2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...
10-dirihleov-princip.doc
1. Prof. Dr Vojislav Andrić
DIRIHLEOV PRINCIP
U prethodnim razredima smo se upoznali sa Dirihleovim principom i naučilin da ako treba kn + 1
kuglicu rasporediti u k kutija , onda pri ma kom rasporedu kuglica postoji bar jedna kutija u kojoj se nalazi
bar n + 1 kuglica. Videli smo i da se Dirihleov princip najefikasnije dokazuje svođenjem na protivurečnost,
jer ako bi bio moguć drugačiji raspored, tj. raspored u kome bi u svakoj od k kutija bilo n ili manje kuglica
onda bi ukupan broj kuglica bio manji, ili eventualno jednak nk, što je suprotno pretpostavci da je raspore-
đena nk + 1 kuglica.
Međutim Dirihleov princip je mnogo više od kutija i kuglica i ima mnogobrojne primene u praksi i
matematičkoj teoriji (brojevi, geometrija, logičko-kombinatorni problemi …). Primeri i zadaci koji slede
imaju za cilj da prikažu neke od mogućih, a raznovrsnih primena Dirihleovog principa.
PRIMER 1.
Dokazati da postoji prirodan broj koji počinje ciframa 9876543210, a deljiv je sa 2011.
Rešenje: Uočimo brojeve: x1 = 9876543210, x2 = 98765432109876543210 ... , x2012 od kojih svaki sadrži
jednu, dve, ..., 2012 grupa cifara 9876543210. Pri deljenju sa 2011 tih 2012 brojeva daju ostatke 0, 1,... 2010.
Kako je 2012 brojeva, a 2011 ostataka to među uočenim brojevima postoje dva broja koji imaju jednake
ostatke. Neka su to brojevi xi = 2011a + r i xj = 2011b + r. Tada je xi – xj = 2011a + r – (2011b + r) =
2011(a – b) = 98765432109876543210 ... (i grupa cifara) – 98765432109876543210 ... (j grupa cifara) =
9876543210 ...0000 (i – j grupa cifara 9876543210 i sve ostalo su nule). Jasno je da dobijeni broj počinje sa
9876543210 i deljiv je sa 2011.
PRIMER 2.
Data su 52 proizvoljna prirodna broja. Dokazati da među njima postoje dva čiji zbir ili razlika je deljiva sa
100. Važi li dato tvrđenje za 51 broj ?
Rešenje: Ako se dati brojevi rasporede u klase prema dvocifrenom završetku tako da jednu klasu čine brojevi
sa dvocifrenim završetkom 00, drugu sa dvocifrenim završetkom 01 i 99, treću sa dvocifrenim završetkom
02 i 98, ..., pedesetu klasu brojevi čiji su dvocifreni završetci 49 i 51 i pedeset i prvu klasu brojevi čiji je
dvocifreni završetak 50, onda ima 52 broja, a 51 klasa pa na osnovu Dirihleovog principa postoji bar jedna
klasa u kojoj se nalaze bar dva broja. Jasno je da je zbir ili razlika ta dva broja deljiva sa 100.
Tvrđenje ne važi ako ima 51 broj. Na primer skup brojeva 50, 51, 52, ..., 99, 100 sadrži 51 broj, ali ne
postoje dva broja čiji je zbir ili razlika deljiva sa 100, jer za zbir brojeva s važi nejednakost 101 s 199, a
za razliku brojeva r nejednakost 1 r 50.
PRIMER 3.
Posmatrajmo niz četvorocifrenih brojeva koji predstavljaju poslednje četiri cifre brojeva 3, 32
, 33
, 34
... To je
niz 0003, 0009, 0027, 0081, 0243, 0729, 2187, ... Dokazati da je počev od nekog člana dati niz periodičan.
2. Rešenje: Niz 3, 32
, 33
, 34
... ima najviše 1010104 = 4000 četvorocifrenih završetaka (za prve tri cifre
kandidati su svaka od 10 cifara, a za poslednju cifru samo 1, 3, 7 i 9). Na osnovu Dirihleovog principa
zaključujemo da već u nizu 3, 32
, 33
, ...34000
, 34001
postoji 4001 broj, a 4000 četvorocifrenih završetaka, što
znači da postoje dva koji imaju jednake četvorocifrene završetke. Neka su to 3a
= 104
m + r i 3b
= 104
n + r.
