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• Number theory/Advanced
  ...e can study the set of points where both coordinates lie in some subfield (like the reals or the rationals). One also needs to add a limit point, called th ...ve, they can be added in a way that satisfies the normal laws of addition, like associativity, commutativity and the existence of an identity and inverses.
  5 KB (849 words) - 16:14, 18 May 2021
• Ellipse
  ...an ellipse with semimajor and semiminor axes <math>a,b</math> is <math>ab\pi</math>. ...re relatively prime positive integers. Find <math>m+n</math>. ([[2001 AIME I Problems/Problem 5|Source]])
  5 KB (892 words) - 21:52, 1 May 2021
• Pi
  ...mber]], as proved by Lindemann in 1882) denoted by the Greek letter <math>\pi </math>. ...^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for pi are <math>\frac{22}{7} \approx 3.14285</math> and <math>\frac{355}{113} \ap
  8 KB (1,469 words) - 21:11, 16 September 2022
• Logarithm
  ..., we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>.
  4 KB (680 words) - 12:54, 16 October 2023
• Roots of unity
  ...as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik}</math>. ...math> ni\theta = 2\pi ik</math>. Solving this gives <math> \theta=\frac{2\pi k}n </math>. Additionally, we note that for each of <math> k=0,1,2,\ldots,
  3 KB (558 words) - 21:36, 11 December 2011
• Complex analysis
  <cmath> f^{(n)}(z_0)=\frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}. </cmath> Let <math>f</math> be an [[entire]] function (i.e. holomorphic on the whole complex plane). If <math>\lvert f(z)\rvert \le
  2 KB (271 words) - 22:06, 12 April 2022
• Riemann Hypothesis
  ...th>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>. ...tion]] to <math>\Re(s)>\frac{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\
  2 KB (425 words) - 12:01, 20 October 2016
• Algebraic number
  ...ic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</math>.
  1,006 bytes (151 words) - 21:56, 22 April 2022
• Prime Number Theorem
  specifically, it states that the functions <math>\pi(x)</math> and <math>x/\log x</math> are [[asymptotically equivalent]], where <math>\pi(x)</math> is the number
  10 KB (1,729 words) - 19:52, 21 October 2023
• LaTeX:Symbols
  ...texify] is an online application which allows you to draw the symbol you'd like and shows you the <math>\text{\LaTeX}</math> code for it! |<math>\pi</math>||\pi||<math>\varpi</math>||\varpi||<math>\rho</math>||\rho||<math>\varrho</math>
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2021 JMC 10
  ...can <math>|\pi - |e - | e - \pi|||</math> be expressed in terms of <math>\pi</math> and <math>e?</math> ...}2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2004 AMC 12A Problems
  (i) <math>f(1) = 1</math>, and ...rt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \text {(E) }4\pi - 2\sqrt3</math>
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2001 AMC 12 Problems
  ...that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</math>. Furthermore, ...</math> are consecutive even digits; <math>G</math>, <math>H</math>, <math>I</math>, and <math>J</math> are consecutive odd
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  ...{C}) 75+100\pi \quad (\mathrm {D}) 100+100\pi \quad (\mathrm {E}) 100+125\pi</math> ...hrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  \mathrm{(A)}\ 80-20\pi \qquad \mathrm{(B)}\ 60-10\pi \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2005 AMC 12B Problems/Problem 22
  ...z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</ma Since <math>|z_0|=1</math>, let <math>z_0=e^{i\theta_0}</math>, where <math>\theta_0</math> is an [[argument]] of <math>z_
  4 KB (660 words) - 17:40, 24 January 2021
• 2006 AMC 10A Problems
  MP('I', (8,-8), (0,0)); ...qquad\mathrm{(D)} \text{II, by}\ 8\pi\qquad\mathrm{(E)}\ \text{II, by}\ 10\pi</math>
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 12
  MP('I', (8,-8), (0,0)); ...{(D) } II,\,\textrm{ by }\,8\pi\qquad \textbf{(E) } II,\,\textrm{ by }\,10\pi </math>
  3 KB (424 words) - 10:14, 17 December 2021
• Newman's Tauberian Theorem
  <cmath> F(0)-F_T(0) = \frac{1}{2\pi i} \int_\Gamma K(z) (F(z)-F_T(z)) \le\int_{\Gamma_+} \frac{2M}{R^2} \lvert dz \rvert = \frac{2\pi M}{R} . </cmath>
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2005 AIME II Problems/Problem 9
  ...or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>? ...al number]]s <math>t</math> and all [[integer]]s <math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...ying using the fact that <math>e^{i \frac{\pi}{4}} = \dfrac{\sqrt{2}}{2} + i \dfrac{\sqrt{2}}{2}</math>, clearing the denominator, and setting the imagi I hope you like expanding
  13 KB (2,080 words) - 21:20, 11 December 2022
• Euler's identity
  '''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is named after the 18th-century mathematician [[Leon ...eta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)</math>.
  3 KB (452 words) - 23:17, 4 January 2021
• 2005 AIME I Problems/Problem 1
  [[Image:2005 AIME I Problem 1.png]] ...ath>. <math>K = 30^2\pi - 6(10^2\pi) = 300\pi</math>, so <math>\lfloor 300\pi \rfloor = \boxed{942}</math>.
  1 KB (213 words) - 13:17, 22 July 2017
• 2005 AIME I Problems/Problem 6
  ...urth root of 2006. Then the roots of the above equation are <math>x = 1 + i^n r</math> for <math>n = 0, 1, 2, 3</math>. The two non-real members of th Starting like before,
  4 KB (686 words) - 01:55, 5 December 2022
• 2004 AIME I Problems/Problem 14
  pair Cxy = 8*expi((3*pi)/2-CE/8); pair L = 8*expi(pi+0.05), R = 8*expi(-0.22); /* left and right cylinder lines, numbers from tr
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  ...4</math> complex roots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \l {{AIME box|year=2004|n=I|num-b=12|num-a=14}}
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 11
  ...the [[Pythagorean Theorem]], we get <math>\ell = 5</math> and <math>A = 24\pi</math>. ...ace area of the original solid, we find that <math>A_f=24\pi - \frac{5}{3}\pi x^2</math>.
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 4
  ...ed by all of the midpoints is <math>4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86</math> to the nearest hundredth. Thus <math>100\cdot k=\boxed{8 ...rea of the square minus the area of the quarter circles, which is <math>4-\pi \approx 0.86</math>, so <math>100k = \boxed{086}</math>. ~Extremelysupercoo
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems
  {{AIME Problems|year=2004|n=I}} [[2004 AIME I Problems/Problem 1|Solution]]
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems
  ...to the smaller can be expressed in the form <math> \frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}, </math> where <math> a, b, c, d, e, </math> and <math> f </math {{AIME box|year = 2004|n=II|before=[[2004 AIME I Problems]]|after=[[2005 AIME I Problems]]}}
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1989 AIME Problems
  ...up heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j^{}_{}</math>. pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1991 AIME Problems
  ...th>x^{}_{}</math> satisfy the equation <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? ...The sum of the areas of the twelve disks can be written in the form <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers a
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1995 AIME Problems
  ...> are the perpendicular bisectors of two adjacent sides of square <math>S_{i+2}.</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, ...math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math>
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1998 AIME Problems
  Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> ...,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math> Find <math>\frac n{100}.</math>
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2003 AIME I Problems
  {{AIME Problems|year=2003|n=I}} [[2003 AIME I Problems/Problem 1|Solution]]
  6 KB (965 words) - 16:36, 8 September 2019
• 2002 AIME II Problems
  ...math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q< {{AIME box|year = 2002|n=II|before=[[2002 AIME I Problems]]|after=[[2003 AIME I Problems]]}}
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...e log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. Find <math>n</math>. ...istinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>a_{k}</math> and <mat
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 15
  ...the minor arc <math>AD</math>, we see that <math>\alpha \in \left(-\frac{\pi}{2}-\alpha_0,\alpha_0\right)</math>. However, since the midpoint of <math>A Let I be the intersection of AD and BC.
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 13
  Recall that <math>\cot^{-1}\theta = \frac{\pi}{2} - \tan^{-1}\theta</math> and that <math>\arg(a + bi) = \tan^{-1}\frac{b ...{2} - \arg x + \frac{\pi}{2} - \arg y + \frac{\pi}{2} - \arg z) = 10\cot(2\pi - \arg wxyz)</cmath>
  3 KB (473 words) - 12:06, 18 December 2018
• 1985 AIME Problems/Problem 12
  ...can generalize the following relationships for all <i><b>nonnegative</b></i> integers <math>k:</math> Let <math>\omega = e^{i\pi / 2}</math>. We have that if <math>G(x) = (x+x^2+x^3)^7</math>, then <cmath
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1987 AIME Problems/Problem 14
  ...eta</math> is the argument of <math>N</math> such that <math>0\leq\theta<2\pi.</math> .../math> so <math>\theta=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}.</math></li><p>
  7 KB (965 words) - 10:42, 12 April 2024
• 1988 AIME Problems/Problem 14
  ...eflection of the asymptote <math>x=0</math> by multiplying this by <math>2-i</math>, getting <math>4+3i</math>. Therefore, the asymptotes of <math>C^*</ ...atrix for a reflection about the polar line <math>\theta = \alpha, \alpha+\pi</math> is:
  4 KB (700 words) - 17:21, 3 May 2021
• 1990 AIME Problems/Problem 12
  ...576}</math> instead-the motivation for this is to make the expression look like the half angle identity, and the fact that <math>\sqrt{576}</math> is an in ...\pi k}{12}}} \\ &= 12 \sqrt{4\sin^2{\frac{\pi k}{12}}} \\ &= 24\sin{\frac{\pi k}{12}}.\end{align*}</cmath> The rest follows as Solution 1.
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 10
  ...math> and <math>48</math> is <math>144</math>, so define <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^8, ...right)</math> respectively, where <math>\text{cis}\,\theta = \cos \theta + i \sin \theta</math> and <math>k_1</math> and <math>k_2</math> are integers f
  3 KB (564 words) - 04:47, 4 August 2023
• 1991 AIME Problems/Problem 15
  .../math> for <math>1 \le i \le n</math> and <math>0 \le \theta_{i} < \frac {\pi}{2}</math>. We then have that
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 11
  ...The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers a for (int i=0; i<12; ++i)
  4 KB (740 words) - 19:33, 28 December 2022
• 1992 AIME Problems/Problem 10
  Let <math>z=a+bi \implies \frac{z}{40}=\frac{a}{40}+\frac{b}{40}i</math>. Since <math>0\leq \frac{a}{40},\frac{b}{40}\leq 1</math> we have th ...c{40}{\overline{z}}=\frac{40}{a-bi}=\frac{40a}{a^2+b^2}+\frac{40b}{a^2+b^2}i</math> so we have <math>0\leq a,b \leq \frac{a^2+b^2}{40}</math>, which lea
  2 KB (323 words) - 12:05, 16 July 2019
• 1994 AIME Problems/Problem 14
  ...a * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*expi(beta * pi/180); ...-(1.5,0));D(r);D(anglemark(C,B,D(A)));D(anglemark((1.5,0),C,C+.4*expi(beta*pi/180)));MP("\beta",B,(5,1.2),fontsize(9));MP("\alpha",C,(4,1.2),fontsize(9))
  2 KB (303 words) - 00:03, 28 December 2017
• 1994 AIME Problems/Problem 13
  Using DeMoivre, <math>13 - t = \text{cis}\left(\frac {(2k + 1)\pi}{10}\right)</math> where <math>k</math> is an integer between <math>0</math ...pi}{10}\right) \Rightarrow \bar{t} = 13 - \text{cis}\left(-\frac {(2k + 1)\pi}{10}\right)</math>.
  3 KB (375 words) - 23:46, 6 August 2021
• 1996 AIME Problems/Problem 15
  pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D); === Solution 3 === <!-- I would put this solution first, but needs a bit of formatting for clarity -
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 11
  ...maginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta ...y <math>z - 1</math> to both sides, noting that then <math>z \neq \cos 0 + i\sin 0</math> since, then we are multiplying by <math>0</math> which makes i
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1997 AIME Problems/Problem 14
  :<math>z^{1997}=1=1(\cos 0 + i \sin 0)</math> Define <math>\theta = 2\pi/1997</math>. By [[De Moivre's Theorem]] the roots are given by
  5 KB (874 words) - 22:30, 1 April 2022
• 1998 AIME Problems/Problem 5
  Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> ...uates to an integer ([[triangular number]]), and the [[cosine]] of <math>n\pi</math> where <math>n \in \mathbb{Z}</math> is 1 if <math>n</math> is even a
  1 KB (225 words) - 02:20, 16 September 2017
• 1999 AIME Problems/Problem 11
  &= \mbox{Im } \frac{2\cos5 + 2i \sin 5}{(1 - \cos 5) - i \sin 5}\\ &= \mbox{Im } \frac{(2 \cos 5 + 2i \sin 5)[(1 - \cos 5) + i \sin 5]}{(1 - \cos 5)^2 + \sin^2 5}\\
  4 KB (614 words) - 04:38, 8 December 2023
• 1999 AIME Problems/Problem 9
  ...finition of <math>f(z)</math>, <cmath>f(1+i) = (a+bi)(1+i) = (a-b) + (a+b)i,</cmath> this image must be equidistant to <math>(1,1)</math> and <math>(0, ...re positive, <math>z</math> lies in the first quadrant and <math>\theta < \pi/2</math>; hence by right triangle trigonometry <math>\sin \theta = \frac{\s
  6 KB (1,010 words) - 19:01, 24 May 2023
• 2000 AIME I Problems/Problem 8
  ...the volume of the liquid can be found by <math>\frac{\pi}{3}r^2h - \frac{\pi}{3}(r')^2h'</math>. ...rac{\pi}{3}\left(\frac{3}{4}r\right)^2 \left(\frac{3}{4}h\right) &= \frac{\pi}{3}\left(r^2h - \left(\frac{rh'}{h}\right)^2h'\right)\\
  4 KB (677 words) - 16:33, 30 December 2023
• 2003 AIME I Problems/Problem 5
  ...5</math> units. Given that the [[volume]] of this set is <math>\frac{m + n\pi}{p}, </math> where <math> m, n, </math> and <math> p </math> are positive [ ...ach of radius <math>1</math>. Together, their volume is <math> \frac{4}{3}\pi </math>.
