Math olympiad books

https://youtu.be/mScpHTIi-kM?si=FIGtRAMeKuFjduXd

😉 https://youtu.be/1TewCPi92ro

https://artofproblemsolving.com/community/c2771h1181729_isogonal_line_lemma_updated

https://drive.google.com/drive/folders/1wHcrzqwhYnXucJ9CPC8X37q2R7FosuLj?usp=sharing

https://blog.evanchen.cc/2017/03/06/on-reading-solutions/

https://blog.evanchen.cc/2019/01/31/math-contest-platitudes-v3/ About olympiad preparation

https://artofproblemsolving.com/community/q1h2120852p15430373

prime-divisors-special.pdf

This principle can also be extended to our eating habits. From an evolutionary perspective, humans are instinctively hunter-gatherers. Our ancestors did not consume meals regularly in constant portions like herbivores. They would often consume large portions when food was available and might go without food for extended periods when nothing could be found. Similarly, in the animal kingdom, beasts hunt when they are hungry, not for the sake of eating regularly. In this context, the idea of consuming a daily portion of seafood may not necessarily be the most beneficial. Instead, it might be more advantageous to have a weekly portion of seafood once a week, as per Jensen's inequality. Therefore, it is perhaps not ideal to eat regularly when we are not hungry, but rather to align our eating habits with our instinctual nature and the principles of convexity.and it also concerns to dine too .We are instinctively hunter-gatherers look at our ancestries ( again ancestries) they did not consume food regular in constant portion instead once in a lot of portions and after that may be not found nothing to eat during a week .Consuming a daily portion of sea food is not as beneficial as eating weekly portion once a week ( again from jensen`s inequality ) that`s why it is not good to eat reguar in small portions when you are not hungry . Like that regular small portions of stress and feeling of anxiety (often due to social media ) that impacts to our mental health very badly than one big portion of stress like attack of beast or snake from under the stone as it was before on our ancestries or nowadays example it can be job loss ,divorce , death of the near person etc . there are tons of examples , that is how life is it is non-linear , i tried to familiarize main of them ,i know many of you thought at the beginning how there is a connection with this simple inequality and life, me too before .But there are more intesersting situations in our daily life connected with jensen`s inequality ( like finances ) yet , may be i will do 2 nd part of this post but those who interested let me introduse Taleb Nassim trader, risk analyst and essayist that his works involves randomness and probability his books impacted me to write this post and everone recommend to read his book especially "antifragile".

Today i want to talk about relationship of jensen inequality with convex functons and our daily life.Jensen's inequality indeed has connections to the real world and can be observed in various aspects of our daily lives. Many objects and processes exhibit asymmetries and non-linear relationships. In the context of convexity, we can define an event or variable as convex when the benefit derived from its growth is non-linear. Conversely, an event or variable is concave when the benefit reduces non-linearly as the scale increases. For example, the results from workouts can be considered concave. If you do 10 push-ups after a week of workouts, you will probably be able to do around 15-16 push-ups after two weeks, not two times more and after three weeks progress will likely be much less than 30. Similarly, throwing a cup 10 times from a height of 20 cm will cause less damage compared to throwing it once from a height of 2 meters. Here, the cup exhibits concavity with respect to the scale of events and variables, which aligns with Jensen's inequality. To delve deeper into this topic, it is important to recognize that significant changes often occur at the limits. Just like the high temperature required to shape a sword perfectly, pushing limits and consistently striving for success are necessary for bringing about changes. From a physiological standpoint, it is interesting to compare our current societal norms with the habits of our ancestors. Our ancestors, in primitive societies, were often mentally and physically healthier. They did not have the need to run for long distances at average speeds. Instead, they walked long distances at a slow pace, often for hunting purposes. Running at maximum speed was typically reserved for situations when they needed to fend off dangerous beasts. However, as our society and lifestyle have evolved, our bodies have not adapted accordingly. So, instead of regularly running for shorter durations at an average speed, you have adopted a different approach. By walking more and designating some days as "marathon" days, where you walk for a very long time and occasionally sprint as fast as you can, you have noticed significant changes in your health. Following these intense sessions, you lead a mostly sedentary lifestyle for 2-3 days or sometimes even a week. This change in routine aligns with the concept of convexity, as our bodies often exhibit a convex response to the surrounding stimuli, as per Jensen's inequality.

What do you know about jensen's inequality especially about this case and about concave/convex functions ?

Thanks to all who maintain the channel and found benefical and we continue

Wow there are 610 people on this channel . I had some olympiad books and I created this channel just for to save my olympiad books and I almost not invited no-one to this channel . I almost can not did high quality content for audience on this channel but I can if you have an interest , there are several reasons for it main of them is time . I am lately not using social networks ( except rare moments) and I have no sufficiently time for this channel too so

Combinatorics of sets

One of the best bots on discord. It choose from 3 to 6 problems from IMO shortlists randomly to solve . Solved problems can mark with /mark command and it also shows statistics. When write slash (/) all commands will pop-up .

https://artofproblemsolving.com/community/c5h2242638p17212597