Let . If , then what is the minimum value of ?



Answer confirmation

If what is the value of ?



Answer Confirmation

There have been all these random but useful theorems
Please post any theorems you know, random or not, but please say whether they are random or not.
I'll start give an example:
Random
I am just looking for some theorems to study.

If you pay for MathDash, specifically lessons, could you please give feedback below?

Let Prove that

Let be a polynomial of degree 4 with integer coefficients, leading coefficient 1, and having as one of its zeros. What is ?



Answer confirmation

Given straight lines in general position in the plane (no three lines pass through the same point and no two are parallel) they divide the plane into several parts. Prove that among these parts there are at least
(a) triangles
(b) triangles
(c) − triangles.

Suppose that the seven-digit number is a multiple of . What is the sum of its distinct prime divisors?



Answer confirmation

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Grok Free: Complete Beginner's Guide to Free AI Tools and Platforms
The artificial intelligence revolution has made advanced technology more accessible than ever before. Grok Free solutions are leading this transformation, offering powerful AI capabilities without the traditional cost barriers that once limited access to cutting-edge technology. This comprehensive guide will help you navigate the world of free AI tools and unlock their full potential for personal and professional use.

Understanding the Grok Free Movement
What Makes Free AI Tools Possible?
The emergence of free AI platforms represents a fundamental shift in how technology companies approach user acquisition and market penetration. By offering robust free tiers, these platforms can:

Build large user bases quickly
Gather valuable feedback for product improvement
Create pathways for eventual premium conversions
Democratize access to advanced AI capabilities

The Competitive Landscape
Major technology companies are competing fiercely to attract users to their AI platforms, resulting in increasingly generous free offerings. This competition benefits users by providing access to sophisticated tools that would have cost thousands of dollars just a few years ago.

Top Categories of Free AI Tools Available Today
Content Creation and Writing
Modern AI writing assistants have revolutionized content creation by offering:
Blog Post Generation: Create engaging articles on virtually any topic with AI assistance that understands context, tone, and audience requirements.
Social Media Content: Generate compelling posts, captions, and hashtags optimized for different social media platforms.

Email Marketing: Craft persuasive email campaigns with personalized messaging and compelling subject lines.
Academic Writing: Receive assistance with research papers, essays, and academic projects while maintaining originality and proper citation practices.
Visual Content and Design
Free AI image generators have opened new possibilities for visual content creation:

Custom Artwork: Generate unique illustrations, paintings, and artistic pieces based on text descriptions
Logo Design: Create professional logos and branding materials for businesses and personal projects
Photo Editing: Enhance and modify existing images with AI-powered editing tools
Infographic Creation: Design informative visual content that effectively communicates complex information

Programming and Development
Developers can leverage free AI coding assistants for:
Code Generation: Transform natural language descriptions into functional code across multiple programming languages.
Bug Detection and Fixing: Identify and resolve coding errors more efficiently with AI-powered debugging assistance.
Code Optimization: Improve existing code for better performance, readability, and maintainability.
Learning Support: Receive explanations and tutorials for complex programming concepts and techniques.

Maximizing Your Free AI Tool Usage
Strategic Approach to Free Tiers
To get the most value from free AI services, develop a strategic approach:
Understand Usage Limits: Most platforms impose daily or monthly limits on free users. Plan your usage accordingly to maximize value.

Combine Multiple Platforms: Use different AI tools for different tasks to extend your overall capabilities without hitting individual platform limits.
Time Your Usage: Some platforms offer reset periods or bonus credits during off-peak hours or specific promotional periods.

Advanced Techniques for Better Results
Prompt Engineering: Learn to craft effective prompts that yield better results from AI tools. This includes being specific, providing context, and using clear instructions.
Iterative Refinement: Use the AI's output as a starting point and refine your requests based on initial results to achieve better outcomes.
Template Development: Create reusable templates and prompts for common tasks to streamline your workflow and maintain consistency.