  2 KB (288 words) - 19:58, 4 July 2013
• 2003 AIME I Problems/Problem 2
  ...\ldots - 1^{2} \pi</math>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is <cmath>\frac{100^{2}\pi - 99^{2}\pi + 98^{2}\pi - \ldots - 1^{2}\pi}{100^{2}\pi}</cmath>
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 14
  ...{3}}{2} i\right) = -\frac{b}{2}-\sqrt{3}+\left(\frac{b\sqrt{3}}{2}-1\right)i</math>. We solve for <math>b</math> and <math>f</math> and find that <math
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2000 AIME II Problems/Problem 9
  ...we have <math>z = \frac{2\cos 3 \pm \sqrt{4\cos^2 3 - 4}}{2} = \cos 3 \pm i\sin 3 = \text{cis}\,3^{\circ}</math>. ...could let <math>z = a + bi</math> and solve to get <math>z=\cos 3^\circ + i\sin 3^\circ</math>.
  4 KB (675 words) - 13:42, 4 April 2024
• Holomorphic function
  <cmath> f(z) = u(x,y) + i v(x,y). </cmath> + i \frac{\partial v}{\partial x}, \qquad
  9 KB (1,537 words) - 21:04, 26 July 2017
• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...athrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6 </cmath> If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007/Problems
  .../math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder when <mat ...et circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math
  8 KB (1,355 words) - 14:54, 21 August 2020
• 2006 Romanian NMO Problems
  ...<math>a_i \in \{ -1, 1 \}</math>, such that <center><math>n = \sum_{1\leq i < j \leq k } a_ia_j</math>.</center> ...ngle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the midpoint of <math>BC</math>. The poin
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2005 Canadian MO Problems/Problem 3
  For <math>i = 1, 2, 3</math>, let <math>m_i</math> be the line perpendicular to <math>l ...\left(\frac{1}{2}, \frac{\pi}{3}\right), V_3 = \left(\frac{1}{2}, \frac{2\pi}{3}\right) \in S</math>. It is easy to see that for any point <math>P</math
  2 KB (460 words) - 13:35, 9 June 2011
• Volume
  ...volume of a [[sphere]] of [[radius]] <math>r</math> is <math>\frac 43 r^3\pi</math>. ...[[cylinder]] of height <math>h</math> and radius <math>r</math> is <math>\pi r^2h</math>. (Note that this is just a special case of the formula for a p
  3 KB (523 words) - 20:24, 17 August 2023
• Right angle
  ...of every right angle is 90 [[degree (geometry) | degrees]] or <math>\frac \pi 2</math> [[radian]]s. When drawing diagrams, we denote right angles with a pair A, B, C, D, I;
  673 bytes (91 words) - 23:59, 11 June 2022
• 1970 IMO Problems/Problem 1
  Let <math>I</math> be the incenter of <math>ABC</math>, and let <math>I_{c}</math> be i c & = &q \left[ \cot\left(\frac{\pi - A}{2}\right) + \cot \left(\frac{\pi - B}{2}\right) \right]\\
  2 KB (380 words) - 22:12, 19 May 2015
• 2007 iTest Problems
  <cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> ...4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math>
  33 KB (5,177 words) - 21:05, 4 February 2023
• Straight angle
  ...e of every straight angle is 180 [[degree (geometry) | degrees]] or <math>\pi</math> [[radian]]s. pair A, B, I;
  568 bytes (78 words) - 13:50, 12 June 2022
• 2005 AMC 10A Problems
  ...D) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \mathrm{(E) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{2} </math> for(int i=0;i<=4;i=i+1)
  14 KB (2,026 words) - 11:45, 12 July 2021
• 1985 IMO Problems/Problem 1
  ...es of arcs, <math> \angle DTA = \angle DCO = \frac{\angle DCB}{2} = \frac{\pi}{2} - \frac{\angle DAB}{2} </math>. It follows that <math>AT = AD </math>. ...</math> such that <math>AT = AD </math>. Then <math>\angle DTA = \frac{ \pi - \angle DAB}{2} = \angle DCO</math>, so <math>DCOT </math> is a cyclic qua
  4 KB (684 words) - 07:28, 3 October 2021
• 2004 AMC 10A Problems
  A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above for(int i=0; i<=5; ++i)
  15 KB (2,092 words) - 20:32, 15 April 2024
• 1985 AIME Problems
  ...rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the ...itive integers which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>.
  7 KB (1,071 words) - 19:24, 23 February 2024
• Asymptote: Advanced Configuration
  There are two things that Asymptote users commonly would like to do with their figures: make the images output in a different format, and I like to make pics with Asymptote like this one:
  12 KB (1,931 words) - 13:53, 26 January 2020
• Asymptote: Reference
  returns the complex number <math>e^{i\theta}</math>, i.e. (cos(theta),sin(theta)) where theta is measured in radians. star=expi(0)--(scale((3-sqrt(5))/2)*expi(pi/5))--expi(2*pi/5)--
  7 KB (1,205 words) - 21:38, 26 March 2024
• 2007 AIME I Problems/Problem 4
  ...30} </math>, <math> b(t) = \frac{t \pi}{42} </math>, <math>c(t) = \frac{t \pi}{70}</math>. ...frac{ 2 t \pi}{105} </math>. These are simultaneously multiples of <math>\pi </math> exactly when <math>t </math> is a multiple of <math>105</math>, so
  1 KB (228 words) - 18:46, 11 March 2021
• 2007 AIME II Problems
  ...<math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are ther ...> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math>
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 AMC 12A Problems
  <math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math> ...bx^{2} + cx + d</math> has real [[coefficient]]s, and <math>f(2i) = f(2 + i) = 0.</math> What is <math>a + b + c + d?</math>
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 USAMO Problems/Problem 6
  pair A=(2,8), B=(0,0), C=(13,0), I=incenter(A,B,C), O=circumcenter(A,B,C), p_a, q_a, X, Y, X1, Y1, D, E, F; real r=abs(I-foot(I,A,B)), R=abs(A-O), a=abs(B-C), b=abs(A-C), c=abs(A-B), x=(((b+c-a)/2)^2)/(r
  7 KB (1,274 words) - 15:11, 31 August 2017
• LaTeX:Commands
  | <math>\textstyle 2^{23}</math>||2^{23}||<math>\textstyle n_{i-1}</math>||n_{i-1} | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2}
  12 KB (1,898 words) - 15:31, 22 February 2024
• 2005 PMWC Problems
  for (int i=0;i<6;i=i+1){ draw(dir(60*i)..3*dir(60*i)..cycle);
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  for(int i = 0; i < n; ++i){ for(int j = n-i; j > 0; --j){
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  for(int i = 0; i < 4; ++i){ if(i < 2){
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2001 PMWC Problems
  real eta=pi/2; for(int i = 0; i < 4; ++i)
  4 KB (641 words) - 21:24, 21 April 2014
• 2002 PMWC Problems
  for(int i = 0; i < nrows; ++i) for(int j = 0; j <= i; ++j)
  5 KB (725 words) - 16:07, 23 April 2014
• 2003 PMWC Problems
  pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); for(int i = -2; i <= 2; ++i)
  7 KB (918 words) - 16:15, 22 April 2014
• Root (operation)
  ...t[n]{x}=\sqrt[n]{|x|}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)</math> , where <math>k=0,1,2,...,n-1</math> and <math>x\in\mat ...h>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot0}{4}+i\sin\frac{0+2\pi\cdot0}{4}\right)\implies\boxed{2}</math>
  3 KB (532 words) - 16:52, 20 May 2020
• 2007 AMC 10A Problems
  <math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math> pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1995 AHSME Problems
  <math>\text{(i)}</math> <math>4,000 \leq N < 6,000;</math> for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1959 AHSME Problems
  ...i p^2}{27}\qquad\textbf{(D)}\ \frac{\pi p^2}{81}\qquad\textbf{(E)}\ \frac{\pi p^2\sqrt3}{27} </math> ...owing is not true for the equation<math> ix^2-x+2i=0</math>, where <math>i=\sqrt{-1}</math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1991 USAMO Problems
  ...<math>\,S\,</math> of numbers, let <math>\,\sigma(S)\,</math> and <math>\,\pi(S)\,</math> denote the sum and product, respectively, of the elements of <m <cmath> \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}
  3 KB (512 words) - 19:17, 18 July 2016
• 1976 IMO Problems/Problem 2
  I shall prove by induction that <math>P_n(x)</math> has <math>2^n</math> dist ...l}{2^n+1}\cdot\pi</math>. As we can choose the range <math>0\leq\theta\leq\pi</math> to ensure no duplications, we get that, upon rearranging, <math>0\le
  3 KB (596 words) - 16:19, 28 July 2015
• 2008 AMC 12A Problems/Problem 23
  The solutions of the equation <math>z^4+4z^3i-6z^2-4zi-i=0</math> are the vertices of a convex polygon in the complex plane. What is ...^{4}=1+i=2^{\frac{1}{2}}\cos \frac {\pi}{4} + 2^{\frac{1}{2}}i\sin \frac {\pi}{4} </cmath>
  1 KB (199 words) - 01:38, 10 November 2019
• 2008 AMC 12A Problems/Problem 25
  ...<math>z_n = 2e^{i\pi/6}z_{n-1}</math>, so <math>z_{n-1} = \frac{1}{2}e^{-i\pi/6}z_n</math>. This is the reverse transformation. We have <cmath> z_{99} = \frac{1}{2}e^{-i\pi/6}z_{100}</cmath>
  5 KB (745 words) - 10:58, 9 December 2022
• 2008 AMC 12B Problems
  ...uad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}</math> ...ers and <math>i^2 = - 1</math>. Suppose that <math>f(1)</math> and <math>f(i)</math> are both real. What is the smallest possible value of <math>| \alph
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AIME I Problems
  {{AIME Problems|year=2008|n=I}} ...e party is now <math>58\%</math> girls. How many students now at the party like to dance?
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such that ...efine a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</ma
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 5
  ...math>\sqrt {r^{2} + h^{2}}</math>. Thus, the length of the path is <math>2\pi\sqrt {r^{2} + h^{2}}</math>. ...of the path is 17 times the circumference of the base, which is <math>34r\pi</math>. Setting these equal gives <math>\sqrt {r^{2} + h^{2}} = 17r</math>,
  1 KB (230 words) - 20:18, 4 July 2013
• 2008 AIME I Problems/Problem 8
  ...\arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.</cmath> We now have <math>\arctan{\dfrac{23}{24}} + \arctan{\dfrac{1}{n}} = \dfrac{\pi}{4} = \arctan{1}</math>. Thus, <math>\dfrac{\dfrac{23}{24} + \dfrac{1}{n}}{
  3 KB (490 words) - 22:36, 28 November 2023
• 2008 AIME I Problems/Problem 10
  ...whose angle at the longer base <math>\overline{AD}</math> is <math>\dfrac{\pi}{3}</math>. The [[diagonal]]s have length <math>10\sqrt {21}</math>, and po pair A=(0,0), D=(4,0), B= A+2 expi(1/3*pi), C= D+2expi(2/3*pi), E=(-4/3,0), F=(3,0);
  4 KB (629 words) - 22:38, 28 November 2023
• 2008 AIME II Problems/Problem 9
  ...efine a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</ma <cmath>x'=r\cos(\pi/4+\theta)+10 = \frac{\sqrt{2}(x - y)}{2} + 10</cmath>
  5 KB (725 words) - 22:37, 28 January 2024
• Cone
  ...ath> and height <math>h</math> has [[volume]] <math>V = \frac{1}{3} \cdot \pi r^2 \cdot h</math>. This is a special case of the general formula for the ...[lateral area]] is <math>\pi rs</math>, and the area of the base is <math>\pi r^2</math>).
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2008 AIME II Problems/Problem 13
  ...\right\rbrace</math>. Then the area of <math>S</math> has the form <math>a\pi + \sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive int ...>S</math> (where [[cis]] denotes <math>\text{cis}\, \theta = \cos \theta + i \sin \theta</math>). Since <math>R</math> is symmetric every <math>60^{\cir
  6 KB (894 words) - 18:56, 25 December 2022
• 2001 Iran NMO (Round 3) Problems/Problem 5
  In triangle <math>ABC</math>, let <math>I</math> be the incenter and <math>I_a</math> the excenter opposite <math>A</ pair I=incenter(A,B,C);
  3 KB (437 words) - 15:47, 27 April 2008
• 2008 USAMO Problems/Problem 4
  int n = 17; real r = 1; real rad = pi/2; return (r*expi(rad-2*pi*k/n));
  8 KB (1,318 words) - 12:37, 20 April 2022
• 1986 IMO Problems/Problem 2
  ...or <math>A_3</math> the rotation sends <math>P</math> to <math>e^{\frac{4i\pi}{3}}(P-a)+a</math>. Thus the result of all three rotations sends <math>P</ <math>e^{\frac{4i\pi}{3}}(e^{\frac{4i\pi}{3}}(e^{\frac{4i\pi}{3}}P-1)+1-a)+a</math>
  2 KB (345 words) - 00:04, 30 January 2021
• 2008 iTest Problems
  for(int i=0;i<=5;i=i+1) draw(circle(7/2+d*i,3/2));
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2005 AMC 8 Problems
  for ( int i = 1; i <= 7; ++i ) draw((i,0)--(i,6));
  13 KB (1,821 words) - 22:18, 5 December 2023
• Van Aubel's Theorem
  ...by the complex number <math>x</math>. Note that <math>\angle PAB = \frac{\pi}{4}</math> and that <math>PA = \frac{\sqrt{2}}{2}AB</math>, and similarly f p &=& \frac{\sqrt{2}}{2}(b-a)e^{i \frac{\pi}{4}}+a \\
  2 KB (410 words) - 14:01, 4 March 2023
• 1974 USAMO Problems
  (i) The weakest player chooses the first two contestants. pair P=(7,0), Q=(14,0), R=P+7expi(pi/3), M=(10,1.2);
  3 KB (427 words) - 18:55, 3 July 2013
• 1962 IMO Problems/Problem 6
  .... Let <math>ABC</math> have circumcenter <math>O</math> and incenter <math>I</math>. Extend <math>AI</math> to meet the circumcircle again at <math>L</m ...cdot IL = PI\cdot QI</math> by Power of a Point. Therefore, <math>2r\rho = PI \cdot QI = (PO + OI)(QO - OI) = (r + d)(r - d)</math>, and we have <math>2r
  2 KB (308 words) - 06:29, 16 December 2023
• 1963 IMO Problems/Problem 3
  ...h>y\textrm{-coordinate} = \sum_{k = 1}^{\frac{n}{2}}a_k \cdot \sin \frac{2\pi(k-1)}{n}.\textbf{ (1)}</cmath> ...y\textrm{-coordinate} = \sum_{k = \frac{n}{2}+1}^{n}a_k \cdot \sin \frac{2\pi(k-1)}{n}</cmath>
  4 KB (836 words) - 17:58, 7 December 2022
• 1963 IMO Problems/Problem 5
  Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>. ...{8\pi}{7}} + \cos{\frac{10\pi}{7}} + \cos{\frac{12\pi}{7}} + \cos{\frac{14\pi}{7}} = 0</cmath>
  5 KB (746 words) - 19:34, 14 October 2023
• 1963 IMO Problems
  == Day I == Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>.