Tada je 3a
– 3b
= 104
(m – n). Množenjem dobijene jednakosti sa 3p
dobija se 3a+p
– 3b+p
= 3p
104
(m – n), a
to znači da i brojevi 3a+p
i 3b+p
za svaki prirodan broj p imaju jednake četvorocifrene ostatke, što dokazuje
da je niz četvorocifrenih ostataka stepena broja 3 periodičan.
PRIMER 4.
Dato je sedam duži od kojih je svaka veća od 10 cm, a manja od 1 m. Dokazati da među datima postoje tri
duži od kojih se može sastaviti trougao.
Rešenje: Neka su dužine datih duži d1, d2, ... , d7. Tada je 10cm d1 d2 ... d7 < 100cm. Od najmanje
tri duži se ne može sastaviti trougao, ako je d3 d1 + d2 10 + 10 = 20cm. Slično trougao se ne može
sastaviti ni ako je d4 d2 + d3 10 + 20 = 30cm. Ako se postupak nastavi dobija se d5 d3 + d4 20 + 30 =
50cm, d6 d4 + d6 30 + 50 = 80cm i konačno ako je d7 d5 + d6 50 + 80 = 130cm. Kako je po
pretpostavci d7 < 100cm dobili smo protivurečnost, što znači da se od datih duži uvek može sastaviti trougao.
PRIMER 5.
U kvadratu stranice 15 cm raspoređeno je 20 disjunktnih kvadrata stranice 1 cm. Dokazati da postoji krug
poluprečnika 1 cm koji ne dodiruje nijedan od datih kvadrata.
Rešenje: Oko, a unutar datog kvadrata formira se ’’zaštitna zona’’ širine 1cm, tako da ’’slobodna zona’’
predstavlja kvadrat čija je stranica 13cm, a površina 169cm2
. Slično, oko svakog od 20 disjunktnih kvadrata
opiše se figura F (vidi sliku) koja sadrži ’’zaštitnu zonu’’ od 1 cm oko svakog od datih kvadrata.
Kako je površina svake od 20 figura F jednaka (5 + )cm2
, to ukupna površina koju prekriva 20 figura F
iznosi najviše 20(5 + ) < 165cm2
< 169cm2
. To znači da postoji bar jedna tačka S koja se nalazi u
’’slobodnoj zoni’’. Krug sa centrom u tački S poluprečnika 1cm je traženi krug, jer se on zbog postojanja
’’zaštitne zone’’ sigurno nalazi unutar datog velikog kvadrata i ne dodiruje nijedan od malih kvadrata.
3. ZADACI
1. Dokazati da je između 100 proizvoljnih celih brojeva uvek moguće izabrati 15 takvih da je razlika
bilo koja dva od njih deljiva sa 7.
2. Dato je proizvoljnih 1111 prirodnih brojeva. Dokazati da bar 124 od datih brojeva počinje istom
cifrom, bar 112 datih brojeva se završava istom cifrom i bar 13 brojeva počinje i završava se istom cifrom.
3. Dokazati da se među 51 proizvoljnih, a različitih prirodnih brojeva manjih od 100 mogu izabrati tri
takva da je jedan od njih jednak zbiru ostala dva.
4. Dato je 20 različitih prirodnih brojeva manjih od 65. Dokazati da među svim mogućim razlikama
tih brojeva postoje bar četiri jednake.
5. Dokazati da postoji prirodan broj koji se u dekadnom zapisu zapisuje samo šesticama i nulama koji
je deljiv sa prirodnim brojem n.
6. Brojevi od 1 do 10 zapisani su u niz u proizvoljnom poretku, a svaki od njih je sabran sa svojim
rednim brojem. Dokazati da među dobijenim zbirovima postoje bar dva čija je poslednja cifra jednaka.
7. Prirodni brojevi od 1 do 2n zapisani su u proizvoljnom poretku, a zatim je ispod svakog od njih
napisan njegov redni broj u tom nizu. Svaki broj je potom sabran sa svojim rednim brojem. Dokazati da
među tako dobijenim brojevima postoje dva čija je razlika deljiva sa 2n.
8. Ako se iz skupa prvih 2n prirodnih brojeva izabere n+1 brojeva, dokazati da postoje bar dva od njih
od kojih je jedan deljiv drugim.
