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• 2000 AMC 12 Problems
  ...00^{4000} \qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000}</math> == Problem 4 ==
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits ...example, <math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  \qquad\mathrm{(C)}\ 4 when <math>x=4</math>?
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2003 AMC 12B Problems
  <cmath>\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?</cmath> == Problem 4 ==
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  == Problem 4 == [[2004 AMC 12B Problems/Problem 4|Solution]]
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? \mathrm{(B)}\ 4 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 14
  ...h>, <math>J</math> and <math>N</math> are all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? <math>253=N(4B+5J)</math>
  1 KB (227 words) - 17:21, 8 December 2013
• 2006 AMC 12B Problems/Problem 18
  ...e object only makes <math>1</math> move, it is obvious that there are only 4 possible points that the object can move to. At this point we can guess that for n moves, there are <math>(n + 1)^2</math> different endpoints. Thus, for 10 moves, there are <math>11^2
  2 KB (354 words) - 16:57, 28 December 2020
• 2006 AMC 12B Problems/Problem 20
  ...th hold at the same time if and only if <math>10^k \leq x < \frac{10^{k+1}}4</math>. ...math>k</math> the length of the interval <math>\left[ 10^k, \frac{10^{k+1}}4 \right)</math> is <math>\frac 32\cdot 10^k</math>.
  3 KB (485 words) - 14:09, 21 May 2021
• 2006 AMC 12B Problems/Problem 22
  ...divisible by <math>10</math>. What is the smallest possible value of <math>n</math>? n{5}\right\rfloor +
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12B Problems/Problem 23
  MP("B",D(B),plain.N,f); MP("B",D(B),plain.N,f);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12A Problems/Problem 17
  pair f = (4.34, 74.58); label("F", f, N);
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 23
  Given a finite sequence <math>S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence <math>\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)</math>
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  <cmath>2(x+x^3+x^5\cdots)(1+x^2+x^4\cdots)(1+x+x^2+x^3\cdots) = \frac{2x}{(1-x)^3(1+x)^2}</cmath> ...n-1}+...+P^{n-1}x+P^n)+(x^n-Px^{n-1}+...-P^{n-1}x+P^n)</math>, where <math>n=2006</math> (we may omit the coefficients, as we are seeking for the number
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  ...any ways are there to choose <math>k</math> elements from an ordered <math>n</math> element [[set]] without choosing two consecutive members? ...n with <math>k</math> elements where the largest possible element is <math>n-k+1</math>, with no restriction on consecutive numbers. Since this process
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  ...h>, and <math>f^{[n + 1]}(x) = f^{[n]}(f(x))</math> for each integer <math>n \geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> ...h>\frac{5}{2^{n+1}}</math>, <math>\cdots</math> ,<math>\frac{2^{n+1}-1}{2^{n+1}}</math>.
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12B Problems/Problem 12
  ...f <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>? ...xtbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qquad \textbf{(D) }\ {{{8}}} \qquad \textbf{(E) }\ {{{16}}}</math>
  2 KB (317 words) - 12:27, 16 December 2021
• 2005 AMC 12B Problems/Problem 21
  ...e greatest integer <math>k</math> such that <math>7^k</math> divides <math>n</math>? ...\mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
  888 bytes (140 words) - 20:04, 24 December 2020
• 2005 AMC 12B Problems/Problem 22
  <cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath> where <math>\overline {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
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 24
  ...th> are relatively prime positive integers. What is the value of <math>m + n</math>? draw((0,-0.5)--(0,4),Arrows);
  4 KB (761 words) - 09:10, 1 August 2023
• 2005 AMC 12B Problems/Problem 25
  ...osing which ant moves to <math>A</math>. Hence, there are <math>2 \times 2=4</math> ways the ants can move to different points. ...ath> can actually move to four different points, there is a total of <math>4 \times 20=80</math> ways the ants can move to different points.
