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| − | ==Introduction==
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| − | SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE PUTTING THEM ON OVERLEAF FOR NOW, MAYBE I'LL POST A LINK OR SOMETHING WHEN I'VE MADE SOME PROGRESS.
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| − | 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)
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| − | ==Algebra==
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| − | ===Problems worth noting/reviewing===
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| − | I'll leave this empty for now, I want to start on HARD stuff yeah!
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| − | ===Inequalities===
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| − | We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems.
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| − | ====Power mean (special case)====
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| − | Statement: Given that <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \mathbb{R}</math> where <math>1 \le i \le n</math>. Define the <math>pm_x(a_1, a_2, \cdots , a_n)</math> as: <cmath>(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},</cmath> where <math>x\neq0</math>, and: <cmath>\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.</cmath> where <math>x=0</math>.
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| − | If <math>x \ge y</math>, then <cmath>pm_x(a_1, a_2, \cdots , a_n) \ge pm_y(a_1, a_2, \cdots , a_n).</cmath>
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| − | ====Power mean (weighted)====
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| − | Statement: Let <math>a_1, a_2, a_3, . . . a_n</math> be positive real numbers. Let <math>w_1, w_2, w_3, . . . w_n</math> be positive real numbers ("weights") such that <math>w_1+w_2+w_3+ . . . w_n=1</math>. For any <math>r \in \mathbb{R}</math>,
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| − | if <math>r=0</math>,
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| − | <cmath>P(r)=a_1^{w_1} a_2^{w_2} a_3^{w_3} . . . a_n^{w_n}</cmath>.
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| − | if <math>r \neq 0</math>,
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| − | <cmath>P(r)=(w_1a_1^r+w_2a_2^r+w_3a_3^r . . . +w_na_n^r)^{\frac{1}{r}}</cmath>.
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| − | If <math>r>s</math>, then <math>P(r) \geq P(s)</math>. Equality occurs if and only if all the <math>a_i</math> are equal.
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| − | ==Combinatorics==
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| − | ==Number Theory==
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| − | ==Geometry==
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