User superagh olympiad notes
SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE UPDATING THIS AS I GO THOUGH :)
Ok, so inspired by master math solver Lcz, I have decided to take Oly notes (for me) online! I'll probably be yelled at even more for staring at the computer, but I know that this is for my good. (Also this thing is almost the exact same format as Lcz's :P ). (Ok, actually, a LOT of credits to Lcz)
Problems worth noting/reviewing I'll leave this empty for now, I want to start on HARD stuff yeah!
We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems.
Power mean (special case)
Statement: Given that , where . Define the as:where , and:where .
If , then Power mean (weighted) Statement: Let be positive real numbers. Let be positive real numbers ("weights") such that . For any ,
If , then . Equality occurs if and only if all the are equal.
Let there be two sets of integers, and , such that is a positive integer, where all members of the sequences are real, then we have:Equality holds if for all , where , , or for all , where , ., or we have some constant such that for all .
Given that , are real numbers such that and , we have: Rearrangement Inequality Given thatandWe have:is greater than any other pairings' sum.
If , , , are nonnegative real numbers and are nonnegative reals with sum of , then:This is a generalization of the Cauchy Swartz Inequality.