  2 KB (342 words) - 21:16, 20 August 2020
• 1976 USAMO Problems
  int i; for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);}
  3 KB (510 words) - 19:01, 3 July 2013
• 2009 AMC 12A Problems
  For what value of <math>n</math> is <math>i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i</math>? Note: here <math>i = \sqrt { - 1}</math>.
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2009 AMC 12A Problems/Problem 21
  <center><cmath>p(2009 + 9002\pi i) = p(2009) = p(9002) = 0</cmath></center> From the three zeroes, we have <math>p(x) = (x - (2009 + 9002\pi i))(x - 2009)(x - 9002)</math>.
  2 KB (322 words) - 10:25, 29 July 2020
• 2001 AMC 12 Problems/Problem 17
  ...h>, <math>B = (4,0)</math>, <math>C = (2 \pi + 1, 0)</math>, <math>D = (2 \pi + 1,4)</math>, and <math>E=(0,4)</math>. What is the probability that <math real pi=3.14159265359;
  5 KB (792 words) - 15:23, 30 November 2021
• 2002 AMC 12A Problems/Problem 24
  ...and when <math>s=1</math>. When <math>s=1</math> and <math>a+bi=\cos\theta+i\sin\theta</math>, ...in(2002\theta) </math> <math> =a-bi= \cos\theta-i\sin\theta= \cos(-\theta)+i\sin(-\theta)</math>
  4 KB (743 words) - 19:54, 14 March 2024
• 2009 AMC 12B Problems
  For how many values of <math>x</math> in <math>[0,\pi]</math> is <math>\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)</math>? int i,j;
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 23
  ...frac34i</math> into polar form as <math>\frac{3\sqrt{2}}{4}e^{\frac{\pi}{4}i}</math>. Restated using geometric probabilities, we are trying to find the We multiply <math>z</math> and <math>(\frac{3}{4}+\frac{3}{4}i)</math> to get <cmath>(\frac{3}{4}x-\frac{3}{4}y)+(\frac{3}{4}xi+\frac{3}{4
  2 KB (422 words) - 13:25, 20 January 2020
• 2009 AIME I Problems/Problem 12
  ...> be the circle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overl First, note that <math>AB=37</math>; let the tangents from <math>I</math> to <math>\omega</math> have length <math>x</math>. Then the perimete
  12 KB (1,970 words) - 22:53, 22 January 2024
• 1966 AHSME Problems/Problem 4
  ...in the given square. If <math>r</math> is the ratio of the area of circle I to that of circle II, then <math>r</math> equals: ...the radius of the smaller circle is <math>x</math>, so it's area is <math>\pi x^2</math>.
  1 KB (191 words) - 22:09, 14 January 2018
• Cauchy's Integral Formula
  <cmath> \frac{1}{2\pi i} \int\limits_C \frac{f(z)}{z- z_0}dz = f(z_0) .</cmath> as <math>h(t) = r e^{it}+ z_0</math>, for <math>t\in [0,2\pi]</math>. Since <math>\int\limits_{C_r}
  4 KB (689 words) - 17:19, 18 January 2024
• Liouville's Theorem (complex analysis)
  <cmath> \lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i}
  2 KB (412 words) - 20:30, 16 January 2024
• 2000 AMC 8 Problems
  ...e numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi?</math> label("I",(0.4,3),E);
  15 KB (2,165 words) - 03:32, 13 April 2024
• 2009 AIME II Problems/Problem 14
  We now discovered that <math>b_4=b_2</math>. And as each <math>b_{i+1}</math> is uniquely determined by <math>b_i</math>, the sequence becomes ...t <math>b_k = \sin (kt)</math>, where <math>kt</math> exceeds <math>\frac{\pi}2</math>. Then we'll have <math>b_{k+1} = \sin(kt - t) = \sin ((k-1)t) = b_
  4 KB (680 words) - 13:49, 23 December 2023
• 2009 AIME II Problems/Problem 13
  ...ath> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac {\pi}{7}</math>. Using the [[Law of Cosines]], <math>\overline {AC_1}^2</math> = <math>8 - 8 \cos \frac {\pi}{7}</math>,
  8 KB (1,266 words) - 20:27, 10 December 2023
• 1998 USAMO Problems/Problem 3
  ..._0,\cdots a_n</math> be real numbers in the interval <math>\left(0,\frac {\pi}{2}\right)</math> such that ...tan{\left(a_1 - \frac {\pi}{4}\right)} + \cdots + \tan{\left(a_n - \frac {\pi}{4}\right)}\ge n - 1</cmath>
  2 KB (322 words) - 13:31, 23 August 2023
• Constructible polygon
  ...primitive <math>n^{th}</math> root of unity <math>\zeta_n = e^{\frac{2\pi i}{n}}</math>. So now the <math>n^{th}</math> roots of unity are <math>1,\zet ...h extension <math>K_i\subseteq K_{i+1}</math> is quadratic (i.e. <math>[K_{i+1}:K_i]=2</math>). We claim that this happens iff <math>\phi(n)</math> (whe
  5 KB (926 words) - 18:47, 4 March 2022
• 2010 AMC 12A Problems
  ...}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math> Brian: "Mike and I are different species."
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 12A Problems/Problem 24
  ...\left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)</math>. The intersection of the domain of <math>f(x)</math> with ...align*}\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x) &= 0\\
  9 KB (1,434 words) - 17:54, 17 August 2022
• 2010 AMC 10A Problems
  The area of a circle whose circumference is <math>24\pi</math> is <math>k\pi</math>. What is the value of <math>k</math>? A <i>palindrome</i>, such as <math>83438</math>, is a number that remains the same when its di
  13 KB (1,902 words) - 11:20, 5 March 2023
• Proofs without words
  Fibonacci: sum_{i=1}^{2n-1} F_iF_{i+1} = F_{2n}^2 pen colors(int i){ return rgb(i/n,0.4+i/(2n),1-i/n); } /* shading */
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems
  {{AIME Problems|year=2010|n=I}} [[2010 AIME I Problems/Problem 1|Solution]]
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 11
  ...x = 15</math>, the [[volume]] of the resulting [[solid]] is <math>\frac {m\pi}{n\sqrt {p}}</math>, where <math>m</math>, <math>n</math>, and <math>p</mat ...6.5,0), C= (9.75,8.25,0), F=(Fxy.x,Fxy.y,0), G=2*F-A, H=(F.x,F.y,abs(F-A)),I=(F.x,F.y,-abs(F-A));
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME I Problems/Problem 13
  label("$I$",(255.242,5.00321),NE/2); <math>Y = \frac {1}{3}\pi(3)^2 = 3\pi</math>
  10 KB (1,418 words) - 23:05, 20 October 2021
• Mock AIME 2 2010 Answer Key
  Notice that removing as many lemmings as possible at each step (i.e., using the greedy algorithm) is optimal. This can be easily proved throu ...{10})</math>, with <math>0 \le b_i \le 2, 3|a_i-b_i</math> for <math>1 \le i \le 10</math>. Given that <math>f</math> can take on <math>K</math> distinc
  36 KB (6,214 words) - 20:22, 13 July 2023
• LaTeX:LaTeX on AoPS
  ...If we do this for our <math>f</math> equation above, we get <cmath>f(x) = \pi\left(\frac{\sqrt{x}}{x-1}\right).</cmath> We can use <code>\left</code> and ...example is <code>$\sum_{i=0}^n a_i$</code>, giving <math>\textstyle \sum_{i=0}^n a_i.</math> Note the use of superscripts and subscripts to obtain the
  8 KB (1,356 words) - 22:35, 26 June 2020
• 2010 USAMO Problems/Problem 3
  the inequality <math>a_ia_j \le i+j</math> for all distinct indices <math>i, j</math>. <cmath> \prod_{i=1}^{1005}(4i-1) = 3\times 7 \times \ldots \times 4019. </cmath>
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2010 USAMO Problems/Problem 4
  <math>CE</math> meet at <math>I</math>. Determine whether or not it is possible for pair I = extension(B, D, C, E); ldot(I, "$I$", A-I);
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2010 AMC 12B Problems/Problem 18
  <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath> ...length where the third jump could land to satisfy the problem. To do this, I can apply average function value with our old buddy calculus,
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2019 IMO Problems/Problem 6
  Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \ne <i><b>Step 1</b></i>
  5 KB (792 words) - 01:52, 19 November 2023
• 2004 AMC 8 Problems
  I. Bill is the oldest. ...extbf{(C)}\ 28-4\pi \qquad \textbf{(D)}\ 28-2\pi \qquad \textbf{(E)}\ 32-2\pi</math>
  13 KB (1,860 words) - 19:58, 8 May 2023
• 2011 AMC 12A Problems
  \textbf{(A)}\ 3 - \frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12B Problems/Problem 24
  So <math>z^4 = 1 \implies z = e^{i\frac{n\pi}{2}}</math> <math>z^2 = \pm i \sqrt{4\sqrt{3} + 7} = e^{i\frac{(2n+1)\pi}{2}} \left(\sqrt{3} + 2\right)</math>
  2 KB (344 words) - 18:12, 22 August 2021
• 2011 AIME I Problems
  {{AIME Problems|year=2011|n=I}} [[2011 AIME I Problems/Problem 1|Solution]]
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AIME I Problems/Problem 9
  Suppose <math>x</math> is in the interval <math>[0, \pi/2]</math> and <math>\log_{24\sin x} (24\cos x)=\frac{3}{2}</math>. Find <ma Like Solution 1, we can rewrite the given expression as
  2 KB (330 words) - 20:47, 10 December 2023
• 2011 AIME I Problems/Problem 14
  ...A_6}</math>, and <math>\overline{A_7 A_8}</math>, respectively. For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</ma ...s we have that <math>\cos 2 \angle A_3 M_3 B_1=\cos \left(2\theta + \frac{\pi}{2} \right)=-\sin2\theta</math>.
