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| + | {{User:Temperal/testtemplate|page 8}} |
| − | {| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
| + | ==<span style="font-size:20px; color: blue;">Geometry</span>== |
| − | |+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
| + | This is the geometry page. Of course, there is much too much to cover here, but this will review the basics. |
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| − | | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 8}}
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| − | ==<span style="font-size:20px; color: blue;">Intermediate Number Theory</span>== | |
| − | These are more complex number theory theorems that may turn up on the USAMO or Pre-Olympiad tests.
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| − | This will also cover diverging and converging series, and other such calculus-related topics. | |
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| − | ===General Mean Inequality===
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| − | Take a set of functions <math>m_j(a) = \left({\frac{\sum a_i^j}{n}}\right)^{1/j}</math>.
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| − | Note that <math>m_0</math> does not exist. The geometric mean is <math>m_0 = \lim_{k \to 0} m_k</math>.
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| − | For non-negative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following holds:
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| − | <math>m_x \le m_y</math> for reals <math>x<y</math>.
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| − | , if <math>m_2</math> is the quadratic mean, <math>m_1</math> is the arithmetic mean, <math>m_0</math> the geometric mean, and <math>m_{-1}</math> the harmonic mean. | |
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| − | ===Chebyshev's Inequality===
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| − | Given real numbers <math>a_1 \ge a_2 \ge ... \ge a_n \ge 0</math> and <math>b_1 \ge b_2 \ge ... \ge b_n</math>, we have
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| − | <math>{\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}</math>.
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| − | ===Minkowsky's Inequality===
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| − | Given real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following holds:
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| − | <math>\sqrt{\sum a_i^2} + \sqrt{\sum b_i^2} \ge \sqrt{\sum (a_i+b_i)^2}</math>
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| − | ===Nesbitt's Inequality===
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| − | For all positive real numbers <math>a</math>, <math>b</math> and <math>c</math>, the following holds:
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| − | <math>{\frac{a}{b+c}} + {\frac{b}{c+a}} + {\frac{c}{a+b}} \ge {\frac{3}{2}}</math>.
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| − | ===Schur's inequality===
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| − | Given positive real numbers <math>a,b,c</math> and real <math>r</math>, the following holds:
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| − | <math>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b)\ge 0</math>.
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| − | ===Fermat-Euler Identitity===
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| − | If <math>gcd(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relatively prime numbers lower than <math>m</math>.
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| − | | |
| − | ===Gauss's Theorem===
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| − | If <math>a|bc</math> and <math>(a,b) = 1</math>, then <math>a|c</math>.
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| − | ===Power Mean Inequality===
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| − | For a real number <math>k</math> and positive real numbers <math>a_1, a_2, ..., a_n</math>, the <math>k</math>th power mean of the <math>a_i</math> is
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| − | <math>M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}</math>
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| − | when <math>k \neq 0</math> and is given by the geometric mean]] of the
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| − | <math>a_i</math> when <math>k = 0</math>.
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| − | ===Diverging-Converging Theorem===
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| − | A series <math>\sum_{i=0}^{\infty}S_i</math> converges iff <math>\lim S_i=0</math>.
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| − | ===Errata===
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| − | All quadratic residues are <math>0</math> or <math>1\pmod{4}</math>and <math>0</math>, <math>1</math>, or <math>4</math> <math>\pmod{8}</math>.
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| | + | This will be completed later. |
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| | [[User:Temperal/The Problem Solver's Resource7|Back to page 7]] | [[User:Temperal/The Problem Solver's Resource9|Continue to page 9]] | | [[User:Temperal/The Problem Solver's Resource7|Back to page 7]] | [[User:Temperal/The Problem Solver's Resource9|Continue to page 9]] |
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