Kyport: Portals. Dimensions

Welcome to Kyport, where you will explore various portals and dimensions in order to collect the essential green matter needed to restore balance to the universe. Your goal is to gather all the green matter in each dimension as quickly as possible. To aid in your journey, you can create portals on flat surfaces to swiftly travel between two points. You also have the ability to open portals to parallel dimensions. However, be prepared to face numerous obstacles and traps in this Android game. Use your skills to move objects and avoid deadly laser beams in order to progress through the 20 challenging levels. With its vibrant graphics, diverse puzzles, and captivating gameplay, Kyport is sure to keep you entertained.<|endoftext|><|endoftext|> # 2015 AMC 10A Problems/Problem 1. ## Contents. 1 Problem 2 Solution 3 Video Solution 4 See Also ## Problem What is the value of \[2-4+6-8+10-12+14-16+18-20?\] $\textbf{(A)}\ -10\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 10$ ## Solution. We can rearrange the terms to get \[2+6+10+14+18-4-8-12-16-20.\] Now, we can pair up the terms to get \[(2-2)+(6-4)+(10-8)+(14-12)+(18-16)-20.\] Simplifying, we get \[0+2+2+2+2-20=10-20=\boxed{\textbf{(A)}\ -10}.\] ## Video Solution. https://youtu.be/8-3XeJXm-q0 ~savannahsolver <|endoftext|>## Mathematical Forums ## Category: High School Olympiads ## Topic: Inequality ## Views: 338 ## [enter: math-user1, num_posts=697, num_likes_received=372] ## [math-user1, num_likes=1] Let $a,b,c>0$. Prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+3\sqrt[3]{abc}\geq 2(a+b+c)$ ## [enter: math-user2, num_posts=545, num_likes_received=72] ## [math-user2, num_likes=0] By AM-GM have that $\frac{a^2}{b}+\frac{a^2}{b}+b\ge3a$ and similar for others so we have that $\sum_{cyc}\frac{a^2}{b}+a+b+c\ge3(a+b+c)$ $\implies