  8 KB (1,344 words) - 18:39, 9 February 2023
• 2011 AIME II Problems/Problem 8
  ...h <math>j</math>, let <math>w_j</math> be one of <math>z_j</math> or <math>i z_j</math>. Then the maximum possible value of the real part of <math>\sum_ ...the real parts of the blue dots is easily seen to be <math>8+16\cos\frac{\pi}{6}=8+8\sqrt{3}</math> and the negative of the sum of the imaginary parts o
  5 KB (805 words) - 18:46, 27 January 2024
• 1998 USAMO Problems
  ...th>\{a_i,b_i\}</math> (<math>1\leq i\leq 999</math>) so that for all <math>i</math>, <math>|a_i-b_i|</math> equals <math>1</math> or <math>6</math>. Pro Let <math>a_0,a_1,\cdots ,a_n</math> be numbers from the interval <math>(0,\pi/2)</math> such that
  3 KB (486 words) - 06:11, 24 November 2020
• 2011 AMC 10B Problems
  ...f{(C)}\ \pi \qquad\textbf{(D)}\ \frac{4\pi}{3} \qquad\textbf{(E)}\ \frac{5\pi}{3}</math> ...s hold for all numbers <math>x, y,</math> and <math>z</math>? <cmath>\text{I. x @ (y + z) = (x @ y) + (x @ z)}</cmath> <cmath>\text{II. x + (y @ z) = (x
  13 KB (2,090 words) - 18:05, 7 January 2021
• Trigonometric Equations
  Solve for <math>x</math> for all answers in the domain <math>[0, 2\pi]</math>. <math>i = sin(x)</math>
  8 KB (1,351 words) - 20:30, 10 July 2016
• 2012 AMC 12B Problems/Problem 23
  ...d, or 4th root of unity. These are among the set <math>\{\pm1,\pm i,(-1\pm i\sqrt{3})/2\}</math>. Since complex roots of polynomials come in conjugate p Now suppose <math>z=i</math>. Then <math>4=(a-c)i+(b-d)</math> whereupon <math>a=c</math> and <math>b-d=4</math>. But then <m
  11 KB (1,979 words) - 17:25, 6 September 2021
• 1967 AHSME Problems
  <math>\textbf{(A)}\ \frac{\pi m^2}{2}\qquad \textbf{(B)}\ \frac{3\pi m^2}{8}\qquad
  20 KB (3,108 words) - 14:14, 20 February 2020
• 2009 AMC 8 Problems
  int i; for(int i=0; i<15; i=i+1) {
  18 KB (2,551 words) - 18:46, 27 February 2024
• 1996 AHSME Problems
  ...-\pi\qquad\text{(C)}\ \frac{\pi} 2\qquad\text{(D)}\ \pi\qquad\text{(E)}\ 2\pi </math> int i;
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1985 AHSME Problems
  ...qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2 </math> int i,j;
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 23
  ...}}{2} \qquad\text{and}\qquad y=\frac{-1-i\sqrt{3}}{2},</cmath> where <math>i^2 = -1</math>, then which of the following is not correct? ...c{2\pi k}{3}\right) = \cos\left(\frac{2\pi k}{3}\right)-i\sin\left(\frac{2\pi k}{3}\right),\end{align*}</cmath> using the fact that <math>\cos</math> is
  3 KB (578 words) - 00:47, 20 March 2024
• Mathtime Version 1 Issue 2
  ...axis. As <math>\theta</math> sweeps from <math>0</math> to <math>\dfrac{\pi}{2}</math>, what is the probability that <math>b\ge 1</math>? (Proposed by I roll 4 dice with 6, 10, 12, and 20 sides each numbered 1-6, 1-10, 1-12, 1-2
  15 KB (2,444 words) - 21:46, 1 January 2012
• 1971 Canadian MO Problems/Problem 9
  ...m statement tells us that < HPI = < KPJ. By transitivity, we must have < H'PI = < KPJ, implying that H', P, and K are collinear by the converse of Vertic
  1 KB (204 words) - 18:09, 19 September 2012
• 1989 AHSME Problems
  for(int i = 1;i<y.length;++i) { pair p = (a,y[i]);
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 12A Problems
  ...up of <math>9</math> congruent circular arcs each of length <math>\frac{2\pi}{3}</math>, where each of the centers of the corresponding circles is among ...extbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 </math>
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AMC 10B Problems
  ...xtbf{(C)}\ 12\pi+\sqrt{3}\qquad\textbf{(D)}\ 10\pi+9\qquad\textbf{(E)}\ 13\pi</math> ...)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi</math>
  18 KB (2,350 words) - 18:48, 9 July 2023
• 2012 AIME I Problems/Problem 6
  ...= z,</math> and the imaginary part of <math>z</math> is <math>\sin{\frac{m\pi}{n}}</math>, for relatively prime positive integers <math>m</math> and <mat ...>z</math> will be of the form <math>\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}},</math> where <math>k \in \{1, 2, \ldots, 70\}.</math> Note that <mat
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• 2015 AIME I Problems
  {{AIME Problems|year=2015|n=I}} [[2015 AIME I Problems/Problem 1|Solution]]
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  {{AIME Problems|year=2012|n=I}} [[2012 AIME I Problems/Problem 1|Solution]]
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME II Problems/Problem 13
  ...th>\sqrt{111}</math>, and <math>\sqrt{11}</math>. That is, let <math>z=e^{i\theta}</math>, and then: ...g <math>D_1</math> and <math>D_2</math> around <math>A</math> by <math>\pm\pi/3</math>:
  13 KB (2,052 words) - 18:02, 5 February 2024
• 2017 AMC 8 Problems
  ...t then I had <math>3</math> gold coins left over. How many gold coins did I have? ...{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}</math>
  12 KB (1,771 words) - 21:13, 20 January 2024
• Mock AIME I 2012 Problems
  [[Mock AIME I 2012 Problems/Problem 1 |Solution]] [[Mock AIME I 2012 Problems/Problem 2| Solution]]
  7 KB (1,309 words) - 11:13, 8 April 2012
• 2015 IMO Problems/Problem 1
  ...that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> <math>\mathcal{S}</math> is <i>centre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</math> in
  4 KB (773 words) - 08:14, 19 July 2016
• 1988 USAMO Problems/Problem 5
  ...n}</math> are all <math>n</math>th roots of unity. If <math>\omega=e^{2\pi i/n}</math>, then the sum of <math>32</math>nd powers of these roots will be
  8 KB (1,348 words) - 09:44, 25 June 2022
• 2006 SMT/Team Problems
  ...e <math> M </math> minus the volume fo cone <math> N </math> is <math> 140\pi </math>, find the length of <math> \overline{BC} </math>. ...h>, for terms <math> n\ge3 </math>, <math> S_n=\sum_{i=1}^{n-1}i\cdot S_{n-i} </math>. For example, if the first two elements are <math> 2 </math> and <
  6 KB (910 words) - 17:32, 27 May 2012
• 2006 SMT/Advanced Topics Problems/Problem 1
  ...assumes she means the square root of himself, or the square root of <math> i </math>. What two answers should he give? ...[[Euler's identity|Euler's Formula]] that <math> \cos\theta+i\sin\theta=e^{i\theta} </math>.
  1 KB (170 words) - 20:12, 27 May 2012
• 1998 USAMO Problems/Problem 6
  ...h exactly <math>k</math> of the quadrilaterals <math>A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.) ...> for all <math>i</math>, then quadrilateral <math>A_{i+1}A_{i+2}A_{i+3}A_{i+4}</math> is not tangential.
  5 KB (871 words) - 18:59, 10 May 2023
• 2012 AMC 8 Problems
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; I=(7,8);
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2013 AMC 10A Problems/Problem 20
  ...qrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8} </math> First, we need to see what this looks like. Below is a diagram.
  4 KB (701 words) - 17:55, 23 July 2021
• 2013 AIME I Problems
  {{AIME Problems|year=2013|n=I}} [[2013 AIME I Problems/Problem 1|Solution]]
  9 KB (1,580 words) - 13:07, 24 February 2024
• 2013 AIME I Problems/Problem 14
  For <math>\pi \le \theta < 2\pi</math>, let ...>\sin \theta = -17/19, 1/3.</math> Since we're given <math>\pi\leq\theta<2\pi,</math> <math>\sin\theta</math> is nonpositive. We therefore use the negati
  10 KB (1,641 words) - 20:03, 3 January 2024
• 1991 AHSME Problems
  <math>|3-\pi|=</math> ...4 \qquad \textbf{(C) }3-\pi \qquad \textbf{(D) }3+\pi \qquad \textbf{(E) }\pi-3 </math>
  16 KB (2,451 words) - 04:27, 6 September 2021
• 2013 AIME II Problems/Problem 4
  ...ian plane, we use complex numbers. Thus A is 1 and B is <math>2 + 2\sqrt{3}i</math>. ...here we require a clockwise rotation, so we multiply by <math>e^{-\frac{i\pi}{3}}</math> to obtain C. Upon averaging the coordinates of A, B, and C, we
  8 KB (1,268 words) - 14:10, 31 January 2024
• 2014 AIME II Problems/Problem 10
  Now, let <math>w = r_w e^{i \theta_w}</math> and likewise for <math>z</math>. Consider circle <math>O</ ...>w = \frac{-2014 \pm \sqrt{2014^2 - 4(2014^2)}}{2} = -1007 \pm 1007\sqrt{3}i</cmath>
  6 KB (1,045 words) - 13:08, 21 January 2024
• Symmetry
  <i><b>Proof</b></i> Given the triangle <math>\triangle ABC, \omega</math> is the incircle, <math>I</math> is the incenter, <math>B' = \omega \cap AC.</math>
  15 KB (2,549 words) - 08:36, 2 September 2023
• ASIA TEAM Problems
  ...t, when he catches a fly, he places it on <math>a_i</math> for which <math>i</math> is the least such number satisfying the following rules: ...4</math>, then he eats all the flies on <math>a_1</math> through <math>a_{i-1}</math> and then puts his newly caught fly on <math>a_i</math>.
  10 KB (1,710 words) - 23:23, 10 January 2020
• 1971 IMO Problems/Problem 5
  I shall prove a more general statement about the unit distance graph(<math>V= ...claim that for some choice of <math>0\leq\theta<2\pi</math>, <math>v_i+e^{i\theta}w_j</math> will do the job(a suitable rotation).
  4 KB (749 words) - 14:09, 29 January 2021
• 1962 AHSME Problems
  <math>\textbf{(A)}\ \pi + 2 \qquad \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad
  17 KB (2,459 words) - 22:40, 10 April 2023
• 2013 AMC 8 Problems
  triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P; I = (1,1,1);
  15 KB (2,162 words) - 20:05, 8 May 2023
• 1994 AHSME Problems
  <math>\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textb If <math>i^2=-1</math>, then <math>(i-i^{-1})^{-1}=</math>
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1981 AHSME Problems
  ...through four distinct points of the form <math>(i,j,k)</math>, where <math>i</math>, <math>j</math>, and <math>k</math> are positive integers not exceed If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each po
  17 KB (2,633 words) - 15:44, 16 September 2023
• 1968 AHSME Problems
  ...increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter. Then <math>P</math> equals: ...pi\quad\text{(C) } \frac{\pi^2}{2}\quad\text{(D) } \pi^2\quad\text{(E) } 2\pi</math>
  16 KB (2,571 words) - 14:13, 20 February 2020
• 2014 AMC 10A Problems
  <math>\textbf{(I)}\ x+y < a+b\qquad</math> for(int i=0;i<=5;i=i+1)
  15 KB (2,190 words) - 15:21, 22 December 2020
• 2014 AMC 10A Problems/Problem 12
  for(int i=0;i<=5;i=i+1) path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle;
  3 KB (482 words) - 11:50, 7 September 2021
• 1994 AHSME Problems/Problem 17
  ...qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 </math> for(int i=0; i<4; i=i+1)
  2 KB (287 words) - 03:07, 28 May 2021
• 2000 PMWC Problems/Problem I8
  for(int i = 0; i < 4; ++i){ if(i < 2){
  702 bytes (115 words) - 17:32, 20 April 2014
• 2014 USAMO Problems/Problem 6
  ...re positive integers such that <math>\gcd(a+i, b+j)>1</math> for all <math>i, j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n ...math> and in each cell place a prime <math>p</math> dividing <math>\gcd (a+i, b+j)</math>.
  2 KB (361 words) - 11:55, 25 June 2020
• 1960 AHSME Problems
  <math>\textbf{(A) }\frac{25\sqrt{2}}{\pi}\qquad \textbf{(B) }\frac{50\sqrt{2}}{\pi}\qquad
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AIME II Problems
  Let <math>f(x)=(x^2+3x+2)^{\cos(\pi x)}</math>. Find the sum of all positive integers <math>n</math> for which {{AIME box|year=2014|n=II|before=[[2014 AIME I Problems]]|after=[[2015 AIME I Problems]]}}
  8 KB (1,410 words) - 00:04, 29 December 2021
• 2014 AIME I Problems/Problem 10
  ...the smaller disk, so <math>6\alpha = 2\pi</math>, or <math>\alpha = \frac{\pi}{3}</math>. ...A}</math>, and call this point <math>X</math>. Since <math>\alpha = \frac{\pi}{3}</math> and <math>\angle DXE</math> is right, <cmath>DE = 6</cmath> <cma
  5 KB (854 words) - 20:02, 4 September 2021
• 2014 AIME I Problems/Problem 15
  ...we can find from angle chasing that <math>\angle ABF = \angle EDF = \frac{\pi}4</math>. Therefore, <math>\overline{BF}</math> is the angle bisector of <m {{AIME box|year=2014|n=I|num-b=14|after=Last Question}}
  10 KB (1,643 words) - 22:30, 28 January 2024
• 1973 AHSME Problems
  Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), <math>\text{I. Averaging is associative }</math>
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1992 AHSME Problems
  If <math>3(4x+5\pi)=P</math> then <math>6(8x+10\pi)=</math> <math>I.\quad y=x-2 \qquad II.\quad y=\frac{x^2-4}{x+2}\qquad III.\quad (x+2)y=x^2-
  16 KB (2,548 words) - 13:40, 19 February 2020
• 1990 AHSME Problems/Problem 22
  This equation is <math>r^6e^{6\theta i}=2^6e^{(\pi\pm 2k\pi) i}</math>. Solving in the usual way, <math>r=2</math> and <math>\theta\in\{\p
  798 bytes (133 words) - 13:17, 4 February 2016
• 1990 AHSME Problems
  draw((0,1)--(cos(pi/14),-sin(pi/14))--(-cos(pi/14),-sin(pi/14))--cycle,dot); draw((-cos(pi/14),-sin(pi/14))--(0,-1/cos(3pi/7))--(cos(pi/14),-sin(pi/14)),dot);
  14 KB (2,099 words) - 01:15, 10 September 2021
• 1970 AHSME Problems
  <math>\textbf{(A) }\frac{4}{\pi}\qquad \textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad
  15 KB (2,366 words) - 07:52, 26 December 2023
• 1980 AHSME Problems/Problem 28
  .../cmath> These roots are <math>w = e^{i \pi /3}</math> and <math>w = e^{2i \pi /3}</math>. ...}{2} + \frac{\sqrt{3}}{2}i + 1 = \frac{1}{2} + \frac{\sqrt{3}}{2}i = e^{i \pi /3}.</cmath>
  3 KB (472 words) - 20:04, 30 October 2021
• 1987 AHSME Problems
  have a solution <math>(x, y)</math> inside Quadrant I if and only if <math>\textbf{(A)}\ 36\pi \qquad
  16 KB (2,291 words) - 13:45, 19 February 2020
• 2015 AMC 10B Problems/Problem 22
  ...(2*cos(pi/5)-sin(pi/10),cos(pi/10)), C=(1,0), D=(0,0), E1=(-sin(pi/10),cos(pi/10)); ..., G=intersectionpoints(A--C,E1--B)[0], H=intersectionpoints(B--D,A--C)[0], I=intersectionpoints(C--E1,D--B)[0], J=intersectionpoints(E1--C,D--A)[0];
  7 KB (1,146 words) - 14:09, 13 April 2024
• 1983 AHSME Problems
  \textbf{(B)} \ \pi \qquad \textbf{(D)} \ 4 \pi \qquad
  15 KB (2,309 words) - 23:43, 2 December 2021
• 1982 AHSME Problems
  <math>\textbf{(A)} \ \pi \qquad \textbf{(B)} \ \frac{2}{\pi} \qquad
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1979 AHSME Problems
  <math>\textbf{(A) }\frac{\pi}{4}\qquad \textbf{(B) }\frac{3\pi}{4}\qquad
  17 KB (2,664 words) - 01:34, 19 March 2022
• 1978 AHSME Problems
  <math>\textbf{(A) }\frac{1}{\pi^2}\qquad \textbf{(B) }\frac{1}{\pi}\qquad
  15 KB (2,432 words) - 01:06, 22 February 2024
• 1976 AHSME Problems
  \textbf{(D) }\frac{\pi}{16}\qquad <math>\begin{array}{l}\textbf{I. }\text{"If the wild pig on planet beta has a long nose, then the pink elep
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1972 AHSME Problems
  The lengths in inches of the three sides of each of four triangles <math>I, II, III</math>, and <math>IV</math> are as follows: \hbox{I}& 3,\ 4,\ \hbox{and}\ 5\qquad &
  17 KB (2,732 words) - 13:54, 20 February 2020
• 2007 iTest Problems/Problem 17
  ...th>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\t <math>\text{(I) }\frac{1}{6}\qquad
  2 KB (266 words) - 21:30, 4 February 2023
• 2007 iTest Problems/Problem 48
  ...math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>.
