ( B_{21}B_{22}ldots B_{2n} ) to another polygon ( B_{i1}B_{i2}ldots B_{in} ). This means that ( B_{i1}B_{i2}ldots B_{in} ) is similar to ( P ). 5. Notice that both ( B_{i1}B_{i2}ldots B_{in} ) and ( AB_{2j}B_{3j}ldots B_{nj} ) are similar to ( P ). 6. By the given condition, the sum of the values assigned to the vertices of any polygon similar to ( P ) is zero. Therefore, we have: [ sum_{i=2,j=1}^n f(B_{ij}) = 0 ] and [ sum_{i=2,j=1}^n f(B_{ij}) + n f(A) = 0 ] 7. Subtracting the first equation from the second equation, we get: [ left( sum_{i=2,j=1}^n f(B_{ij}) + n f(A) right) - sum_{i=2,j=1}^n f(B_{ij}) = 0 ] Simplifying this, we obtain: [ n f(A) = 0 ] 8. Since ( n ) is the number of vertices of the polygon ( P ) and is a positive integer, we conclude that: [ f(A) = 0 ] 9. Since ( A ) was an arbitrary point in the plane, it follows that ( f(K) = 0 ) for all points ( K ) in the plane. (blacksquare)

answer:When developing your detective and killer, it's essential to consider their motivations and backstories, as these will drive their actions and decisions throughout the novel. However, the most important thing to focus on is creating a symbiotic relationship between the two characters. Think of them as two sides of the same coin. Your detective should have a personal connection or a deep understanding of the motivations that drive the killer. This connection could stem from a shared past experience, a similar personality trait, or even a contrasting worldview that sparks a fascination in the detective. This connection will create tension and raise the stakes, making their cat-and-mouse game more believable and engaging. On the other hand, your killer should be a reflection of the detective's darker side or a manifestation of their deepest fears. This will create an unsettling sense of mirroring, where the detective is forced to confront their own demons through the killer's actions. This dynamic will also allow you to explore themes such as morality, justice, and the blurred lines between good and evil. By establishing this symbiotic relationship, you'll create a sense of intimacy and urgency between the two characters, driving the plot forward and keeping your readers invested in their twisted game of hide and seek.

answer:Creating a likable yet flawed detective is a delicate balance. You want your readers to root for them, but also be aware of their imperfections, making them more relatable and human. To achieve this, consider giving your detective a complex mix of traits. Start by making them good at their job, but not perfect. Perhaps they have a keen instinct for solving crimes, but their unorthodox methods often land them in trouble. This blend of competence and recklessness will make them more believable and interesting. Next, give them a relatable personal life. Your detective could be struggling with a past trauma, a difficult relationship, or a personal addiction. This vulnerability will make them more sympathetic and accessible to your readers. However, be careful not to make their personal struggles so overwhelming that they overshadow the mystery itself. It's also essential to make your detective likable, not just in spite of their flaws, but because of them. Maybe they have a witty sense of humor, a dry sarcasm, or a quirky habit that makes them endearing. This will help your readers connect with them on a personal level and forgive them for their mistakes. Another approach is to make your detective an outsider, someone who doesn't quite fit in with the traditional law enforcement mold. This could be due to their background, their methods, or their unconventional perspective on justice. This outsider status will make them more intriguing and allow you to explore themes of identity and belonging. Lastly, don't be afraid to make your detective wrong sometimes. Let them misinterpret clues, follow dead leads, or make mistakes that put them in danger. This fallibility will make them more human and raise the stakes, making the mystery more suspenseful and engaging. By combining these elements, you'll create a detective that's both likable and flawed, someone your readers can root for, even when they're not always right.

answer:A traumatic event from the past can have a profound impact on a character, shaping their personality, motivations, and actions in profound ways. The aftermath of such an event can leave emotional scars, influencing their relationships, worldview, and even their sense of identity. The character may develop coping mechanisms to deal with the trauma, such as avoidance, denial, or self-destructive behaviors. They might become withdrawn, isolated, or defensive, struggling to form close relationships or trust others. Alternatively, they may become overly dependent on others, seeking constant reassurance or validation. The traumatic event can also influence their motivations, driving them to seek justice, revenge, or redemption. This desire for closure can become an all-consuming force, leading them to take risks or make decisions that put themselves or others in danger. Moreover, the trauma can change their perception of themselves and the world around them. They may feel a sense of guilt, shame, or responsibility for the event, leading to self-blame or self-doubt. They may also develop a distorted view of reality, perceiving threats or dangers that aren't there, or feeling a constant sense of hypervigilance. However, the impact of a traumatic event can also be a catalyst for growth and transformation. As the character navigates their pain and struggles to come to terms with their past, they may develop resilience, resourcefulness, and a newfound sense of purpose. They may learn to forgive themselves or others, finding a way to heal and move forward. In the context of your detective, a traumatic event from their past could be the driving force behind their determination to solve the mystery. Perhaps the event is connected to the current case, or maybe it's a personal demon they're trying to exorcise through their work. Whatever the connection, the traumatic event can add depth and nuance to your character, making their journey more compelling and relatable. As you develop your character's backstory, consider how the traumatic event has shaped them, and how it continues to influence their actions and decisions. This will help you create a richer, more complex character, and add layers to your story that will resonate with your readers.