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The Problem Solver's Resource

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Advanced Number Theory

These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions.

Jensen's Inequality

For a convex function $f(x)$ and real numbers $a_1,a_2,a_3,a_4\ldots,a_n$ and $x_1,x_2,x_3,x_4\ldots,x_n$ , the following holds:

$\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)$

Holder's Inequality

For positive real numbers $a_{i_{j}}, 1\le i\le m, 1\le j\le n be$ , the following holds:

$\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}$

Muirhead's Inequality

For a sequence $A$ that majorizes a sequence $B$ , then given a set of positive integers $x_1,x_2,\ldots,x_n$ , the following holds:

$\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}$

Rearrangement Inequality

For any multi sets ${a_1,a_2,a_3\ldots,a_n}$ and ${b_1,b_2,b_3\ldots,b_n}$ , $a_1b_1+a_2b_2+\ldots+a_nb_n$ is maximized when $a_k$ is greater than or equal to exactly $i$ of the other members of $A$ , then $b_k$ is also greater than or equal to exactly $i$ of the other members of $B$ .

Newton's Inequality

For non-negative real numbers $x_1,x_2,x_3\ldots,x_n$ and $0 < k < n$ the following holds:

$d_k^2 \ge d_{k-1}d_{k+1}$ ,

with equality exactly iff all $x_i$ are equivalent.

Mauclarin's Inequality

For non-negative real numbers $x_1,x_2,x_3 \ldots, x_n$ , the following holds:

$x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}$

with equality iff all $x_i$ are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition!

@@ Line 14: / Line 14: @@
 ===Holder's Inequality===
 For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n be</math>, the following holds:
-<math>
 <cmath>\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}</cmath>
 ===Muirhead's Inequality===
-For a sequence </math>A<math> that majorizes a sequence </math>B<math>, then given a set of positive integers </math>x_1,x_2,\ldots,x_n<math>, the following holds:
+For a sequence <math>A</math> that majorizes a sequence <math>B</math>, then given a set of positive integers <math>x_1,x_2,\ldots,x_n</math>, the following holds:
 <cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath>
 ===Rearrangement Inequality===
-For any multi sets </math>{a_1,a_2,a_3\ldots,a_n}<math> and </math>{b_1,b_2,b_3\ldots,b_n}<math>, </math>a_1b_1+a_2b_2+\ldots+a_nb_n<math> is maximized when </math>a_k<math> is greater than or equal to exactly </math>i<math> of the other members of </math>A<math>, then </math>b_k<math> is also greater than or equal to exactly </math>i<math> of the other members of </math>B<math>.
+For any multi sets <math>{a_1,a_2,a_3\ldots,a_n}</math> and <math>{b_1,b_2,b_3\ldots,b_n}</math>, <math>a_1b_1+a_2b_2+\ldots+a_nb_n</math> is maximized when <math>a_k</math> is greater than or equal to exactly <math>i</math> of the other members of <math>A</math>, then <math>b_k</math> is also greater than or equal to exactly <math>i</math> of the other members of <math>B</math>.
 ===Newton's Inequality===
-For non-negative real numbers </math>x_1,x_2,x_3\ldots,x_n<math> and </math>0 < k < n<math> the following holds:
+For non-negative real numbers <math>x_1,x_2,x_3\ldots,x_n</math> and <math>0 < k < n</math> the following holds:
 <cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>,
-with equality exactly iff all </math>x_i<math> are equivalent.
+with equality exactly iff all <math>x_i</math> are equivalent.
 ===Mauclarin's Inequality===
-For non-negative real numbers </math>x_1,x_2,x_3 \ldots, x_n<math>, the following holds:
+For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, the following holds:
 <cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath>
-with equality iff all </math>x_i$ are equivalent.
+with equality iff all <math>x_i</math> are equivalent.
 [[User:Temperal/The Problem Solver's Resource10|Back to page 10]] | Last page (But also see the
 [[User:Temperal/The Problem Solver's Resource Tips and Tricks|tips and tricks page]], and the
 [[User:Temperal/The Problem Solver's Resource Competition|competition]]!
 |}<br /><br />

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