9. Dokazati da postoji stepen broja 3 čija se reprezentacija u dekadnom zapisu završava ciframa 0001.
10. U skupu od 10 bilo kojih dvocifrenih brojeva postoje dva disjunktna podskupa takva da je zbir
elemenata u jednom podskupu jednak zbiru elemenata u drugom podskupu. Dokazati.
11. Dokazati da se među 39 uzastopnih prirodnih brojeva nalazi bar jedan broj čiji je zbir cifara deljiv
sa 11.
12. Mase 50 mašina iznose 370kg, 372kg, 374kg, ... ,466kg i 468kg. Mogu li se tih 50 mašina
transportovati sa 7 kamiona nosivosti tačno 3 tone?
13. Čvorovi beskonačne kvadratne mreže obojeni su crvenom ili plavom bojom. Dokazati da postoje
dve horizontalne i dve vertikalne linije koje grade u kvadratnoj mreži pravougaonik čija su sva temena iste
boje.
14. U kocki čija je ivica 13 cm na slučajan način je raspoređeno 2011 tačaka. Dokaži da postoji
’’kockica’’ ivice 1cm unutar koje se ne nalazi ni jedna od datih tačaka.
15. U staklenoj kocki ivice 1 m nalazi se 2011 muva. Dokazati da postoji sfera poluprečnika 1/11
unutar koje se u svakom trenutku, bez obzira na raspored muva, nalaze bar 3 muve.
16. Dato je 9 pravih od kojih svaka deli dati kvadrat ABCD na dva trapeza čije se površine odnose kao
2:3. Dokazati da postoji tačka kroz koju prolaze bar 3 date prave.
17. Unutar konveksnog 2n-tougaonika data je tačka M. Kroz tačku M i svako teme uočena je prava.
Dokazati da postoji strana mnogougla s kojom ni jedna prava nema zajedničkih unutrašnjih tačaka.
18. Unutar kvadrata stranice 1 cm raspoređeno je nekoliko krugova čiji zbir obima je 10 cm. Dokazati
da postoji prava koja preseca bar 4 data kruga.
19. Unutar kvadrata stranice 1 cm nalazi se konveksan mnogougao sa 100 stranica. Dokazati da postoji
trougao, čija su temena - temena datog mnogougla i čija je površina manja od 0,0008 cm2.
4. 20. Svaka tačka ravni na proizvoljan način je obojena jednom od tri boje. Dokazati da u datoj ravni
postoje dve tačke iste boje čije je rastojanje 1 cm .
REŠENJA
1. Proizvoljnih 100 celih brojeva pri deljenju sa 7 daje sedam mogućih ostataka: 0, 1, 2, 3, 4, 5, 6.
Kako je 100 : 7 = 14(2), to postoji bar 14 + 1 = 15 celih brojeva koji pri deljenju sa 7 daju jednake ostatke.
Tada je razlika bilo koja dva od njih deljiva sa 7, jer ako je xi = 7a + r i xj = 7b + r, onda je xi – xj = 7(a – b).
2. Datih 1111 prirodnih brojeva počinje sa 9 mogućih cifara. Kako je 1111:9 = 123(4), to znači da
postoji prva cifra koja zastupljena bar 123 + 1 = 124 puta. Slično, datih 1111 prirodnih brojeva završava se
sa 10 mogućih cifara. Kako je 1111:10 = 111(1), to znači da postoji poslednja cifra koja zastupljena bar
111 + 1 = 112 puta. Kombinacija prva-poslednja cifra ima tačno 910 = 90. Kako je 1111:90 = 12(31), to
postoji bar 13 brojeva koji počinju i završavaju se jednakim ciframa.
3. Kako su dati brojevi različiti, to se uvek mogu rasporediti u niz x1 < x2 … < x50 < x51 < 100. Ako
se uoči niz brojeva x51 – x1 > x51 – x2 > … > x51 – x50 > 0 onda postoji ukupno 51 + 50 = 101 broj, koji
uzimaju vrednosti od 1 do 99, tj. 99 vrednosti. Na osnovu Dirihleovog principa postoje bar dva broja koja su
međusobno jednaka. Očigledno je da ta dva broja nisu u istom nizu, jer je prvi niz rastući, a drugi opadajući.
Dakle postoje brojevi xi i x51 – xj takvi da je xi = x51 – xj što znači da je xi + xj = x51.
