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- Up to 2021, the top 12 scorers on the [[USAMO]] were designated as USAMO winners. In early years, this number was smaller. Starting 2022, the **In Sung Na10 KB (1,317 words) - 08:16, 23 April 2024
- ...as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything after the decimal point," but this is ''not'' true for * [[Hermite's Identity]]: <cmath>\lfloor na\rfloor = \left\lfloor a\right\rfloor+\left\lfloor a+\frac{1}{n}\right\rfloo3 KB (508 words) - 21:05, 26 February 2024
- ...c structure called '''the integers modulo <math>n</math>''' (usually known as "the integers mod <math>n</math>," or <math>\mathbb{Z}_n</math> for short). ...h> the third. The symbol <math>\overline{3}</math> refers to the same set as <math>\overline{0}</math>, and so on.14 KB (2,317 words) - 19:01, 29 October 2021
- ...proofs of the weighted [[AM-GM Inequality]]. The inequality's statement is as follows: for all nonnegative reals <math>a_1, \dotsc, a_n</math> and nonneg ...we may disregard any <math>a_i</math> for which <math>\lambda_i= 0</math>, as they contribute to neither side of the desired inequality. We also note tha12 KB (2,171 words) - 07:55, 11 May 2023
- ...(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)] ...coordinate plane at <math>(5,0)</math>. Define a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin7 KB (1,166 words) - 21:56, 28 June 2024
- ...(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)] &= -\sin(0) + \sin(2a)- \sin(2a) + \sin(6a) - \cdots - \sin((n-1)na) + \sin(n(n+1)a)\\4 KB (621 words) - 18:26, 16 January 2023
- Since <math>\widehat{MN}\equiv \widehat{NA}</math>, it follows that ...(\widehat{NA} + \widehat{AS}) \equiv \tfrac{1}{2} (\widehat{MN} + \widehat{AS}) \equiv \angle MIS \equiv \angle I_a IS . </cmath>3 KB (437 words) - 15:47, 27 April 2008
- ...</math> but it does not imply that for every <math>n\in N</math>, <math>an=na</math>. Since any element of <math>{\rm H}</math> can be expressed as <math>xy^{-1}</math>, the statement "<math>{\rm H}</math> is normal in <mat15 KB (2,840 words) - 12:22, 9 April 2019
- As <math>NV + MV =MW + NW = 1</math> we compute <math>NW = \frac{1}{1+\frac{3} ...es the quadratic inequality <math>d^2 - 14d + 1 \ge 0</math>, which solves as<cmath>d \in \left(-\infty, 7-4\sqrt3\right] \cup \left[7+4\sqrt3, \infty\ri11 KB (1,862 words) - 21:23, 23 May 2024
- .../math>, and <math>UB = 168</math>. Then <math>DA</math> can be represented as <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positi As in the above solutions we discover that <math>\angle AON = 60^\circ</math>,10 KB (1,418 words) - 23:05, 20 October 2021
- ...have the ability to attach images, we do not generally recommend doing so, as we can not guarantee the images will be available through upgrades, restora ...of problems in the topic the Alcumus engine believes you will get correct. As you get more and more correct, the rating will go up slower and slower. How25 KB (4,192 words) - 22:38, 12 August 2024
- ...n(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0</math>, show that <math>x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0</math> for any positive integer <math>n</math ...th>\frac{1}{2}\left((a+b+c)^2-(a^2+b^2+c^2)\right)=ab+bc+ac</math> is real as well.2 KB (347 words) - 11:20, 6 May 2017
- Let <math>BN=x</math> and <math>NA=y</math>. From the conditions, let's deduct some convenient conditions that ...des by <math>AM</math>. Doing so will make it so that <math>AM = 1</math>, as desired, and doing so will allow us to get the length of <math>BN</math>, w11 KB (1,876 words) - 00:08, 12 October 2023
- ...ngth of the line segment along which the triangle is folded can be written as <math>\frac{m\sqrt{p}}{n}</math>, where <math>m</math>, <math>n</math>, and Proceed with the same labeling as in Solution 1.11 KB (1,778 words) - 20:30, 11 August 2024
- ...g <math>a=1</math> and <math>r=\frac12</math>, we can rewrite the sequence as <math>a, a+r, \dots, a+(n-1)r</math>. ...e sum of the first <math>n</math> terms of an arithmetic sequence is <math>na+\binom{n}2r</math>.905 bytes (157 words) - 20:50, 31 May 2018
- ...ath>r=BO</math>. Drawing perpendiculars, <math>BM=MC=2</math> and <math>BN=NA=2.5</math>. From there, <cmath>OM=\sqrt{r^2-4}</cmath> Thus, <cmath>OQ=\fra ...el lines. From here, <math>\angle P'PB = \angle PP'B = \angle A'P'Q</math> as <math>P, P', Q</math> collinear. From here, <math>A'P'QC</math> is cyclic,9 KB (1,249 words) - 21:59, 26 November 2023
- ...+DX</math>. The least possible value of <math>f(X)</math> can be expressed as <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positiv ...e CDA\cong\triangle DCB</math>, and therefore <math>MC=MD</math> and <math>NA=NB</math>. It follows that <math>M</math> and <math>N</math> both lie on th6 KB (971 words) - 02:08, 22 January 2024
- ...that if <math>a=0</math> then <math>g(x)=g(x)+b</math> to <math>b=0</math> as well, so similarly if <math>b=0</math> then <math>a=0</math>, so now assume ...x-a</math>, so <math>|a|=|b|</math> does work and are the only solutions, as desired.4 KB (816 words) - 18:53, 7 April 2021
- ...math> excenter. Denote the B excenter as <math>I_B</math> and the incenter as <math>I</math>. ...excenter lemma, <math>II_B=2IM</math> so <math>II_b=\frac{2b*IB}{a}</math> as desired.21 KB (3,915 words) - 19:55, 10 October 2023
- ...b} = n</math> means that the ordered pair can be rewritten as <math>(a,b,c,na,nb,nc)</math>. ...he ordered pairs that are solutions are <math>\boxed{(0,0,0,x,y,z), (a,b,c,na,nb,nc)}</math>.2 KB (273 words) - 13:34, 17 December 2019