Challenging Romanian Olympiad Problem Solved In 4 Ways

How can we sum square roots without using a calculator? 0:00 problem 1:14 method 1 10:50 method 2 18:18 method 3 23:53 method 4 @arthursmith482 notes a typo around time 16:50: (b+x) is written as (b-x) in method 2 Prime Newtons video (proof 1 algebraic) https://www.youtube.com/watch?v=Cqlpr0W0-34 proof 1 https://graphicmaths.substack.com/p/sum-of-square-roots-romanian-olympiad comment for proof 2 https://www.youtube.com/watch?v=Cqlpr0W0-34&lc=Ugz4Wlm0cewY86byMKZ4AaABAg comment for proof 3 https://www.youtube.com/watch?v=Cqlpr0W0-34&lc=UgwsIPRsfanZBwyQGml4AaABAg proof 4 references https://en.wikipedia.org/wiki/QM%E2%80%93AM%E2%80%93GM%E2%80%93HM_inequalities https://www.desmos.com/calculator/suv9cf31pr Help me make more videos! A monthly membership directly supports the channel and allows me to make more videos that inspire students and teachers around the world. You also get early access to new videos. Support with a YouTube membership https://www.youtube.com/channel/UCHnj59g7jezwTy5GeL8EA_g/join Support with a Patreon membership https://www.patreon.com/mindyourdecisions Want to support with a tip or one-time donation? Every "coffee" powers me to make more videos. As they say, a mathematician is a machine for turning caffeine into theorems! Send a tip on Ko-Fi https://ko-fi.com/mindyourdecisions Send a tip on PayPal https://www.paypal.com/donate/?hosted_button_id=REZRY3QRQPYQA Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas! If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. Book ratings are from January 2025. My Books (worldwide links) https://mindyourdecisions.com/blog/my-books/#worldwide My Books (US links) Mind Your Decisions: Five Book Compilation https://amzn.to/2pbJ4wR A collection of 5 books: "The Joy of Game Theory" rated 4.2/5 stars on 564 reviews http://amzn.to/1uQvA20 "The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4.2/5 stars on 55 reviews http://amzn.to/1o3FaAg "40 Paradoxes in Logic, Probability, and Game Theory" rated 4.2/5 stars on 81 reviews http://amzn.to/1LOCI4U "The Best Mental Math Tricks" rated 4.3/5 stars on 148 reviews http://amzn.to/18maAdo "Multiply Numbers By Drawing Lines" rated 4.5/5 stars on 57 reviews http://amzn.to/XRm7M4 Mind Your Puzzles: Collection Of Volumes 1 To 3 https://amzn.to/2mMdrJr A collection of 3 books: "Math Puzzles Volume 1" rated 4.4/5 stars on 138 reviews http://amzn.to/1GhUUSH "Math Puzzles Volume 2" rated 4.2/5 stars on 45 reviews http://amzn.to/1NKbyCs "Math Puzzles Volume 3" rated 4.3/5 stars on 38 reviews http://amzn.to/1NKbGlp 2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter. My Blog https://mindyourdecisions.com/blog/ Twitter https://twitter.com/preshtalwalkar Instagram https://www.instagram.com/preshtalwalkar/ Merch https://teespring.com/stores/mind-your-decisions Patreon https://www.patreon.com/mindyourdecisions Press https://mindyourdecisions.com/blog/press