  427 bytes (72 words) - 12:24, 6 April 2024
• 1965 AHSME Problems
  \textbf{(C) }\ 1: \pi \qquad \textbf{(D) }\ 3: \pi \qquad
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1957 AHSME Problems
  The area of a circle inscribed in an equilateral triangle is <math>48\pi</math>. The perimeter of this triangle is: If <math>S = i^n + i^{-n}</math>, where <math>i = \sqrt{-1}</math> and <math>n</math> is an integer, then the total number
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2013 UNCO Math Contest II Problems
  small circle is <math>9\pi</math>. Find the area of the shaded <math>\textit{lune}</math>, the region ...d_1d_2 \ldots d_6}</math> have period exactly six? Do not include patterns like <math>0.323</math> and <math>0.17</math> that have shorter periods.
  11 KB (1,648 words) - 09:55, 20 December 2021
• 2007 UNCO Math Contest II Problems
  where <math>i^2 = -1</math> as usual. your answer as an exact multiple of <math>\pi</math> (and not as a
  6 KB (890 words) - 22:14, 7 November 2014
• 2014 AMC 8 Problems
  string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"}; for(int i=0;i<=11;i=i+1)
  13 KB (1,957 words) - 12:08, 13 January 2024
• 2015 AMC 10A Problems
  for(int i=0; i<3; i+=1) draw((0,i+0.05)--(0,i+0.95));
  12 KB (1,897 words) - 22:45, 18 March 2024
• 2015 AMC 12A Problems
  ...nce between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integer <cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath>
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 10A Problems/Problem 14
  ...aw(0.9*dir(90-30*i)--dir(90-30*i));label("$"+(string) i+"$",0.78*dir(90-30*i));} ...s <math>20\pi</math>, so that is <math>20\pi</math> traveled on a <math>60\pi</math> arrow path. This is a ratio of 1/3, so the angle it carves is 120 de
  9 KB (1,380 words) - 09:00, 1 December 2023
• 2015 AMC 12A Problems/Problem 24
  <cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath> Let <math>\cos(a\pi) = x</math> and <math>\sin(b\pi) = y</math>. Consider the binomial expansion of the expression:
  5 KB (793 words) - 20:32, 26 May 2022
• 2015 AMC 12A Problems/Problem 23
  ...nce between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integer ...t{0.25 - x^2} dx = 4\left(\frac{1}{4} - \frac{\pi}{16}\right) = 1 - \frac{\pi}{4}.</cmath>
  12 KB (1,981 words) - 18:33, 3 September 2023
• 2015 AMC 10B Problems
  ...{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} </math> ...C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A); D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle
  14 KB (2,247 words) - 11:38, 26 March 2024
• 2015 AMC 12B Problems
  ...c{\sqrt{2}}{2} \qquad\textbf{(D)}\; 4(\pi-\sqrt{3}) \qquad\textbf{(E)}\; 2\pi-\dfrac{\sqrt{3}}{2}</math> <cmath>f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AMC 12B Problems/Problem 25
  Let <math>x=e^{i\pi/6}</math>, a <math>30^\circ</math> counterclockwise rotation centered at th <cmath>1-x=1-\frac{\sqrt{3}}{2}-\frac{1}{2}i</cmath>
  13 KB (2,117 words) - 11:30, 8 April 2023
• 2015 AIME I Problems/Problem 15
  ...ce on each half. The area of one of these unpainted faces is <math>a\cdot\pi + b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math [[File:2015 AIME I 15.png|400px|right]]
  9 KB (1,407 words) - 19:37, 17 February 2024
• 2015 AIME I Problems/Problem 13
  Let <math>x = \cos 1^\circ + i \sin 1^\circ</math>. Then from the identity <cmath>\sin 1 = \frac{x - \frac{1}{x}}{2i} = \frac{x^2 - 1}{2 i x},</cmath>
  9 KB (1,351 words) - 17:26, 16 January 2024
• 2015 AIME I Problems/Problem 6
  .../math> are equally spaced on a minor arc of a circle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on a minor arc of a second cir pair A,B,C,D,E,F,G,H,I,O;
  5 KB (782 words) - 16:04, 21 July 2023
• 2015 AIME II Problems/Problem 13
  ...dth such value will be when <math>k = 100</math> and <math>n = \lfloor 200\pi \rfloor = \lfloor 628.318 \rfloor = \boxed{628}</math>. Thus, we need <math>-1<n<0.</math> So we have <math>-1+2\pi k <n <0+2\pi k.</math>
  6 KB (1,034 words) - 21:29, 14 January 2024
• 2019 AMC 12A Problems
  for(int i=0;i <= 4;i=i+1) draw(shift((4*i,0)) * P);
  14 KB (2,180 words) - 22:25, 25 April 2024
• 2016 AMC 10B Problems/Problem 21
  ...{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math> ...;++i){draw(rotate(i*90,(0,0))*arc((1/2,1/2),sqrt(1/2),-45,135));dot(rotate(i*90,(0,0))*(1/2,1/2));}</asy>
  5 KB (761 words) - 23:26, 7 September 2022
• 2016 AMC 10B Problems/Problem 14
  ...t are integers lie entirely within the region bounded by the line <math>y=\pi x</math>, the line <math>y=-0.1</math> and the line <math>x=5.1?</math> for(int i=0;i<6;++i)for(int j=0;j<=3*i;++j)dot((i,j));
  3 KB (497 words) - 08:50, 9 March 2024
• 2016 AMC 12B Problems/Problem 18
  ...C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math> <cmath>A = \frac{\pi}{4} +\frac{1}{2}</cmath>
  1 KB (188 words) - 22:57, 9 January 2024
• 2016 AMC 12B Problems
  ...t are integers lie entirely within the region bounded by the line <math>y=\pi x</math>, the line <math>y=-0.1</math> and the line <math>x=5.1?</math> ...C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math>
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME I Problems/Problem 9
  real r = (pi/2 - asin(1/5))/2; B = 40*dir(r*180/pi);
  9 KB (1,526 words) - 02:31, 29 December 2021
• Gmaas
  -1. GMAAS looks like this when he is mad: https://cdn.artofproblemsolving.com/images/7/4/b/74b21 ...eorem. You wouldn't waste that much money to solve one problem. Therefore, I proved that you cannot use the Games theorem to solve problems. But using t
  69 KB (11,805 words) - 20:49, 18 December 2019
• 2006 iTest Problems
  ...TEST = 2006</math>. What is the largest possible value of the sum <math>I + T + E + S + T + 2006</math>? ...}\,-\frac{\sqrt{26}}{26}\quad\mathrm{(H)}\,\frac{\sqrt{2}}{2}\quad\mathrm{(I)}\,\text{none of the above}</math>
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAJMO Problems/Problem 5
  ...c{\pi}{2} - \beta.</math> (One could probably cite this as well-known, but I have proved it here just in case.) ...\frac{AO}{AQ}.</math> We also have <math>\angle PAH = \angle OAQ = \frac{\pi}{2} - \beta,</math> so <math>\triangle PAH\sim\triangle OAQ</math> by SAS s
  10 KB (1,733 words) - 19:15, 14 June 2020
• 1990 IMO Problems/Problem 6
  ...1,a_2,\cdots,a_{1990}\}=\{1,2,\cdots,1990\}</math> and <math>\theta=\frac{\pi}{995}</math>. ...th>a_1,a_2,\cdots,a_{1990}</math> such that <math>\sum_{i=1}^{1990}a_i^2e^{i\theta}=0</math>.
  3 KB (522 words) - 13:54, 30 January 2021
• 1984 USAMO Problems/Problem 5
  ...p+1}sin^{p}\left ( \frac{\pi}{3} \right )(i)^{p}\left ( cos\left ( \frac{p\pi}{3} \right ) \right )}{3}</math> ...1}sin^{p}\left ( \frac{\pi}{3} \right )(i)^{p+1}\left ( sin\left ( \frac{p\pi}{3} \right ) \right )}{3}</math>
  3 KB (443 words) - 21:54, 22 November 2023
• 1986 USAMO Problems/Problem 5
  ...ondecreasing order. (E.g., if <math>n=4,</math> then the partitions <math>\pi</math> are <math>1+1+1+1,</math> <math>1+1+2,</math> <math>1+3, 2+2,</math> ...artition <math>1+1+2+2+2+5,</math> then <math>A(\pi)=2</math> and <math>B(\pi) = 3</math>).
  5 KB (975 words) - 14:32, 30 August 2018
• 1975 Canadian MO Problems
  <div class=li><span class=num>(i)</span> <math>a_1 = \frac{1}{2}</math></div> ...less than or equal to <math>r</math>, <math>e.g.,</math> <math>[6] = 6, [\pi] = 3, [-1.5] = -2.</math> Indicate on the <math>(x,y)</math>-plane the set
  3 KB (499 words) - 12:17, 11 August 2016
• 2017 AMC 12A Problems/Problem 17
  ...these <math>z</math> such that <math>z^{24}=1</math> are <math>e^{\frac{ni\pi}{12}}</math> for integer <math>0\leq n<24</math>. So <math>z^6=e^{\frac{ni\pi}{2}}</math>
  3 KB (459 words) - 04:19, 2 February 2021
• 2017 AMC 10A Problems
  ...ath> units of line segment <math>\overline{AB}</math> has volume <math>216\pi</math>. What is the length <math>\textit{AB}</math>? ...t{3}-\frac{2\sqrt{3}\pi}{9}\qquad\textbf{(E)}\ \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}</math>
  15 KB (2,285 words) - 18:02, 28 October 2023
• 2017 AMC 12A Problems/Problem 25
  ...+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.</cmath> ...e of odd multiples of <math>\pi/4</math>, i.e. <math>\pi/4,3\pi/4,5\pi/4,7\pi/4</math>. When we draw these 6 complex numbers out on the complex plane, we
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 12A Problems
  ...th> units of line segment <math>\overline{AB}</math> has volume <math>216 \pi</math>. What is the length <math>AB</math>? ...+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.</cmath> For each <math>j</math>, <math>1\leq j\leq 12</math>, an
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2017 AMC 12B Problems
  ...(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math> ...i}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)}</math> It's not periodic.
  15 KB (2,343 words) - 18:26, 25 December 2020
• 2017 AMC 12B Problems/Problem 12
  ...f{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3}) + (1+\sqrt{3})i</math> ...12} = 64</math>, it is easy to see <math>\pm\sqrt{2}</math> and <math>\pm {i} \sqrt{2}</math> as roots. Graphing these in the complex plane, we have fou
  3 KB (449 words) - 01:54, 11 February 2019
• 2017 AIME I Problems/Problem 8
  Q = (10*cos(pi/3), 10*sin(pi/3)); R = (10*cos(5*pi/6), 10*sin(5*pi/6));
  9 KB (1,539 words) - 15:47, 17 February 2024
• 2017 AIME I Problems/Problem 10
  ...th>z_1=18+83i,~z_2=18+39i,</math> and <math>z_3=78+99i,</math> where <math>i=\sqrt{-1}.</math> Let <math>z</math> be the unique complex number with the ...f thinking of complex numbers as purely a real plus a constant times <math>i</math>, let’s graph them and hope that the geometric visualization adds i
  13 KB (2,252 words) - 15:46, 6 January 2024
• 2017 AIME I Problems/Problem 15
  ...lie on the hypotenuse <math>\frac{x}{5} + \frac{y}{2\sqrt{3}} = 1</math>, i.e. <math>a,b</math> must satisfy ...+bi)\cdot\cos(-\frac{\pi}{3})=(-\frac{a+\sqrt{3}b}{2}+\frac{\sqrt{3}a+b}{2}i)</math>. We know that the slope of <math>AC</math> is <math>-\frac{2\sqrt{3
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 USAMO Problems/Problem 5
  ...distance between any two points labeled <math>i</math> is at least <math>c^i</math>. ...For <math>c\le \sqrt[4]{2},</math> we can make a "checkerboard" labeling, i.e. label <math>(x, y)</math> with <math>1</math> if <math>x+y</math> is eve
  8 KB (1,495 words) - 12:19, 17 July 2023
• Reaper Archives
  | 86 || I-Can-Do-Math || 6 || 3059.612 || 509.935 | 76 || math-pi || 79 || 45682.709 || 578.262
  187 KB (10,824 words) - 18:27, 3 February 2022
• 2020 AMC 12A Problems
  for (int i = 0; i < 3; ++i) { pair A = (j,i);
  14 KB (2,073 words) - 15:15, 21 October 2021
• 2020 AMC 12B Problems
  ...f{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> ...textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math>
  16 KB (2,417 words) - 01:03, 28 April 2022
• MIE 2016
  ...th>. With <math>\beta</math> being a real number such that <math>0< \beta<\pi/8</math> and <math>x\neq0</math>, the value of <math>\beta</math> is: (a) <math>\frac{\pi}{9}</math>
  8 KB (1,278 words) - 09:46, 11 January 2018
• MIE 2015
  ...k=1}^{15}</math> Img<math>\left(\right.</math>cis<math>\left.^{2k-1}\frac{\pi}{36}\right)</math> (a) <math>\frac{2+\sqrt3}{4\sin\frac{\pi}{36}}</math>
  7 KB (1,127 words) - 18:23, 11 January 2018
• 2018 AMC 12A Problems/Problem 22
  ...ath>z^2=4+4\sqrt{15}i</math> and <math>z^2=2+2\sqrt 3i,</math> where <math>i=\sqrt{-1},</math> form the vertices of a parallelogram in the complex plane <li><math>z^2=4+4\sqrt{15}i</math><p>
  10 KB (1,662 words) - 12:45, 13 September 2021
• 2018 AMC 12A Problems
  ...st subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which ...+\sin(y)</cmath>for every <math>x</math> between <math>0</math> and <math>\pi</math>, inclusive?