Industry Applications and Use Cases
Small Business and Entrepreneurship
Entrepreneurs and small business owners are leveraging Grok Free tools to compete with larger organizations:
Marketing Materials: Create professional advertising content, product descriptions, and promotional materials without hiring expensive agencies.
Customer Service: Implement AI-powered chatbots and automated response systems to improve customer support efficiency.

Business Planning: Generate business plans, market analysis, and strategic documents with AI assistance.
Financial Modeling: Create financial projections and analyze business metrics using AI-powered analytical tools.
Education and Research
Educational institutions and individual learners benefit from free AI tools through:
Personalized Learning: AI tutors that adapt to individual learning styles and pace, providing customized educational experiences.
Research Assistance: Comprehensive literature reviews, data analysis, and hypothesis generation for academic projects.
Language Learning: Interactive conversation practice, grammar correction, and vocabulary building exercises.
Skill Development: Professional development courses and training materials generated specifically for career advancement.

Creative Industries
Artists, writers, and content creators are pushing creative boundaries with free AI assistance:
Music Composition: Generate original melodies, harmonies, and complete musical compositions across various genres.

Storytelling: Develop plot outlines, character development, and narrative structures for novels, screenplays, and other creative writing projects.
Game Development: Create game assets, storylines, and interactive elements for indie game developers working with limited budgets.
Technical Considerations and Best Practices
Quality Control and Verification
While AI tools are incredibly powerful, maintaining quality control is essential:

Always review and verify AI-generated content for accuracy and appropriateness
Cross-reference important information with reliable sources
Maintain human oversight in critical applications
Understand the limitations and potential biases of AI systems

Data Privacy and Security
When using free AI platforms, prioritize data protection:
Information Sharing: Be cautious about sharing sensitive personal or business information with AI platforms.
Account Security: Use strong passwords and enable two-factor authentication where available.
Terms of Service: Carefully review how platforms use and store your data and generated content.
Backup Strategies: Maintain copies of important AI-generated content on your own systems.

Future Trends and Developments
Emerging Technologies
The free AI landscape continues to evolve with exciting developments:
Multimodal AI: Tools that can process and generate text, images, audio, and video content simultaneously.
Specialized Industry Tools: AI platforms designed specifically for healthcare, legal, finance, and other professional sectors.

Improved Integration: Better connectivity between different AI tools and existing software ecosystems.
Enhanced Personalization: AI systems that learn and adapt to individual user preferences and working styles.
Market Predictions
Industry experts anticipate continued expansion of free AI offerings as competition intensifies and technology costs decrease. This trend suggests that users will have access to increasingly sophisticated tools without financial barriers.

Getting Started: Your First Steps
Recommended Approach for Beginners

Start Small: Begin with one or two AI tools that address your most immediate needs
Learn the Basics: Invest time in understanding how to effectively communicate with AI systems
Experiment Freely: Take advantage of free tiers to explore different capabilities and find what works best for you
Join Communities: Connect with other users to share tips, techniques, and discoveries
Stay Updated: Follow AI news and platform updates to discover new features and opportunities

Building Your AI Toolkit
As you become more comfortable with AI tools, gradually expand your toolkit to include specialized applications for your specific needs and interests.

Conclusion
The Grok Free movement represents more than just cost savings – it's about democratizing access to transformative technology that can enhance human creativity, productivity, and problem-solving capabilities. By understanding how to effectively leverage these tools, users can achieve professional-quality results across a wide range of applications without significant financial investment. As the AI landscape continues to evolve, those who master these free tools today will be well-positioned to take advantage of even more advanced capabilities in the future. The key is to start exploring, experimenting, and learning how AI can enhance your personal and professional endeavors while remaining mindful of best practices for security, privacy, and quality control.

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can you still get accepted if you dont get all the application questions right

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As many people (whether from the competitor side or the problem-writing side) get ready for the 2025-2026 school year, we often try to find nice and interesting problems to solve and hone our skills. Many of these problems are classics which have passed through years or generations. But this often leaves us to throw problems from recent years to the sides. While not all problems from any given year are great, there are some that do deserve praise and are undeservedly overlooked.

