Ormen Lange: Pipe Rider

The Ormen Lange Pipe Rider offers thrilling races through underwater tunnels that were specifically built for travel along the ocean floor to England! Along the way, you will encounter various obstacles that you must either navigate around or eliminate using the weapons at your disposal on board. With its 3D graphics, user-friendly controls, and numerous bonuses to aid you on your journey, be sure to keep an eye on the clock as time runs out if you move too slowly!<|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 \[2^{2015}-2^{2013}+2^{2011}?\] $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2^{2011}\qquad\textbf{(C)}\ 2^{2013}\qquad\textbf{(D)}\ 2^{2014}\qquad\textbf{(E)}\ 2^{2015}$ ## Solution. We can factor out $2^{2011}$ from the expression to get \[2^{2015}-2^{2013}+2^{2011}=2^{2011}(2^4-2^2+1)=2^{2011}(16-4+1)=2^{2011}\cdot 13=\boxed{\textbf{(C)}\ 2^{2013}}.\] ## Video Solution. https://youtu.be/8WrdYLw9_ns ~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 $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+3\geq 2(a^2+b^2+c^2)$ ## [enter: math-user2, num_posts=467, num_likes_received=180] ## [math-user2, num_likes=0] By AM-GM, $\frac{a}{b}+\frac{a}{b}+b\ge3a$. Summing cyclically, we get $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge a+b+c=3$. So we need to prove $3+3\ge2\left(a^2+b^2+c^2\right)$ or $a^2+b^2+c^2\le3$. By Cauchy, $\left(a^2+b^2+c^2