4. Razlika r bilo koja dva izabrana broja ispunjava nejednakost 1 r 63, pa r može da uzme najviše
63 vrednosti. Ako se formiraju sve moguće razlike 20 izabranih različitih prirodnih brojeva, onda se dobija
tačno 190 razlika. Kako je 190 : 63 = 3(1), to postoje bar 3 + 1 = 4 jednake razlike izabranih brojeva.
5. Zadatak se rešava veoma slično primeru 2, s tim što se posmatra niz brojeva x1 = 6, x2 = 66, ...
xn+1 = 666 ... 666 (n + 1 šestica).
6. Neka su zapisani brojevi x1, x2, ..., x10 i neka su njihovi redni brojevi 1, 2, ..., 10. Dobijeni zbirovi
su x1 + 1, x2 + 2, ..., x10 + 10. Suma dobijenih zbirova je x1 + 1 x2 + 2 + ... +x10 + 10 = 55 + 55 = 110. Ako
bi se zbirovi x1 + 1, x2 + 2, ..., x10 + 10 završavali različitim ciframa, onda bi poslednja cifra njihove sume
bila petica (jer je .0 + .1 + .2 + ... + .8 + .9 = .5). Protivurečnost.
7. Neka su zapisani brojevi x1, x2, ..., x2n i neka su njihovi redni brojevi 1, 2, ..., 2n. Dobijeni zbirovi
su x1 + 1, x2 + 2, ..., x2n + 2n. Neka dobijeni zbirovi pri deljenju sa 2n daju redom količnike q1, q2, ..., q2n i
ostatke r1, r2, ..., r2n. Tada je x1 + 1 = 2nq1 + r1, x2 + 2 = 2nq2 + r2 ... , x2n + 2n = 2nq2n + r2n. Sledi da je
S = x1 + 1 + x2 + 2 + ... +x2n + 2n = 2(1 + 2 + ...+ 2n) = 2n(2n + 1). S druge strane je S = 2nq1 + r1 + 2nq2 +
r2 + ... + 2nq2n + r2n = 2n(q1 + q2 + ... + q2n) + (r1 + r2 + ... + r2n). Ako su svi ostaci pri deljenju sa 2n različiti
onda je r1 + r2 + ... + r2n = 0 + 1 + 2 + ... + 2n – 1 = n(2n + 1). Dobija se 2n(2n + 1) = 2nQ + n(2n – 1) ili
posle deljenja sa n jednakost 2(2n + 1) = 2Q + (2n – 1). što nije moguće, jer je sa leve strane paran, a sa
desne strane neparan broj. To znači da nisu svi ostaci različiti, već da postoje bar dva jednaka. Iz ove
činjenice neposredno sledi i traženo tvrđenje.
8. U skupu od 2n prirodnih brojeva postoji n neparnih 1, 3, ...,2n – 1 i n parnih brojeva 2, 4, 6, ... ,2n.
Uočimo skupove koje čine neparni brojevi i njihovi proizvodi sa 2, 4, 8, 16 ... dakle stepenima broja 2. Ti
skupovi su: S1 = 1, 2, 4, 8, 16 ..., S2 = 3, 6, 12, 24, 48 ..., S3 = 5, 10, 20, 40, 80 ... ... Sn-1 = 2n - 3 i
Sn = 2n - 1. Uočeni skupovi S1, S2 ... Sn su disjunktni i sadrže sve prirodne brojeve od 1 do 2n. Kako je
izabran n + 1 prirodan broj, a ima n skupova, to postoji skup Sk = 2k – 1, 2(2k -1), 4(2k – 1) ... unutar
5. koga se nalaze dva od n + 1 izabranih prirodnih brojeva. Jasno je da se u svakom od skupova Sk nalaze
brojevi u koji zadovoljavaju traženi uslov da je jedan od njih deljiv drugim.
9. U primeru 3 videli smo da postoje stepeni broja 3 za koje važi jednakost 3a
– 3b
= 104
(m – n). Sledi
da je 3a
(3a-b
– 1) = 104
(m – n), odnosno b
b
a n
m
3
)
(
10
1
3
4
. Kako su 104
i 3b
uzajamno prosti brojevi, to
je m – n deljivo sa 3b
, pa je 3a-b
– 1 = 104
p. Tada je 3a-b
= p104
+ 1 = ...0001, čime je dokaz završen.