So let's fix this! The question: what were some of your favorite problems from competitions that occurred in the 2024-2025 school year?

Just to clarify: this is not necessarily asking for problems which you may have solved this year for practice. Instead, I'm talking about e.g. problems which came from contests hosted within the past 12 months - so for example AMC/AIME/USA(J)MO, ARML, PUMaC, HMMT, CMIMC, CHMMC, Duke, etc., as well as any national olympiads or TST's from this year. (2024 ISL problems also count since they were released to the public in 2025.) Problems from mocks released this year are eligible, too. Also, to make things easier to read, I highly recommend writing the original problem statements in your posts, as opposed to just stating the sources.

I also think this is a good way to officially end the 2024-2025 school year of contests. :)

I'm interested in seeing what some of your responses are!

is an acute triangle and its orthocenter is .
The circumcircle of intersects line at .
Lines and meets at , and the circumcenter of is .
Prove that the circumcenter of lies on the circumcircle of .

You are given a word consisting of letters You can apply 3 operations on this word:

1) You can write any letter long in reverse. ()

2) You can add same letters between any consecutive letters. (zz)

3) You can remove any in the given form (yzzy)

Given these operations, can you go from to ?

(Note: A is a string of consecutive letters from the given word)

Given . Prove that:

There are at least subway stations in a city.
In this city, there exists a route that passes through more than subway stations, without revisiting.
Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A.
Prove that at least one of the two holds.

. There exists three subway stations , , such that there does not exist a route from to which doesn't pass through .

. There is a cycle passing through at least stations, without revisiting a same station more than once.

If for . Prove that is an irrational number.
In the problem above is arbitrary sequence with elements and .

Q) prove that is countable.
Proof:- I have constructed a bijective function like such that

Now how to visualise this function in this given visualise way
https://youtube.com/shorts/M1dQNwyBjg0?si=cXOM9NR2rv3QIzYq

And extended to Z*Z
How to make guess for the function help please

Let be two continuous and strictly positive functions. We assume that .
Show that there exists a closed interval, such that

does there exist a differentiable function such that for all real x
f(f(x))+f(x)=x^3


recursive formula for power series of f(x) is obtainable, but chances are that the radius of convergence is 0.
or is there another way to prove only the existence of such function without explicit form solution?

Summary of the problem: Some error is preventing me from finishing my homework. My brother is encountering the same problem.
Page URL: Your class homework
Steps to reproduce:
1. Open the classes tab.
2. Open homework
3. A window pops up.
Expected behavior: You should be able to access your homework.
Frequency: 100% after 18:00 PST
Operating system(s): Windows 10
Browser(s), including version: Firefox (and my brother tried google)
Additional information: Screenshot below (cannot upload screenshot for some reason...). As a side note, recently, I've been experiencing extreme lag on AoPS that doesn't seen to occur with other websites.

Does anybody know all the codes for AOPS emojis?

Greetings, Site Support,

I've been working on a TeXeR file. However, all of a sudden, it stopped saving my work, or rendering it. No matter how many times I pressed the save button, I wasn't being notified of my work being saved. Upon reloading the page, it wouldn't render my work (don't worry, I didn't actually lose any of my work; I have it saved in a backup Google Document). The "Render With BBCode" button isn't working either.

Thank you. :)

After going onto the homework page of my introduction to geometry class, I found a strange message popping up, and I don't what in the world it means. It keeps showing up when I refresh it, and I've never seen it before. For a second I thought it stopped because it showed me week 17 homework, but when I switched to week 16, the message popped again, and now it's doing it for week 17 too. I was going to attach a screen shot, but now it's not letting me do that either, (whenever I click on the screen shot I want to upload it shows it's loading for a split second and then disappears, like I never chose a file) so maybe the whole site just really hates me. Anywho, I'm gonna paste in the message below instead, for anyone to try and decipher, because I am fully stumped.


(I'm gonna edit this part out bc there's already another discussion on it and I think it had my IP so better safe than sorry)