In a chess tournament with n ≥ 5 players, each player played all other players. One gets a point for a
win, half a point for a draw, and zero points for a loss. At the end of the tournament, each player had
a different number of points. Prove that the second and third ranked players had together more points
than the winner of the tournament.

Let be a positive integer. There are circles drawn on a chalkboard such that any two circles intersect at distinct points and no circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd there exists at least one configuration of circles with a single snail orbit, and find all such that there is exactly one of the aforementioned configuration type.

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.

Determine all integers such that for any two infinite sequences of positive integers and , such that for all , there always exists a real number such that for all .

Proposed by Wong Jer Ren & Ivan Chan Kai Chin