  15 KB (2,380 words) - 18:52, 7 April 2022
• 2018 AMC 12B Problems
  \textbf{(A) }25\pi \qquad \textbf{(B) }50\pi \qquad
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 12B Problems/Problem 16
  ...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> ...{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4}.</math></li><p>
  11 KB (1,708 words) - 12:01, 18 March 2023
• 2018 AMC 12B Problems/Problem 25
  ...<math>P_iP_{i+1}</math> is tangent to <math>\omega_i</math> for each <math>i=1,2,3</math>, where <math>P_4 = P_1</math>. See the figure below. The are Let <math>O_i</math> be the center of circle <math>\omega_i</math> for <math>i=1,2,3</math>, and let <math>K</math> be the intersection of lines <math>O_1
  13 KB (2,080 words) - 19:09, 21 October 2023
• 1960 AHSME Problems/Problem 7
  ...er of, and is tangent to, circle <math>II</math>. The area of circle <math>I</math> is <math>4</math> square inches. \textbf{(C) }8\sqrt{\pi}\qquad
  2 KB (306 words) - 18:57, 17 May 2018
• 2018 AIME I Problems/Problem 6
  ...heta = \sin (\pi - \theta)</math> and <math>\sin \theta = \sin (\theta + 2\pi)</math>. We can use these facts to create two types of solutions: <cmath>\sin \theta = \sin ((2m + 1)\pi - \theta)</cmath>
  7 KB (1,211 words) - 00:23, 20 January 2024
• 2018 AIME I Problems/Problem 12
  <math>W_S = \sum_{i=1}^{|D|} \tbinom{6}{x_i}\tbinom{6}{y_i}</math> if and only if there exists ...>S</math> is divisible by 3. Therefore, by the fact that <math>W_S = \sum_{i=1}^{|D|} \tbinom{6}{x_i}\tbinom{6}{y_i}</math>, we have that;
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2018 AIME I Problems/Problem 10
  label("$I$", (2+r) * dir(162), dir(162)); ...ounter-Clockwise}</math>, and <math>\text{Switching}</math>. Let an <math>I</math> signal going clockwise (because it has to be in the ''inner'' circle
  11 KB (1,934 words) - 12:18, 29 March 2024
• 2018 AIME II Problems/Problem 5
  .../math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers < ...you could do the same thing with <math>xy</math> but <math>zx</math> looks like it's easier due to it being smaller. Anyway you get <math>x=20+12i</math>.
  11 KB (2,077 words) - 20:15, 12 January 2024
• 2008 iTest Problems/Problem 56
  ...enewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?" ...Oh," she thinks, "my wormhole allotment was <math>\textit{two}</math>, and I used it up already!"
  7 KB (1,092 words) - 19:05, 17 December 2021
• 2018 AMC 8 Problems
  for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray);
  14 KB (2,191 words) - 03:19, 2 April 2024
• 2006 iTest Problems/Problem 17
  \text{(I) }\frac{97}{98}\qquad \text{(N) }\pi\qquad
  2 KB (290 words) - 18:27, 30 November 2018
• 2019 AMC 10A Problems
  for(int i=0;i <= 4;i=i+1) draw(shift((4*i,0)) * P);
  13 KB (2,024 words) - 16:07, 22 April 2024
• Cyclotomic polynomial
  ...rac{2\pi k}{n}\right)=\cos\left(\frac{2\pi k}{n}\right)+i\sin\left(\frac{2\pi k}{n}\right)</cmath></center> <center><cmath>(e^{2\pi i k/n})^n=e^{n(2\pi i k/n)}=e^{2\pi i k}=1.</cmath></center>
  8 KB (1,438 words) - 14:50, 23 June 2022
• 2019 AIME II Problems/Problem 8
  ...19</math>, and <math>f\left(\tfrac{1+\sqrt{3}i}{2}\right)=2015+2019\sqrt{3}i</math>. Find the remainder when <math>f(1)</math> is divided by <math>1000< ...math>\frac{1+\sqrt{3}i}{2} = \omega</math> where <math>\omega = e^{\frac{i\pi}{3}}</math> is a primitive 6th root of unity. Then we have
  4 KB (706 words) - 22:18, 28 December 2023
• 2019 AIME II Problems/Problem 13
  D= \frac{1}{8} \pi r^2 - [A_1 A_2 O]=\frac{1}{8} \pi \left(4+2\sqrt{2}\right)- \left(\sqrt{2}+1\right) ...i r^2}{7} - D= \frac{1}{7} \pi \left(4+2\sqrt{2}\right)-\left(\frac{1}{8} \pi \left(4+2\sqrt{2}\right)- \left(\sqrt{2}+1\right)\right)
  7 KB (1,051 words) - 20:45, 27 January 2024
• 1993 IMO Problems
  ...ide acute triangle <math>ABC</math> such that <math>\angle ADB=\angle ACB+\pi/2</math> and <math>AC\cdot BD=AD\cdot BC</math>. ...>s_0, s_1, \ldots</math> as follows: at step <math>s_i</math>, if <math>L_{i-1}</math> is lit, switch <math>L_i</math> from on to off or vice versa, oth
  2 KB (452 words) - 17:59, 24 March 2019
• 2020 AMC 10B Problems
  ...f{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> ...textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math>
  16 KB (2,497 words) - 21:14, 12 November 2023
• 1981 AHSME Problems/Problem 24
  If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each po <cmath>x=\cos(\theta) + i\sin(\theta)</cmath>
  1 KB (220 words) - 15:24, 6 July 2021
• 2019 AMC 12A Problems/Problem 21
  Let <cmath>z=\frac{1+i}{\sqrt{2}}.</cmath>What is <cmath>\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{1 ...e^{i\pi}+...+e^{2i\pi}\right)\left(e^{\frac{i7\pi}{4}}+e^{i7\pi}+...+e^{2i\pi}\right)</cmath> which can easily be computed as <math>\boxed{36}</math>.
  5 KB (836 words) - 12:36, 3 December 2023
• 2019 AMC 10A Problems/Problem 21
  surface s=surface(f,(0,0),(pi,2pi),70,Spline); label("$I$",(12,4),SE);
  7 KB (1,000 words) - 15:03, 23 October 2021
• 2019 AMC 10A Problems/Problem 7
  Like in Solution 1, we determine the coordinates of the three vertices of the tr Like in the other solutions, solve the systems of equations to see that the tria
  7 KB (1,079 words) - 22:24, 10 November 2023
• 2019 AMC 10B Problems/Problem 20
  <cmath>\frac{a}{b}\cdot\pi-\sqrt{c}+d,</cmath> The area of <math>A</math> is <math>\frac{1}{2} \pi \cdot 2^2 = 2\pi</math>.
  6 KB (984 words) - 23:52, 11 November 2023
• 2019 AMC 12B Problems
  for (int i = 0; i < 3; ++i) { pair A = (j,i);
  16 KB (2,477 words) - 15:41, 9 September 2023
• 2019 AMC 12B Problems/Problem 17
  ...< \theta < \pi</math>. By symmetry, the interval <math>\pi \leq \theta < 2\pi</math> will also give <math>2</math> solutions. The answer is thus <math>2 ...rm an equilateral triangle, their difference in angle must be <math>\frac{\pi}{3}</math>, so
  6 KB (987 words) - 19:19, 12 November 2022
• 2019 AMC 12B Problems/Problem 24
  Let <math>\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.</math> Let <math>S</math> denote all points in the complex plane of ...c{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</math>
  4 KB (729 words) - 21:23, 15 November 2022
• 2019 AIME I Problems/Problem 12
  ...st have <cmath>\frac{f(f(z))-f(z)}{f(z)-z}=-\frac{f(f(z))-f(z)}{z-f(z)}\in i\mathbb R .</cmath>However, <math>f(t)-t=t(t-20)</math>, so ...perpendicular in the complex plane if and only if <math>\frac{a-b}{b-c}\in i\mathbb{R}</math>. To prove this, note that when dividing two complex number
  8 KB (1,534 words) - 22:17, 28 December 2023
• Euler's formula and It's history
  i (the unit imaginary number) π (the famous number pi that turns up in many interesting areas)
  3 KB (543 words) - 15:24, 13 June 2019
• What is the definition of Pure Mathematics?
  ===== pi and e ===== ...the limit <math>\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}...=\frac{\pi}{4}</math>. e is the limit <math>\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}...<
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2006 SMT/Calculus Problems/Problem 6
  <cmath> \int_{0}^{\pi/2}\frac{\sin^3x}{\sin^3x+\cos^3x}dx </cmath> Let <math>u=\frac{\pi}{2}-x</math> and <math>du=-dx</math>, the integral than turns into:
  1 KB (196 words) - 18:32, 14 January 2020
• 2006 SMT/Team Problems/Problem 15
  Let <math> c_i </math> denote the <math> i\text{th} </math> composite integer so that <math> \{c_i\}=4, 6, 8, 9\cdots <cmath> \prod_{i=1}^{\infty}\frac{c_i^2}{c_i^2-1} </cmath>
  3 KB (445 words) - 18:37, 14 January 2020
• 2020 AIME I Problems/Problem 8
  pair O, A, B, C, D, F, G, H, I, P, X; I = rotate(-120, H) * ((G + H) / 2);
  9 KB (1,380 words) - 16:12, 2 January 2024
• Brachistochrone
  ...then it moves <math>2\pi a</math> units, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second). the x-coordinate of the point relative to the center is: <math>a\cos(-\frac{\pi}{2}-t)=-a\sin t</math>
  2 KB (357 words) - 02:41, 27 July 2019
• Set of Complex Numbers
  The set of complex numbers is a set of all numbers ever existed, from pi to 1. ...a real part and an imaginary part, usually written in the from a + b<math>i</math>.
  2 KB (270 words) - 16:28, 13 June 2022
• Mock AMC 10B Problems
  ...\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad</math> ...math>. The area of Region <math>A</math> can be expressed as <math>\frac{a\pi+b\sqrt{c}}{d}</math>, where <math>a, b, c, d</math> are positive integers,
  13 KB (2,059 words) - 02:59, 21 January 2021
• GMAAS
  4. The Infinity Numeral, PI, is the second deity of the Almighty Gmaas's heaven.
  85 KB (13,954 words) - 17:25, 22 March 2024
• 2000 SMT/Advanced Topics Problems/Problem 2
  Simplify <math>\left(\frac{-1+i\sqrt{3}}{2}\right)^6+\left(\frac{-1-i\sqrt{3}}{2}\right)^6</math> to the form <math>a+bi</math>. ...s6\cdot\frac{4\pi}{3}+\sin6\cdot\frac{4\pi}{3}\right)=\left(\cos8\pi+\sin8\pi\right)=1+0=1.</math> Thus, the total sum is <math>1+1=\mathbf{2}.</math>
  1 KB (169 words) - 09:23, 24 July 2020
• 2000 SMT/Calculus Problems
  A rectangle of length <math>\frac{1}{4} \pi</math> and height 4 is bisected by the x-axis and is in the first and fourt For what value of <math>x</math> <math>(0 < x < \frac{\pi}{2})</math> does <math>\tan x + \cot x</math> achieve its minimum?
  3 KB (413 words) - 13:10, 21 January 2020
• 2008 Indonesia MO Problems
  ...ch the sum of its elements, when divided by 7, leaves the remainder <math> i</math>. Prove that <math> N(1) - N(2) + N(3) - N(4) + N(5) - N(6) = 0</math <cmath> \frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}.</cmath>
  3 KB (478 words) - 14:09, 23 June 2021
• 2020 AMC 12A Problems/Problem 9
  ...n <math>\tan(2x)=\cos(\tfrac{x}{2})</math> have on the interval <math>[0,2\pi]?</math> ...<math>x=\frac{\pi}{4}+\frac{k\pi}{2},</math> and zeros at <math>x=\frac{k\pi}{2}</math> for some integer <math>k.</math> <p>
  4 KB (615 words) - 04:07, 8 July 2022
• 2020 AMC 12A Problems/Problem 12
  ...get <math>-4\sqrt 2 - 16 \sqrt2 i</math> and <math>-3\sqrt 2 - 12 \sqrt 2 i</math>. This line has a slope of <math>4</math>. Now, back to the cartesian
  7 KB (1,145 words) - 20:27, 5 November 2023
• 2020 AMC 12A Problems/Problem 15
  ...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> <p> ...<math>3\theta=0,2\pi,4\pi,</math> or <math>\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}.</math> </li><p>
  4 KB (696 words) - 12:38, 13 September 2021
• 2020 AMC 12B Problems/Problem 25
  <cmath>\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1</cmath> <cmath>\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1</cmath>
  8 KB (1,412 words) - 06:17, 30 December 2023
• 2020 AMC 10B Problems/Problem 14
  ...quad \textbf{(D) } 3\sqrt3 - \pi \qquad \textbf{(E) } \frac{9\sqrt3}{2} - \pi</math> pair G,H,I,J,K;
  17 KB (2,392 words) - 12:36, 24 December 2023
• 2020 AMC 12B Problems/Problem 17
  ...root, so is <math>\frac{-1+i\sqrt{3}}{2} \cdot r</math>? (Note that <math>i=\sqrt{-1}</math>) ...3}}{2}</math> = <math>\cos(120^\circ) + i \cdot \sin(120^\circ) = e^{ 2\pi i / 3}</math>.