  10 KB (1,840 words) - 21:35, 7 September 2023
• Power of a Point Theorem/Introductory Problem 4
  ...sect at a right angle at <math>E</math> . Given that <math> BE = 16, DE = 4, </math> and <math> AD = 5 </math>, find <math> CE </math>. pair D = (0,4);
  1 KB (177 words) - 02:14, 26 November 2020
• Root (polynomial)
  ...wice, triple roots three times, and so on, there are in fact exactly <math>n</math> complex roots of <math>P(x)</math>. ...1}x^{n-1}}{c_n} + \dots + \frac{c_1x}{c_n} + \frac{c_0}{c_n} = \sum_{j=0}^{n} \frac{c_jx^j}{c_n}.</cmath>
  8 KB (1,427 words) - 21:37, 13 March 2022
• 2006 AMC 10A Problems
  == Problem 4 == [[2006 AMC 10A Problems/Problem 4|Solution]]
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 12
  MP('8', (16,-4), W); MP('8', (20,-8), N);
  3 KB (424 words) - 10:14, 17 December 2021
• 2006 AMC 10A Problems/Problem 16
  D((0,0)--(4*t,0)--(2*t,8)--cycle); D('B', (0,0),SW); D('C',(4*t,0), SE); D('A', (2*t, 8), N);
  5 KB (732 words) - 23:19, 19 September 2023
• 2006 AMC 10A Problems/Problem 17
  D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
  6 KB (1,066 words) - 00:21, 2 February 2023
• Set
  ...multiple indistinct elements, such as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset. ...t of size <math>n</math> then <math>\mathcal{P}(A)</math> has size <math>2^n</math>.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  label("\Large{$\Gamma_-$}",(-.45,.4)); Now, let <math>N(R)</math> be an upper bound for the quantity
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2006 USAMO Problems
  ...that there exist integers <math>m</math> and <math>n</math> with <math>0<m<n<p</math> and ...th>N </math> but every subset of size <math>k</math> has sum at most <math>N/2</math>.
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  <math>\frac{16+8}{4-2}=</math> <math>\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \te
  17 KB (2,246 words) - 13:37, 19 February 2020
• Olimpiada de Mayo/1997
  * [[1997 I Problems/Problem 4|Problem 4]] * [[1997 II Problems/Problem 4|Problem 4]]
  856 bytes (98 words) - 14:53, 3 July 2009
• 2022 USAJMO Problems/Problem 1
  ...<math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>; ...\nmid d</math> and <math>m|a+(n-1)d-gr^{n-1}</math> for all integers <math>n>1</math>.
  4 KB (792 words) - 00:29, 13 April 2024
• 2006 AIME II Problems/Problem 2
  .../math> is a positive integer. Find the number of possible values for <math>n</math>. <math>\log_{10} 12 + \log_{10} n > \log_{10} 75 </math>
  1 KB (164 words) - 14:58, 14 April 2020
• 2005 AIME II Problems/Problem 1
  ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...<math>{n \choose 6} = \frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1}</math>.
  1 KB (239 words) - 11:54, 31 July 2023
• 2005 AIME II Problems/Problem 2
  ...and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </math> *Person 2: <math>\frac{6 \cdot 4 \cdot 2}{6 \cdot 5 \cdot 4} = \frac 25</math>
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 3
  ...and <math> n </math> are [[relatively prime]] [[integer]]s. Find <math> m+n. </math> ...-r} = 2005</math>. Then we form a new series, <math>a^2 + a^2 r^2 + a^2 r^4 + \ldots</math>. We know this series has sum <math>20050 = \frac{a^2}{1 -
  3 KB (581 words) - 07:54, 4 November 2022
• 2005 AIME II Problems/Problem 5
  ...th power are distinct, so they are not redundant. (For example, the pairs (4, 64) and (8, 64).) {{AIME box|year=2005|n=II|num-b=4|num-a=6}}
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems
  {{AIME Problems|year=2005|n=II}} ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math>
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math>(x+1)^{48}</math ...\sqrt[2^n]{5} + 1)(\sqrt[2^n]{5} - 1) = (\sqrt[2^n]{5})^2 - 1^2 = \sqrt[2^{n-1}]{5} - 1 </math>.
