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  • Introduction To Number Theory: https://thepuzzlr.com/math-courses ...mp covering all the essential concepts: https://thepuzzlr.com/courses/amc-8-bootcamp/
    25 KB (3,445 words) - 20:21, 17 October 2021
  • ...+maths+kawasaki&qid=1581288606&sprefix=after+school+maths+%2Caps%2C268&sr=8-2 100 Challenging Maths Problems] * [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.
    22 KB (2,976 words) - 14:07, 15 October 2021
  • The U.S. National Chemistry Olympiad national exam (USNCO) is a 3-part, 4 hour and 45 minute exam administered in mid or late April by ACS Loc ...www.amazon.com/gp/product/0618221565/ref=pd_lpo_k2_dp_k2a_1_img/102-5655201-2084940?%5Fencoding=UTF8 ''Chemistry''] by Steven S. Zumdahl, Susan A. Zumda
    2 KB (253 words) - 11:19, 29 September 2019
  • ...b</math> if <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive. ...b</math> if <math>a</math> is smaller than <math>b</math>, that is, <math>a-b</math> is negative.
    12 KB (1,804 words) - 20:37, 11 June 2020
  • <math>\mathcal{S}</math> is <i>centre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</ma <ol style="list-style-type: lower-latin;">
    4 KB (692 words) - 22:33, 15 February 2021
  • ...wider population. Students who spend time studying maths can develop proof-writing skills over time. ...proof|two-column format]], as favored by many geometry teachers. In higher-level mathematics (taken as meaning an advanced undergraduate level of mathe
    3 KB (502 words) - 18:16, 18 January 2016
  • ...linear term on the other side. A extortive example would be: <cmath>xy+66x-88y=23333</cmath>where <math>23333</math> is the constant term, <math>xy</ma ...e previous example, <math>xy+66x-88y=23333</math> is the same as: <cmath>(x-88)(y+66)=(23333)+(-88)(66)</cmath>
    4 KB (631 words) - 13:42, 14 July 2021
  • The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary [[inequality]], and is generally one o AM-GM states that for any [[set]] of [[nonnegative]] [[real number]]s, the [[ar
    4 KB (562 words) - 01:30, 6 March 2021
  • ...equality''', and the '''Cauchy-Bunyakovsky-Schwarz Inequality''') is a well-known [[inequality]] with many elegant applications. It has an elementary fo [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy|Augustin Louis Cauchy] wrote the first paper about the elementa
    11 KB (1,952 words) - 08:43, 9 August 2021
  • ...ath> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>. ...> denotes [[Euler's totient function]]. In particular, <math>\varphi(p) = p-1</math> for prime numbers <math>p</math>. In turn, this is a special case o
    11 KB (1,827 words) - 02:02, 15 October 2020
  • ...imes(0,1)}\frac {x^ky^\ell}{1-xy}\,dx\,dy</math>. Expanding <math>\frac 1{1-xy}=\sum_{j=0}^\infty x^jy^j</math> and integrating term by term, we get <ma <math>I(k,\ell)=\frac 1{\ell-k}\left[\frac 1{k+1}+\frac 1{k+2}+\dots+\frac 1\ell\right]</math>
    8 KB (1,467 words) - 16:15, 31 December 2010
  • Here are some more precise statements for the single-variable and multi-variable cases. ===Multi-dimensional Chain Rule===
    12 KB (2,377 words) - 11:48, 22 July 2009
  • ...ected line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abst ...h (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at
    7 KB (1,265 words) - 13:22, 14 July 2021
  • 787 bytes (118 words) - 19:20, 23 October 2010
  • <center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> <cmath>a_nP_1 + a_{n-1} = 0</cmath>
    4 KB (633 words) - 02:58, 6 August 2021
  • <cmath> \lvert P(z) \rvert \ge \lvert z^n \rvert - (R-1) \lvert z^{n-1} \rvert = \lvert z^{n-1} \rvert \cdot \bigl[ \lvert z \rvert - (R-1) \bigr]
    5 KB (832 words) - 12:16, 15 July 2021
  • ...bda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...he left-hand side of the desired inequality but may contribute to the right-hand side.
    4 KB (774 words) - 12:12, 29 October 2016
  • Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to {| class="wikitable" style="text-align:center;width:100%"
    56 KB (6,752 words) - 23:43, 9 October 2021
  • ** [[Mock AIME 1 2006-2007]] ** [[Mock AIME 2 2006-2007]]
    4 KB (510 words) - 11:18, 11 June 2021
  • Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possi {{AIME box|year=2006|n=I|num-b=14|after=Last Question}}
    6 KB (911 words) - 14:48, 14 June 2021

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