  4 KB (726 words) - 16:55, 11 September 2023
• 2020 AMC 12B Problems/Problem 23
  ...th> to be equal to <math>e^{i\frac{2\pi}{3}}</math> and <math>e^{-i\frac{2\pi}{3}}</math>, meaning that all three are equally spaced along the unit circl
  2 KB (306 words) - 17:45, 28 January 2024
• 2020 AIME I Problems/Problem 1
  A = (0, tan(3 * pi / 7)); [[File:2020 AIME I 1.png|450px|left]]
  6 KB (968 words) - 15:01, 24 January 2024
• 2022 AMC 10A Problems
  ...but exterior to the quadrilateral can be written in the form <math>\frac{a\pi-b}{c},</math> where <math>a,b,</math> and <math>c</math> are positive integ for(int i = 0; i < 3; ++i){
  18 KB (2,662 words) - 02:08, 9 March 2024
• AMC 12C 2020 Problems
  ...e equation <math>(x + i)^{10} = 1</math> on the complex plane, where <math>i = \sqrt -1</math>. <math>2</math> points from <math>K</math> are chosen, su ...\pi}{13}</math>, <math>\cos \frac{5\pi}{13}</math>, and <math>\cos \frac{7\pi}{13}</math>. What is the least possible sum of the coefficients of <math>P(
  8 KB (1,223 words) - 15:02, 27 November 2022
• Mathcounts Week/Countdown Round
  ...s, according to one of the 20 moderators. The activity began at 3:14 PM or pi time on the west coast and ended 106 minutes later at 5:00 PT. ...guarantee a point. The moderators could see who sent in the answer first, I would say that luck plays a incredibly small role in this." - [[User:Mathan
  11 KB (1,044 words) - 17:36, 11 January 2021
• 1995 IMO Problems/Problem 5
  ...the interior of the hexagon such that <math>\angle AGB=\angle DHE=\frac{2\pi}{3}</math>. Prove that <math>AG+GB+GH+DH+HE\ge CF</math>. ...\triangle EFA</math> and <math>\triangle BCD</math>, and draw points <math>I</math> and <math>J</math> such that <math>IA=IB</math>, <math>JD=JE</math>,
  1 KB (214 words) - 20:19, 5 July 2020
• 2021 AIME I Problems/Problem 7
  ...a</math> is <math>1</math>, is achieved at <math>\theta = \frac{\pi}{2}+2k\pi</math> for some integer <math>k</math>. ...that <math>mx = \frac{\pi}{2}+2a\pi</math> and <math>nx = \frac{\pi}{2}+2b\pi</math>, for integers <math>a, b</math>.
  9 KB (1,523 words) - 09:12, 3 December 2023
• 2021 AMC 12A Problems
  ...3+i \qquad \textbf{(C) }{-}\sqrt2+\sqrt2 i \qquad \textbf{(D) }{-}1+\sqrt3 i\qquad \textbf{(E) }2i</math> ...ft( \frac{\pi}2 \sin x\right)</math> have in the closed interval <math>[0,\pi]</math>?
  15 KB (2,383 words) - 09:49, 25 June 2023
• 2021 AMC 12A Problems/Problem 13
  ...3+i \qquad \textbf{(C) }{-}\sqrt2+\sqrt2 i \qquad \textbf{(D) }{-}1+\sqrt3 i\qquad \textbf{(E) }2i</math> Thus, the answer is <math>\boxed{\textbf{(B) }{-}\sqrt3+i}</math>.
  6 KB (872 words) - 17:36, 4 December 2021
• 2023 AIME II Problems
  ...mega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product <cmath>\prod_{k=0}^6 \lef ...[2023 AIME I Problems|2023 AIME I]]|after=[[2024 AIME I Problems|2024 AIME I]]}}
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2021 AMC 10B Problems
  How many integer values of <math>x</math> satisfy <math>|x| < 3\pi</math>? ...\qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math>
  17 KB (2,418 words) - 12:52, 5 November 2023
• 2022 AIME I Problems/Problem 4
  ...ve integers not exceeding <math>100</math> that satisfy the equation <math>i \cdot w^r = z^s.</math> w &= e^{i\cdot\frac{\pi}{6}}, \\
  5 KB (773 words) - 14:37, 23 February 2023
• 2022 AIME I Problems/Problem 11
  [[File:AIME-I-2022-11.png|530px|right]] ...erefore, <math>\frac{CI}{MC}=\frac{AI}{AN}</math>. Let the length of <math>PI=l</math>, then <math>\frac{25-l}{20}=\frac{3+l}{6}</math>. Solving we get <
  16 KB (2,517 words) - 20:22, 31 January 2024
• 2022 AIME I Problems/Problem 14
  ...th>r^2+r+1=(r-\omega)(r-{\omega}^2)</math>, where <math>\omega=e^{i\frac{2\pi}{3}}</math>. Thus, [[File:AIME-I-2022-14a.png|400px|right]]
  16 KB (2,730 words) - 02:56, 4 January 2023
• 2022 AIME I Problems/Problem 15
  ...>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math>, the system should look like this: Taking the inverse sine (<math>0\leq\theta\frac{\pi}{2}</math>) of each equation yields a simple system:
  15 KB (2,208 words) - 01:25, 1 February 2024
• 2019 CIME I Problems/Problem 12
  ...<math>z'=\frac{i}{z}</math> where <math>z \in \mathcal{T}</math> and <math>i=\sqrt{-1}</math>. If the area enclosed by <math>\mathcal{T}'</math> is <mat ...<math>75\pi+50</math>. The greatest integer less than or equal to <math>75\pi+50</math> is <math>\boxed{285}</math>.
  887 bytes (154 words) - 17:37, 4 October 2020
• 2022 AMC 12B Problems
  ...} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n </math>, where <math>i = \sqrt{-1}</math>. What is <math>f(2022)</math>? for (int i = 0; i <= 5; ++i) {
  15 KB (2,233 words) - 13:02, 10 November 2023
• 2021 Fall AMC 10B Problems
  for (int i=1; i<7; ++i) draw((i,0)--(i,5), gray+dashed);
  16 KB (2,450 words) - 00:13, 12 November 2023
• 2021 Fall AMC 10A Problems/Problem 15
  ...25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi</math> ...ath> is <math>\pi\cdot\left(\frac{AO_1}{2}\right)^2=\boxed{\textbf{(C) }26\pi}.</math>
  7 KB (1,026 words) - 21:23, 15 June 2023
• 2021 AMC 12A Problems/Problem 10
  \textbf{Narrow Cone} & 3 & h_1 & & \frac13\pi(3)^2h_1=3\pi h_1 & \\ [2ex] \textbf{Wide Cone} & 6 & h_2 & & \hspace{2mm}\frac13\pi(6)^2h_2=12\pi h_2 &
  9 KB (1,503 words) - 15:09, 1 August 2023
• 2021 AMC 12A Problems/Problem 19
  ...ft( \frac{\pi}2 \sin x\right)</math> have in the closed interval <math>[0,\pi]</math>? ...<math>\frac{\pi}2 \cos x</math> are both <math>\left[-\frac{\pi}2, \frac{\pi}2 \right],</math> which is included in the range of <math>\arcsin,</math> s
  7 KB (1,110 words) - 20:10, 5 November 2022
• 2021 AMC 12A Problems/Problem 22
  ...e <math>\cos \frac{2\pi}7,\cos \frac{4\pi}7,</math> and <math>\cos \frac{6\pi}7</math>, where angles are in radians. What is <math>abc</math>? ...\right)=\operatorname{Re}\left(z^{-k}\right)</math> and <math>\sin\frac{2k\pi}{7}=\operatorname{Im}\left(z^k\right)=-\operatorname{Im}\left(z^{-k}\right)
  9 KB (1,484 words) - 02:25, 21 September 2023
• 2020 INMO Problems/Problem 5
  ...ided regular polygon . Suppose <math>P_i =z .\zeta ^i</math> for all <math>i \in \{0,1,2,\cdots ,n -1\}</math>. Where <math>z</math> is a complex number ...that <math>l_k \equiv \ Im ({P_k})=y \cos \frac{2\pi k}{n} +x \sin \frac{2\pi k}{n}</math> .
  3 KB (607 words) - 03:12, 22 September 2023
• 2021 AMC 12B Problems
  How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math> How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath>
  15 KB (2,302 words) - 12:31, 27 October 2023
• 2021 CMC 12A Problems
  ...of <math>7\pi</math> and the total area covered by the circles is <math>25\pi</math>. What is the value of <math>r</math>? i. The perpendicular bisectors of the sides of <math>\mathcal{P}</math> all s
  5 KB (776 words) - 09:35, 8 August 2023
• 2022 AMC 12A Problems
  has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written ...nect points <math>(0, a_i)</math> to <math>(b_i, 0)</math> for <math>1 \le i \le 14</math> and are tangent to the circle, where <math>a_i</math>, <math>
  15 KB (2,250 words) - 00:32, 9 March 2024
• Functional equation for the zeta function
  \zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s) B_1(x)\triangleq\{x\}-\frac12=-\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}
  4 KB (682 words) - 03:56, 13 January 2021
• 1977 IMO Problems/Problem 1
  ...a = 1</math>, <math> b = i</math>, <math> c = - 1</math> and <math> d = - i</math>. ...t {3} - 1}{2}(1 + i)</math> and <math> n = \frac {\sqrt {3} - 1}{2}( - 1 + i)</math>.
  1 KB (199 words) - 16:40, 29 January 2021
• 1984 IMO Problems
  ==Day I== ...ere <math>\angle AOX</math> is measured in radians in the range <math>[0,2\pi)</math>. Prove that we can find a point <math>X</math>, not on <math>OA</ma
  2 KB (410 words) - 23:45, 29 January 2021
• 1984 IMO Problems/Problem 3
  ...ere <math>\angle AOX</math> is measured in radians in the range <math>[0,2\pi)</math>. Prove that we can find a point <math>X</math>, not on <math>OA</ma ...e the circle centered at <math>O</math> with radius <math>\sum_{i=1}^{n}a_{i}</math>.
  2 KB (388 words) - 23:49, 29 January 2021
• 2021 AMC 12B Problems/Problem 20
  ...We have our <math>z^{2021}</math> term but we have these unnecessary terms like <math>z^{2020}</math>. We can get rid of these terms by adding <math>-z^{20 z^3 &= e^{i 0} \\
  8 KB (1,315 words) - 11:43, 24 October 2023
• Laurent Series
  <center><math>C_m=\mathcal{C}(t;z_0+r_me^{it},0,2\pi)</math></center> <center><math>2\pi f(w)=\oint_{C_2}\frac{f(z)}{z-w}\,dz-\oint_{C_1}\frac{f(z)}{z-w}\,dz</math>
  8 KB (1,471 words) - 22:02, 12 April 2022
• 2021 GCIME Problems
  ...ss than or equal to <math>n</math>. Suppose <math>\pi(a)^{\pi(b)}=\pi(b)^{\pi(a)}=c</math>. For some fixed <math>c</math> what is the maximum possible nu ...ll positive integers <math>n<1000</math> such that <cmath>\sum_{i=1}^{n}a_{i}<4n.</cmath>
  4 KB (651 words) - 20:18, 6 March 2021
• 1976 AHSME Problems/Problem 24
  pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25),I=(2*sqrt(2)/3,1/3),E=(sqrt(2)/3,1/3),P=(0,0.25); draw(K--I,red);
  3 KB (462 words) - 07:58, 6 September 2021
• 2021 GMC 12
  ...} \qquad\textbf{(D)} ~\frac{\pi+2\sqrt{3}-3}{4} \qquad\textbf{(E)} ~\frac{\pi+3-2\sqrt{2}}{4}</math> ...f{(C)} ~\frac{(2-\sqrt{2})\pi}{8-2\pi} \qquad\textbf{(D)} ~\frac{\pi}{32-8\pi}\qquad</math>
  13 KB (2,097 words) - 17:38, 29 April 2021
• 2021 GMC 12B
  ...)} ~\frac{3\sqrt{3}}{2}-\frac{\pi}{3} \qquad\textbf{(E)} ~2\sqrt{3}-\frac{\pi}{3}</math> Let <math>n=(e^{({\sin(\pi)+\cos(\pi)})\pi})^{e^{\frac{i\pi}{2}}}</math>, find the remainder when<cmath>\left \lfloor{\sum_{k=0}^{20} 2
  11 KB (1,691 words) - 18:56, 25 April 2022
• 2021 Fall AMC 12B Problems/Problem 21
  <cmath>P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)</cmath> ...qrt{-1}</math>. For how many values of <math>x</math> with <math>0\leq x<2\pi</math> does
  6 KB (1,019 words) - 04:02, 9 October 2023
• 2022 AIME II Problems/Problem 11
  <b><i>Claim</b></i> <b><i> Proof</b></i>
  14 KB (2,254 words) - 18:26, 8 February 2024
• 2022 AIME II Problems/Problem 15
  ...}} e^{i \frac{\beta}{2}} = e^{i \frac{\alpha + \beta}{2}} = \frac{3}{5} + i \frac{4}{5}</math>. & = \frac{1}{2i} \left( e^{i \frac{\alpha}{2}} - e^{-i \frac{\alpha}{2}}
  14 KB (2,217 words) - 00:28, 29 June 2023
• 2022 AMC 10B Problems
  for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i));
  15 KB (2,224 words) - 13:10, 20 February 2024
• 2021 JMPSC Sprint Problems
  ...math>8</math> units tall. The volume of the cup can be written as <math>k \pi</math> cubic units. Find <math>k</math>. #David: Chandler told two truths. I am the oldest person in the room.
  7 KB (1,100 words) - 18:40, 11 July 2021
• Northeastern WOOTers Mock AIME I Problems/Problem 10
  <cmath> \theta_c = \pm \frac{\pi}{3} - \theta_a </cmath> From here, we determine <math>\theta_c = \pm \frac{\pi}{3}</math> and <math>c = \frac{1}{2} \pm \frac{\sqrt{3}}{2}</math>. Then we
  2 KB (377 words) - 14:08, 9 August 2021
• 2021 IMO Problems/Problem 4
  Let <math>\Gamma</math> be a circle with centre <math>I</math>, and <math>ABCD</math> a convex quadrilateral such that each of ...AB - \angle IAC</cmath> <cmath>\angle IZX = \angle IAB</cmath> Since <math>I</math> is the incenter of quadrilateral <math>ABCD</math>, <math>AI</math>
  7 KB (1,196 words) - 10:30, 18 June 2023
• 2021 Fall AMC 12A Problems
  pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)* .../math>, where <math>a</math> and <math>b</math> are real numbers and <math>i = \sqrt{-1}</math>, is the complex number <math>\overline{w} = a - bi</math
  15 KB (2,452 words) - 19:37, 7 June 2023
• 2021 Fall AMC 12B Problems
  for (int i=1; i<7; ++i) draw((i,0)--(i,5), gray+dashed);
  14 KB (2,191 words) - 19:57, 12 November 2023
• 2021 WSMO Accuracy Round Problems
  ...centimeters. If the surface area that is iced can be expressed as <math>m\pi,</math> find <math>m.</math> ..._i\right),</math> where <math>r_i</math> is the remainder when <math>2^i+3^i</math> is divided by 10.