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 8
  ...> n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> pair C1 = (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-acos(3/7));
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 9
  ...ath> less than 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>? ...</math> for all [[real 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
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ..., D=(0,0), E=(2.5-0.5*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), G=(4.5,9), O=(4.5,4.5); draw(A--B--C--D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^ ...90); draw(A--B--C--D--A);draw(E--O--F);draw(G--O--J);draw(F--G,linetype("4 4")); dot(A^^B^^C^^D^^E^^F^^G^^J^^O); label("\(A\)",A,(-1,1));label("\(B\)",B
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...</math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> import three; currentprojection = perspective(4,-15,4); defaultpen(linewidth(0.7));
  3 KB (436 words) - 03:10, 23 September 2020
• Euler's identity
  ...states that for any [[real number]] <math>\theta</math> and integer <math>n</math>, ...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
  {{AIME Problems|year=2005|n=I}} ...<math> k. </math> For example, <math> S_3 </math> is the sequence <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> does <math> S_
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 2
  ...h> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> does <math> S_ ...th>(12,167)</math>, <math>(167,12)</math>,<math>(334,6)</math>, <math>(501,4)</math>, <math>(668,3)</math>, <math>(1002,2)</math> and <math>(2004,1)</ma
  2 KB (303 words) - 01:31, 5 December 2022
• 2005 AIME I Problems/Problem 3
  ...s,, so <math>n</math> must be in the form <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math> In the first case, the three proper divisors of <math>n</math> are <math>1</math>, <math>p</math> and <math>q</math>. Thus, we nee
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 4
  ...<math>n \leq 14</math>. In fact, when <math>n = 14</math> we have <math>n(n + 7) = 14\cdot 21 = 294 = 17^2 + 5</math>, so this number works and no larg ...</math>. The [[quadratic formula]] yields <math>r = \frac{7 \pm \sqrt{49 - 4(1)(-s^2 - 5)}}{2} = \frac{7 \pm \sqrt{4s^2 + 69}}{2}</math>. <math>\sqrt{4s
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on o ...and what coins are silver, so the solution is <math>\boxed{9\cdot \binom 8 4=630}</math>.
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 6
  Let <math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>. Thus, we add 1 to each side in order to complete the fourth power
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  label("$10$",(2.5,4.5),W); label("$10$",(18.37,4.5),E);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...h> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...hen the previous statement says that <math>2^{111\cdot(x_1 + x_2 + x_3)} = 4</math> so taking a [[logarithm]] of that gives <math>111(x_1 + x_2 + x_3) =
  1 KB (161 words) - 19:50, 2 January 2022
• 2005 AIME I Problems/Problem 9
  ...cube, we need that they show an orange face. This happens in <math>\frac{4}{6} = \frac{2}{3}</math> of all orientations, so from these cubes we gain a ...the corner cubes together contribute a probability of <math>\left(\frac{1}{4}\right)^8 = \frac{1}{2^{16}}</math>
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 10
  == Solution 4 == {{AIME box|year=2005|n=I|num-b=9|num-a=11}}
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  ...mum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> === Solution 4 ===
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...e the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math> .... So <math>S(1), S(2)</math> and <math>S(3)</math> are odd, while <math>S(4), S(5), \ldots, S(8)</math> are even, and <math>S(9), \ldots, S(15)</math>
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 13
  ..., and <math>3</math> <math>D</math>'s, so the string is divided into <math>4</math> partitions (<math>-D-D-D-</math>). ...<math>R</math>'s and <math>U</math>'s stay together, then there are <math>4 \cdot 3 = 12</math> places to put them.
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME I Problems/Problem 14
  Consider the [[point]]s <math> A(0,12), B(10,9), C(8,0),</math> and <math> D(-4,7). </math> There is a unique [[square]] <math> S </math> such that each of ...th>AE = BD</math>, we have <math>9 - 7 = x_E - 0</math> and <math>10 - ( - 4) = 12 - y_E</math>
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  ...| divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math> D(MP("A",A,s)--MP("B",B,s)--MP("C",C,N,s)--cycle); D(cir);
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  r + 4 &= \sqrt{(x-5)^2 + (y-12)^2} \\ D(CR(A,16));D(CR(B,4));D(shift((0,12)) * yscale(3^.5 / 2) * CR(C,10), linetype("2 2") + d + red)
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  dotfactor = 4; label("$A$",A,N);
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  ...itive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> an ...icting our assumption that <math>20</math> was the smallest value of <math>n</math>. Using [[complementary counting]], we see that there are only <math>
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...> feet. The unicorn has pulled the rope taut, the end of the rope is <math>4</math> feet from the nearest point on the tower, and the length of the rope real x = 20 - ((750)^.5)/3, CE = 8*(6^.5) - 4*(5^.5), CD = 8*(6^.5), h = 4*CE/CD;
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  ...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> ...he form <math>\frac m{19}</math> or <math>\frac n {17}</math> for <math>m, n > 0</math>.