10. Neka su izabrani dvocifreni prirodni brojevi n1, n2 , ... , n10 . Od 10 izabranih brojeva može se
konstruisati 210
– 1 = 1023 neprazna podskupa. Najmanji zbir bojeva u jednom podskupu je 10, a najveći
90 + 91 + ... + 99 = 945. Dakle, moguće je napraviti 945 – 9 = 936 raznih vrednosti. Kako je podskupova
1023, a 936 vrednosti to postoje bar dva podskupa čiji elementi imaju isti zbir. Neka su Si i Sj podskupovi
koji imaju jednak zbir elemenata. Ako su Si i Sj disjunktni, problem je rešen. Ako nisu, onda i iz jednog i iz
drugog treba odstraniti zajedničke elemente.
11. Neka je n prirodan broj. Među brojevima n, n + 1, ..., n + 19 postoje dva kod kojih je poslednja
cifra nula. Bar kod jednog od tih brojeva pretposlednja cifra nije jednaka 9. Neka je taj broj jednak a i neka
je zbir njegovih cifara jednak s. Tada je a + 19 n + 19 + 19 = n + 38 i brojevi a, a + 1, a + 2, ..., a + 19
imaju zbir cifara s, s + 1, s + 2, ... , s + 10. Kako među jedanaest uzastopnih prirodnih brojeva postoji broj
koji je deljiv sa 11, to je zbir cifara bar jednog od brojeva n, n + 1, ..., n + 38 deljiv sa 11.
12. Ako 50 mašina treba da rasporedimo na 7 kamiona, onda na osnovu Dirihleovog principa postoji
kamion koji treba da transportuje bar 8 mašina. Najmanja moguća masa 8 mašina je 370 + 372 + ... + 382 +
384 = 3016kg > 3t, što znači da 7 kamiona nije dovoljno za prevoz mašina.
13. Na osnovu Dirihleovog principa od tri izabrana čvora na istoj horizontali, bar dva su iste boje. Zato
se uoče se tri vertikale i posmatraju svi mogući rasporedi obojenosti čvorova koji se na njima nalaze.
Mogućih, a različitih rasporeda je 23
= 8 i to su:
P P P
1 ---------------o---------------o---------------------------o----------------
P P C
2 ---------------o---------------o---------------------------o----------------
P C P
3 ---------------o---------------o---------------------------o----------------
C P P
4 ---------------o---------------o---------------------------o----------------
P C C
5 ---------------o---------------o---------------------------o----------------
C P C
6 ---------------o---------------o---------------------------o----------------
C C P
7 ---------------o---------------o---------------------------o----------------
C C C
8 ---------------o---------------o---------------------------o----------------
Jasno je da se i u okviru postojećih rasporeda mogu uočiti oni koji ’’zatvaraju’’ pravougaonik čija
su sva temena iste boje, ali i da će se svaka sledeća horizontala ponoviti jedan od 8 mogućih rasporeda, što je
dovoljno za dokaz da postoji pravougaonik čija su sva temena iste boje.
14. Zapremina kocke čija je ivica 13 cm je 2179 cm3
. Kako je 2179 kubnih centimetara, a 2011 tačaka,
to znači da ma kako da su tačke raspoređene postoji ’’kockica’’ ivice 1cm unutar koje se ne nalazi ni jedna
tačka.
15. Kako je 2011 muva, a 1000 kubnih decimetara i kao je 2011:1000 = 2(11), to postoji kubni
decimetar unutar koga se nalaze bar 2 + 1 = 3 muve. Ostaje da se dokaže da se oko kubnog decimetra može
6. opisati sfera poluprečnika 1/11m = 10/11cm. Dokaz sledi direktno iz činjenice da je prečnik sfere veći od
dijagonale kubnog decimetra, tj. 20/11 > 3 .
16. Ako prava deli dati kvadrat ABCD na dva trapeza čije se površine odnose kao 2:3, onda prava
prolazi kroz tačku koja deli srednju liniju kvadrata u odnosu 2:3. Kako takvih tačaka ima 4 (na svakoj
srednjoj liniji kvadrata po 2), to na osnovu Dirihleovog principa postoji tačka kroz koju prolaze bar tri date
prave, jer je 9:4 = 2(1).
17. Ako je data tačka M na nekoj, na primer dijagonali AiAj konveksnog 2n-tougaonika, onda prave
AiM i AjM nemaju zajedničkih unutrašnjih tačaka ni sa jednom od stranica 2n-tougaonika.