  4 KB (695 words) - 12:37, 6 June 2022
• Differential equation
  ...<math>a = 0</math> are <math>C \cos x</math>; those with <math>a = -\frac{\pi}{2}</math> are <math>C \sin x</math>. The initial condition <math>f'(0) = 0 ...1})</math> can be computed in terms of <math>c_{i-1}</math> and <math>f(c_{i-1})</math> using the given differential equation), until <math>b = c_n</mat
  6 KB (969 words) - 20:33, 15 December 2023
• 2002 IMO Problems/Problem 2
  I like to make pics with Asymptote like this one: since I can still type my normal LaTeX stuff around it:
  2 KB (253 words) - 00:28, 19 November 2023
• 2022 AMC 12A Problems/Problem 21
  ...three, we need to make sure the roots are in the form of <math>e^{i\frac{k\pi}{9}}</math>, so we only have to look at <math>D,E</math>. If we look at choice <math>E</math>, <math>x=e^{i\frac{\pm2\pi}{9}}</math> which works perfectly, the answer is just <math>E</math>
  8 KB (1,325 words) - 02:14, 11 April 2024
• 2022 AMC 12A Problems/Problem 22
  Like the solutions above we can know that <math>|z_1| = |z_2| = \sqrt{10}</math> ...\sqrt{10}}e^{-i\theta} </math>, <math>\frac{1}{z_2}= \frac{1}{\sqrt{10}}e^{i\theta}</math>.
  7 KB (1,164 words) - 11:20, 1 January 2024
• 2022 AMC 10A Problems/Problem 21
  for(int i = 0; i < 3; ++i){ lVs.push(rho*lVs[i]);
  13 KB (2,080 words) - 11:27, 25 October 2023
• 2022 AMC 12B Problems/Problem 24
  for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) {
  21 KB (3,265 words) - 17:06, 15 November 2023
• 2022 AMC 12B Problems/Problem 11
  ...} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n </math>, where <math>i = \sqrt{-1}</math>. What is <math>f(2022)</math>? -1 + i\sqrt{3} &= 2e^{\frac{2\pi i}{3}}, \\
  5 KB (866 words) - 22:17, 27 October 2023
• 2022 AMC 12B Problems/Problem 5
  ...+i</math> in the complex plane, by an angle of <math>270^{\circ} = \frac{3\pi}{2} \text{ rad}</math> counterclockwise. ...)e^{\frac{3\pi i}{2}} + (3+i) = (-4-3i)(-i) + (3+i) = 4i-3i^2+3+i = 4i-3+3+i = 5i,</math> which corresponds to <math>\boxed{\textbf{(B)}\ (0,5)}</math>
  2 KB (278 words) - 13:32, 5 April 2023
• 2021 EGMO Problems
  Let <math>ABC</math> be a triangle with incenter <math>I</math> and let <math>D</math> be an arbitrary point on the side <math>BC</m (i) no three points in <math>P</math> lie on a line and
  3 KB (492 words) - 14:07, 24 December 2022
• 2023 AMC 12B Problems
  I. If <math>\gcd(a,14)=1</math> or <math>\gcd(b,15)=1</math> or both, then <m ...xt{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II a
  13 KB (2,107 words) - 22:19, 20 April 2024
• 2023 AMC 8 Problems/Problem 12
  int i, j; for(i=0; i<7; i=i+1)
  3 KB (452 words) - 14:33, 21 January 2024
• 2023 AIME I Problems/Problem 8
  AF + AE e^{i \left( \pi - \theta \right)} + EP e^{i \left( \frac{3 \pi}{2} - \theta \right)} - PF i .
  17 KB (2,612 words) - 14:54, 3 July 2023
• 2023 AIME II Problems/Problem 8
  ...mega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product<cmath>\prod_{k=0}^6 \left ...>z_n = \left(\textrm{cis }\frac{2n\pi}{7}\right)^3 + \textrm{cis }\frac{2n\pi}{7} + 1</math>.
  9 KB (1,284 words) - 23:37, 31 January 2024
• 2023 AMC 10A Problems/Problem 24
  for (int i = 0; i < 6; ++i) { pair next = current + side * dir(angle * i);
  13 KB (1,886 words) - 22:08, 10 April 2024
• 2023 AMC 10A Problems
  ...number of circles needed to make the total shaded area at least <math>2023\pi</math>? for(int i = 0; i < 4; ++i) {
  16 KB (2,411 words) - 15:11, 10 February 2024
• 2022 SSMO Relay Round 5 Problems
  ...})=70T</math>. Also, <math>\text{gcd}(a_i,a_j)=1</math> for all <math>1\le i<j\le2022</math>. Find the number of possible sequences <math>S</math>. ...<math>a+b\cos(\frac{\pi}{c})</math>, where <math>\text{cis}(x) = \cos(x) + i\sin(x)</math>. Find <math>a+b+c</math>.
  1 KB (217 words) - 21:59, 31 May 2023
• 2024 AIME I Problems
  {{AIME Problems|year=2024|n=I}} [[2024 AIME I Problems/Problem 1|Solution]]
  8 KB (1,307 words) - 20:00, 6 February 2024
• 2024 AIME I Answer Key
  ...educes two complicated equations to one linear and one quadratic equation. I can then easily find a non-zero solution and even get the closed form. ...s <math>\left( 1 - \frac{1}{8^2 \cdot 18 \pi^4} , 1 - \frac{1}{8 \cdot 18 \pi^3} \right) = \left( 1 - 8.9 \cdot 10^{-6}, 1 - 2.2 \cdot 10^{-4} \right)</m
  3 KB (515 words) - 03:57, 4 February 2024
• 2019 Mock AMC 10B Problems/Problem 12
  ...en the hypotenuse of the triangle and the curve of the semicircle be <math>I</math>. Then <math>OI</math> and <math>OB</math> are the radii of the semic ...area <math>= \frac{75\sqrt{3}}{4} + \frac{25}{2} \pi = \frac{75\sqrt{3}+50\pi}{4}</math>. Our answer <math>= 75 + 3+ 50 + 4 = \boxed{132}</math>.
  2 KB (276 words) - 20:22, 1 July 2023
• Know the Facts: Gmaas
  ...Pre-algebra class. He enters every class by posting a meme and announcing "I have arrived!" Sseraj is the human servant of Gmaas and is one of only 5 en - 2024 Multiverse war I erupts and Zzgurkk 3T is destroyed, along with many of Gmaas's bibles.
  88 KB (14,928 words) - 13:54, 29 April 2024
• Gmaas number
  1. The system has the digits 3,1,4,(one),5,9,p,i, and (gmaas). It is in base eight. i-7
  824 bytes (144 words) - 20:58, 3 November 2023
• 1999 IMO Problems/Problem 1
  ...\right),Rsin\left( \frac{2\pi}{n}i \right) \right\rangle</math> for <math>i=0,1,2,...,(n-1)</math>. <math>A=\left\langle Rcos\left( \frac{2\pi}{n}a \right),Rsin\left( \frac{2\pi}{n}a \right) \right\rangle</math>
  5 KB (955 words) - 02:21, 21 November 2023
• 2023 AMC 10B Problems
  I. If <math>\gcd(a,14)=1</math> or <math>\gcd(b,15)=1</math> or both, then <m ...xt{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II a
  13 KB (1,959 words) - 10:29, 4 April 2024
• 2023 AMC 10A Problems/Problem 15
  ...number of circles needed to make the total shaded area at least <math>2023\pi</math>? ...(2^2\pi-1^2\pi)+(4^2\pi-3^2\pi)+(6^2\pi-5^2\pi)+ \cdots + (n^2\pi-(n-1)^2 \pi)</math> for any even number <math>n</math>.
  6 KB (967 words) - 07:01, 28 January 2024
• 2023 AMC 12A Problems/Problem 16
  <cmath>|(1+a+a^2-b^2)+ (b+2ab)i|=4</cmath> Thus <math>z=i\sqrt{19}/2</math>, and so the answer is <math>\boxed{\textbf{(B)}~21}</math
  6 KB (1,055 words) - 21:58, 28 February 2024
• 2023 AMC 12B Problems/Problem 25
  pair F, G, H, I, J; I = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 - 3*72 + 180);
  19 KB (2,967 words) - 16:56, 24 February 2024
• Polya’s method for extremums
  <i><b>Solution</b></i> <i><b>Solution</b></i>
  12 KB (2,104 words) - 14:11, 24 February 2024
• 2023 AMC 12B Problems/Problem 20
  ...c{\arctan \frac{1}{2}}{\pi}\qquad\textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}</math> ...<math>1</math> unit away from <math>S</math> is <math>\frac{4 \alpha }{ 2 \pi}</math>.
  4 KB (587 words) - 18:15, 2 January 2024
• 2024 AIME I Problems/Problem 7
  &=(75a-117b)+(117a+75b)i+48\left(\dfrac{2+3i}{a+bi}\right) \\ &=(75a-117b)+(116a+75b)i+48\left(\dfrac{(2+3i)(a-bi)}{(a+bi)(a-bi)}\right) \\
  12 KB (1,842 words) - 19:26, 23 February 2024
• Mock AIME 6 2006-2007 Problems/Problem 7
  ...ts of <math>P_n(x)</math> will be in the form <math>x=e^{\frac{k}{n+1}2\pi i}</math> for <math>k=1,2,\cdots,n</math> with the only real solution when <m ...c{2}{1005}2\pi i},e^{\frac{3}{1005}2\pi i},\cdots,e^{\frac{1004}{1005}2\pi i}</math> for a total of <math>\textbf{1004}</math> complex roots.
  3 KB (442 words) - 20:51, 26 November 2023
• 2022 SSMO Relay Round 5 Problems/Problem 3
  ...<math>a+b\cos(\frac{\pi}{c})</math>, where <math>\text{cis}(x) = \cos(x) + i\sin(x)</math>. Find <math>a+b+c</math>.
  377 bytes (75 words) - 13:13, 14 December 2023
• 2009 OIM Problems/Problem 4
  ...e <math>A</math> with the circumcircle of <math>ABC</math>. The line <math>PI</math> intersects for the second time the circumcircle of <math>ABC</math>
  1 KB (191 words) - 04:59, 26 March 2024
• 2023 SSMO Relay Round 1 Problems
  ...and let <math>a_n = a_{n-1} - \frac{a_{n-3}}{8}</math>. Find <cmath>\sum_{i=0}^\infty a_i.</cmath> such <math>0 \le x \le \pi</math>
  640 bytes (94 words) - 21:32, 15 December 2023
• 2024 AIME I Problems/Problem 12
  ...of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath> ...th> as Function 1 and function <math>x = 4g \left( f \left( \cos \left( 3 \pi y \right) \right) \right)</math> as Function 2.
  8 KB (1,315 words) - 19:41, 18 February 2024
• 2002 AMC 12P Problems
  ...sin}^n x + \text{cos}^n x.</math> For how many <math>x</math> in <math>[0,\pi]</math> is it true that Find <math>i + 2i^2 +3i^3 + . . . + 2002i^{2002}.</math>
  10 KB (1,606 words) - 01:46, 31 December 2023
• Proofs of trig identities
  <math>\sin</math> and <math>\cos</math> are easy to define. I prefer the unit circle definition as it makes these proofs easier to unders Note: I've omitted <math>\theta</math> because it's unnecessary and might clog thin
  16 KB (2,796 words) - 13:12, 21 January 2024
• 2024 AIME II Problems/Problem 15
  for(int i=0; i<360; i+=30) { dot(dir(i), 4+black);
  8 KB (1,395 words) - 17:26, 9 February 2024
• DVI exam
  <i><b>Solution</b></i> ...BC</math> equal to <math>\frac {3}{2},</math> distance from incenter <math>I</math> to vertex <math>C</math> is <math>IC = 4 - \frac {3}{2} = \frac {5}{
  31 KB (5,191 words) - 08:10, 30 April 2024
• Weblogs
  ...^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}=2\underset{I}{\underbrace{\int_0^1{\frac{\ln ^2\left( \dfrac{2}{1+x} \right) \ln ^3\left ...\left( 1+x \right) ^2}}\text{d}x=\frac{1}{4}\zeta \left( 3 \right) +\frac{\pi ^2}{6}-\frac{1}{3}\ln ^32-\ln ^22-\text{2}\ln 2</cmath>
  17 KB (2,704 words) - 04:58, 29 January 2024
• 2024 AIME II Problems/Problem 12
  ...\frac{\sqrt{3}-2\sin \theta}{2\sqrt{3}-2\tan \theta}=\lim_{\theta\to\frac{\pi}{3}}cos^3{x}=\frac{1}{8}</math>. This means that <math>y=\frac{3\sqrt{3}}{8 pair A=(0.5,0); pair B=(0,sin(pi/3));
  10 KB (1,655 words) - 00:31, 11 April 2024
• 2024 AIME II Problems/Problem 13
  ...- \omega^k)^2 + 1\right) = \prod_{k=0}^{12} \left((1 + i) - \omega^k)((1 - i) - \omega^k\right)</cmath> <cmath>((1 + i)^{13} - 1)(1 - i)^{13} - 1)</cmath>
  5 KB (771 words) - 10:36, 19 April 2024
• Sharygin Olympiads, the best
  <i><b>Igor Fedorovich Sharygin</b></i> (13/02/1937 - 12/03/2004, Moscow) - Soviet and Russian mathematician and t <i><b>Solution</b></i>
  23 KB (4,003 words) - 16:17, 21 April 2024