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 12
  ...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> The notation <math> [z] </math> denotes the [[floor function|great <cmath>x \in \left(\frac{1}{2},1\right) \cup \left(\frac{1}{8},\frac{1}{4}\right) \cup \left(\frac{1}{32},\frac{1}{16}\right) \cup \cdots</cmath>
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 11
  ...h> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...lid has volume equal to <math>V = \frac13 \pi r^2 h = \frac13 \pi 3^2\cdot 4 = 12 \pi</math> and has [[surface area]] <math>A = \pi r^2 + \pi r \ell</ma
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 9
  ...<math> n</math> are [[relatively prime]] positive integers. Find <math> m+n. </math> ...ABC</math>. Thus <math>U_1</math>, and hence <math>U_2</math>, are <math>3-4-5\,\triangle</math>s.
  4 KB (618 words) - 20:01, 4 July 2013
• 2004 AIME I Problems/Problem 1
  ...from left to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...+ 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
  2 KB (374 words) - 14:53, 27 December 2019
• 2004 AIME I Problems/Problem 2
  Solving for <math>f+l</math>, we find the sum of the two terms is <math>4</math>. <cmath>2(x-(-x+4)+1) = 1+(x+99)-(-x-99+1)</cmath>
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 3
  {{AIME box|year=2004|n=I|num-b=2|num-a=4}}
  1 KB (156 words) - 17:56, 1 January 2016
• 2004 AIME I Problems/Problem 4
  ...h> and <math>(0,y)</math>. Because the segment has length 2, <math>x^2+y^2=4</math>. Using the midpoint formula, we find that the midpoint of the segmen \sqrt{\frac{1}{4}\left(x^2+y^2\right)}=\sqrt{\frac{1}{4}(4)}=1</math>. Thus the midpoints lying on the sides determined by vertex <mat
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 5
  ...<math> n </math> are relatively prime positive integers. What is <math> m+n </math>? ...'s success ratio <math>\frac{349}{500}</math>. Thus, the answer is <math>m+n = 349 + 500 = \boxed{849}</math>.
  3 KB (436 words) - 18:31, 9 January 2024
• 2004 AIME I Problems/Problem 6
  ...3</math>, <math>x_3x_4x_1x_2</math>. Thus there are <math>5\cdot {9\choose 4}=630</math> snakelike numbers which do not contain the digit zero. ...ath>2\cdot{9 \choose 3}</math>. Thus our answer is <math>5\cdot{10 \choose 4} - 2\cdot{9 \choose 3} = \boxed{882}</math>
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  ...ath> <math>= (1 - x^2)(1 - 4x^2)\cdots(1 - 225x^2)</math> <math>= 1 - (1 + 4 + \ldots + 225)x^2 + R(x)</math>. Equating coefficients, we have <math>2C - 64 = -(1 + 4 + \ldots + 225) = -1240</math>, so <math>-2C = 1176</math> and <math>|C| =
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME I Problems/Problem 8
  Define a regular <math> n </math>-pointed star to be the union of <math> n </math> line segments <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> such tha * each of the <math> n </math> line segments intersects at least one of the other line segments at
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  {{AIME Problems|year=2004|n=I}} ...from left to right. What is the sum of the possible remainders when <math> n </math> is divided by 37?