Ako M nije ni na jednoj dijagonali posmatramo dijagonalu A1An+1 koja dati 2n-tougaonik deli tako
da se i sa jedne i sa druge strane uočene dijagonale nalazi po n – 1 teme 2n-tougaonika. Neka je tačka M
unutar mnogougla A1A2...An+1. Tada prave MAn+1, MAn+2, ... MA2n, MA1 ne seku stranice An+1An+2, ...
A2nA1, dakle n stranica, što znači da mogu seći najviše n preostalih stranica. Slično prave MA2, MA3, ...
MAn, dakle n – 1 prava može seći najviše n – 1 stranicu. Kako imamo 2n stranica, a najviše n + n -1 = 2n – 1
preseka, to postoji bar jedna stranica koju ne preseca ni jedna prava.
18. Neka je u kvadrat upisano n krugova čiji su poluprečnici r1, r2, ... rn. Tada je zbir obima svih kru-
gova jednak S = 2r1 + 2r2 + ... + 2rn = 2(r1 + r2 + ... + rn) = 10, pa je r1 + r2 + ... + rn =
2
3
2
10
.
Kako prečnik svakog od upisanih krugova manji od 1, to je 2ri < 1, tj. ri <
2
1
. To znači da n
2
1
>
2
3
2
10
,
pa je broj upisanih krugova n > 3, tj.broj upisanih krugova je 4 ili više. Ako prečnike svih upisanih krugova
projektujemo na srednju liniju kvadrata, onda je zbir svih projektovanih duži, tj. prečnika jednak 3
10
.
Sledi da na srednjoj liniji kvadrata postoji tačka M koja je unutrašnja tačka bar 4 projektovana prečnika.
Tražena prava sadrži tačku M, normalna je na srednju liniju kvadrata i preseca bar 4 upisana kruga.
7. 19. Obim konveksnog mnogougla je manji od obima kvadrata, pa je Om < 4cm. Ako stranice mnogo-
ugla imaju dužinu A1A2 = a1, A2A3 = a2, …, A100A1 = a100, onda je (a1 + a2) + (a2 + a3) + … + (a99 + a100) +
(a100 + a1) = 2(a1 + a2, …, + a100) < 8. Kako među sumama (a1 + a2), (a2 + a3), … (a99 + a100), (a100 + a1)
postoji najmanja, to za tu najmanju sumu ai + ai+1 važi nejednakost ai + ai+1 < 8/100 = 0,08.
Površina trougla AiAi+1Ai+2 jednaka je
2
i
i h
a
. Kako je P =
2
2
1
i
i
i
i a
a
h
a
to na osnovu nejednakosti
između geometrijske i aritmetičke sredine važi P =
0008
.
0
8
08
,
0
2
2
1
2
2
2
2
1
1
i
i
i
i
i
i a
a
a
a
h
a
cm2.
20. Neka je data ravan obojena crvenom (I), plavom (II) ili belom bojom (III). Uočimo u proizvoljno
obojenoj ravni tačku X koja je, na primer boje (I). Oko tačke X konstruišemo kružnicu k (X, r = 3 cm), tj.
kružnicu poluprečnika 3 cm. Na dobijenoj kružnici izaberemo proizvoljnu tačku T.
Ako je T boje (II), onda nad duži XT kao dijagonalom konstruišemo romb XYTZ čije su sve
stranice 1cm, a i dijagonala YZ iznosi 1cm (XTY = XTZ = 30o
). Ako je tačka Y boje (I), onda je XY
tražena duži,jer su i X i Y boje (I). Ako je Y boje (II) onda su YT tražena duž, jer su i Y i T boje (II), Zato
neka je Y boje (III). Na osnovu Dirihleovog principa od 4 tačke XYZT koje su 3 boje, bar 2 su iste boje, tj.
tada je Z boje (I), (II) ili (III). Ako je Z boje (I), onda je XZ tražena duž. Ako je Z boje (II) onda je TZ
tražena duž, a ako je Z boje III, onda je YZ tražena duž.
Ako je T boje (III), onda je postupak potpuno identičan, a razmatranje analogno.
Ako je T boje (I), onda se na kružnici uoči tačka T1 koja je od tačke T udaljena 1cm. Ako je T1 boje
(I), onda je duž TT1 tražena duž, a ako T1 nije boje (I) onda se već opisani postupak odvija nad duži XT1.