Let such that . Prove that

A point lies inside a triangle on the bisector of angle . Let and be the circles touching and at and passing through ; and be the common points of and with the circumcircle of distinct from . Prove that the circumcircles of the triangles and are tangent.
Proposed by: L Shatunov

Let a,b,c be real numbers. Prove that a^6+b^6+c^6\ge 2(a+b+c)(ab+bc+ca)(a-b)(b-c)(c-a).

Hello Aops! Since IMO 2024 is less than a week away,
What are your predictions for the category of each problem at IMO 2024?

If you want to write your prediction, please write it in the form ABC DEF
Where A,B,C,D,E,F are problems 1,2,3,4,5,6 respectively. Each letter should be A,C,G or N.

Rules:
1. Problems 1,2,4,5 are distinct categories.
2. Each day consists of 3 distinct categories.

Edit : the answer is ANC GCA

Consider the problem of expressing as , where are integers. Determine the number of ways to represent in this form and prove your conclusion.

Find the largest positive real number such that for any positive integer , satisfies .

Determine all functions such that for all real numbers and

Let be an integer. Let , , ..., be positive real numbers such that Show that , , are side lengths of a triangle for all , , with .

Fix positive integers and , and positive (not neccesarily distinct) real numbers , . An equation is written on a whiteboard: where is a fixed positive real number, with exactly asterisks.

Ebi fills each asterisk with a number from , while Rubi fills each asterisk with a number from , so that the equation on the whiteboard is correct. Suppose for every positive real number , the number of ways for Ebi and Rubi to do so are equal.

Prove that the sequences and are permutations of each other.

(Note: and are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example .)

Proposed by Wong Jer Ren

Hello Math Enthusiasts!

I'm excited to announce a series of free Math Olympiad Workshops designed to help you sharpen your problem-solving skills in preparation for competitions. Whether you're a beginner or a seasoned competitor, these workshops aim to provide a supportive, challenging, and collaborative environment to explore advanced math topics.

Workshop Overview

Duration: 6 months (with the possibility of extending based on participant interest)

Structure: Weekly cycles, each dedicated to one of the main areas of Math Olympiad:
Week 1: Number Theory
Week 2: Geometry
Week 3: Algebra
Week 4: Combinatorics

Weekly Format
Monday: Problem Set Release: Approximately 30 problems will be posted covering the week's topic, which you will have chance to discuss.
Throughout the Week:
Theory Notes: I will share helpful theory and insights relevant to the problem set, giving you the tools you need to approach the problems.
Submission Opportunity: You can work on the problems and submit your solutions. I’ll review your work and provide feedback.
End of the Week: Solutions Post: I’ll release detailed solutions to all problems from the problem set.
Leaderboard: For those interested, we can maintain a table tracking participants who solve the most problems during the week.

Cycle Finale – Mock Contest
At the end of each 4-week cycle, we’ll host a Mock Contest featuring 4 problems (one from each topic). This is a great chance to simulate the competition environment and test your skills in a timed setting. I will review and provide feedback on your contest submissions.

Starting date: June 2

How to participate? Just write /signup under this post.

I believe these workshops will provide a comprehensive, engaging, and collaborative way to tackle Math Olympiad problems. I'm looking forward to seeing your creativity and problem-solving prowess!
If you have any questions or suggestions, please leave a comment below.

Circles and with centres and are externally tangent at point and internally tangent to a circle at points and respectively. Line is the common tangent of and at . Let be the diameter of perpendicular to , so that are on the same side of . Prove that lines , , and are concurrent.