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...olution. Call <math>s_{n, k}</math> the number of squares below the <math>n</math> square after the final fold in a strip of length <math>2^{k}</math>. ...the <math>s_{n, k}</math> value of the pairs correspond with the <math>s_{n, k - 1}</math> values - specifically, double, and maybe <math>+ 1</math> (i
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...d from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulti ...1000</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 12
  ...> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> ...lack; pathpen = black+linewidth(0.7); pen d = linewidth(0.7) + linetype("4 4"); pen f = fontsize(8);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 11
  ...ctor, which gives a distance <math>\sqrt{(-375-125)^2+(-375-0)^2}=125\sqrt{4^2+3^2}=\boxed{625}</math>. {{AIME box|year=2004|n=II|num-b=10|num-a=12}}
  2 KB (268 words) - 22:20, 23 March 2023
• 2004 AIME II Problems/Problem 10
  ...and <math>y-x \equiv 3 \pmod 6 = 6n +3</math> for some whole number <math>n</math>. ...+31+25+\ldots+1 = 7 + 36 + 30 + 24 + \ldots + 6 + 0 = 7 + 6 \cdot (6 + 5 + 4\ldots + 1) </math>
  7 KB (1,091 words) - 18:41, 4 January 2024
• 2004 AIME II Problems/Problem 9
  ...est term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...(n-1)f(n)</math>, <math>a_{2n+1} = f(n)^2</math>, where <math>f(n) = nx - (n-1)</math>.{{ref|1}}
  3 KB (538 words) - 21:33, 30 December 2023
• 2004 AIME II Problems/Problem 8
  ...t to partitioning two items in three containers. We can do this in <math>{4 \choose 2} = 6</math> ways. We can partition the 3 in three ways and likew 4, 3, 167
  2 KB (353 words) - 18:08, 25 November 2023
• 2004 AIME II Problems/Problem 7
  ...and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> draw(E--B--C--F, linetype("4 4"));
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 6
  ...c{11}{24}b_3</math>, the second monkey got <math>\frac{1}{8}b_1 + \frac{1}{4}b_2 + \frac{11}{24}b_3</math>, and the third monkey got <math>\frac{1}{8}b_ <math>x = \frac{1}{4}b_1 + \frac{1}{8}b_2 + \frac{11}{72}b_3 = \frac{1}{16}b_1 + \frac{1}{8}b_2
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  ...e initial problem statement, we have <math>1000w\cdot\frac{1}{4}t=\frac{1}{4}</math>. ...lete the same amount of work, which is <math>\frac{1000}{900}\cdot\frac{1}{4}\cdot t=\frac{5}{18}t</math>.
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  ...>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> such numbers (or we can list them: <math>AB, BA, AAB, ABA, ...he other digit. For each choice, we have <math>2^{n - 1} - 1</math> <math>n</math>-digit numbers we can form, for a total of <math>(2^0 - 1) + (2^1 - 1
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems/Problem 3
  ...the 1-cm cubes cannot be seen. Find the smallest possible value of <math> N. </math> ...xtra layer makes the entire block <math>4\times8\times12</math>, and <math>N= \boxed{384}</math>.
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems/Problem 1
  ...rac{1}{2} \cdot \frac{r}{2} \cdot \frac{r\sqrt{3}}{2} = \frac{r^2\sqrt{3}}{4}</math>. The [[central angle]] which contains the entire chord is <math>60 ...{4}}{\frac{1}{3}r^2\pi - \frac{r^2\sqrt{3}}{4}} = \frac{8\pi + 3\sqrt{3}}{4\pi - 3\sqrt{3}}</math>
  2 KB (329 words) - 23:20, 4 July 2013
• 2004 AIME II Problems
  {{AIME Problems|year=2004|n=II}} ...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math>
  9 KB (1,410 words) - 05:05, 20 February 2019
• 2000 AIME II
  * [[2000 AIME II Problems/Problem 4|Problem 4]] {{AIME box|year=2000|n=II|before=[[2000 AIME I]]|after=[[2001 AIME I]], [[2001 AIME II | II]]}}
  1 KB (139 words) - 08:41, 7 September 2011
• 2001 AIME I
  * [[2001 AIME I Problems/Problem 4|Problem 4]] {{AIME box|year=2001|n=I|before=[[2000 AIME I]], [[2000 AIME II|II]]|after=[[2001 AIME II]]}}
  1 KB (139 words) - 08:41, 7 September 2011
• 2000 AIME I
  * [[2000 AIME I Problems/Problem 4|Problem 4]] {{AIME box|year=2000|n=I|before=[[1999 AIME]]|after=[[2000 AIME II]]}}
  1 KB (135 words) - 18:05, 30 May 2015
• 1983 AIME Problems
  == Problem 4 == [[1983 AIME Problems/Problem 4|Solution]]
  7 KB (1,104 words) - 12:53, 6 July 2022
• 2019 AIME I Problems/Problem 2
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...2</math> elements are adjacent. Using the well-known formula <math>\dbinom{n-k+1}{k}</math>, there are <math>\dbinom{20-2+1}{2} = \dbinom{19}{2} = 171</
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ...h>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. ...<math>t_{2}</math>, and <math>t_{3}</math> in the figure, have areas <math>4</math>, <math>9</math>, and <math>49</math>, respectively. Find the area of
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  ...m of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>? == Problem 4 ==
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  .../math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of sim == Problem 4 ==
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...ts of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. == Problem 4 ==
  6 KB (902 words) - 08:57, 19 June 2021

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