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• AMC historical results
  ...at a consistent number of students qualify for AIME each year, rather than a fixed percentage. Usually, 6000-7000 competitors from the AMC 10 and 12 qua *Distinguished Honor Roll: 123
  17 KB (1,921 words) - 20:53, 10 May 2024
• Composite number
  A comprehensive video explaining composite numbers: https://youtu.be/SMOGYNYD A '''composite number''' is a [[positive integer]] with at least one [[divisor]] different from 1 and its
  6 KB (350 words) - 12:58, 26 September 2023
• 2006 AIME I Problems/Problem 6
  ...cimal| decimals]] of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct [[digit]]s. Find the sum of the elements of <mat ...erline{999}</math>, or, more precisely, 1. As an example, <math>.\overline{123}+.\overline{876}=.\overline{999} \Longrightarrow 1</math>.
  2 KB (237 words) - 19:14, 20 November 2023
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 1989 AIME Problems
  Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1987 AIME Problems/Problem 11
  ...h>m</math> we do indeed have the valid [[expression]] <math>3^{11} = 122 + 123 + \ldots + 607</math>, for which <math>k=\boxed{486}</math>. ...e of the two factors will be a power of three, and the other will be twice a power of three. <math>(2n + m + 1)</math> will represent the greater factor
  3 KB (418 words) - 18:30, 20 January 2024
• 1989 AIME Problems/Problem 8
  9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. ...tic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1993 AIME Problems/Problem 3
  :(a) the winner caught <math>15</math> fish; ...(9+5+7)=y-21</math> fishers that caught <math>3</math> or more fish caught a total of <math>x - \left(0\cdot(9) + 1\cdot(5) + 2\cdot(7)\right) = x - 19<
  2 KB (364 words) - 00:05, 9 July 2022
• 2005 AMC 10B Problems/Problem 25
  A [[subset]] <math>B</math> of the set of integers from <math>1</math> to <ma <math>\textbf{(A) } 50 \qquad \textbf{(B) } 51 \qquad \textbf{(C) } 62 \qquad \textbf{(D) }
  3 KB (517 words) - 19:15, 15 October 2023
• 2005 AMC 10A Problems/Problem 14
  <math> \textbf{(A) } 41\qquad \textbf{(B) } 42\qquad \textbf{(C) } 43\qquad \textbf{(D) } 44\ If the middle digit is <math>2</math>, possible numbers range from <math>123</math> to <math>420</math>. So there are <math>4</math> numbers in this cas
  4 KB (694 words) - 19:25, 13 December 2021
• Stirling number
  ...<math>n</math>-[[element]] [[set]] with exactly <math>k</math> [[cycle of a permutation | cycles]]. ...mutations <math>\{(1)(234), (1)(243), (134)(2),(143)(2),(124)(3),(142)(3),(123)(4),(132)(4), (12)(34), (13)(24), (14)(23)\}</math>.
  2 KB (253 words) - 00:50, 15 May 2024
• 2004 AMC 10A Problems
  {{AMC10 Problems|year=2004|ab=A}} ...and five friends need to raise <math>1500</math> dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of y
  15 KB (2,092 words) - 20:32, 15 April 2024
• 2007 AMC 12A Problems
  {{AMC12 Problems|year=2007|ab=A}} ...h> discount. Pam buys <math>5</math> tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  11 KB (1,750 words) - 13:35, 15 April 2022
• LaTeX:Help
  ...ase consult the AoPS [http://www.artofproblemsolving.com/Forum/index.php?f=123 message board]. ==== I'm using a Macintosh. What do I do? ====
  7 KB (1,109 words) - 20:01, 8 September 2022
• 2007 AMC 12A Problems/Problem 11
  ...ctively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term <math>\mathrm{(A)}\ 3\qquad \mathrm{(B)}\ 7\qquad \mathrm{(C)}\ 13\qquad \mathrm{(D)}\ 37\qq
  2 KB (243 words) - 01:15, 27 January 2024
• 2007 AMC 12A Problems/Problem 25
  Call a set of integers ''spacy'' if it contains no more than one out of any three <math>\mathrm{(A)}\ 121 \qquad \mathrm{(B)}\ 123 \qquad \mathrm{(C)}\ 125 \qquad \mathrm{(D)}\ 127 \qquad \mathrm{(E)}\ 129<
  9 KB (1,461 words) - 23:07, 27 January 2024
• 2004 AMC 12A Problems/Problem 22
  ...ually tangent spheres of radius <math>1</math> rest on a horizontal plane. A sphere of radius <math>2</math> rests on them. What is the distance from th ...ext {(B)}\ 3 + \frac {\sqrt {69}}{3} \qquad \text {(C)}\ 3 + \frac {\sqrt {123}}{4}\qquad \text {(D)}\ \frac {52}{9}\qquad \text {(E)}\ 3 + 2\sqrt2</math>
  2 KB (382 words) - 19:59, 29 January 2023
• 2008 AIME I Problems/Problem 4
  Since <math>244</math> is even, one of the factors is even. A [[parity]] check shows that if one of them is even, then both must be even. So <math>y^2-244</math> is a perfect square. Since 244 is even, the difference <math>\sqrt{y^2-244} -y^2
  4 KB (732 words) - 22:17, 28 November 2023
• 1988 AJHSME Problems
  The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of <math>\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)
  14 KB (1,872 words) - 15:23, 17 January 2023
• 2009 AIME I Problems/Problem 1
  ...s <math>3</math> distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest g ...onal of the form <math>k/3</math> for some integer <math>k</math>, because a whole number should be attained for the 3rd term as well. When <math>k = 1<
  2 KB (360 words) - 03:32, 16 January 2023
• 2009 AIME I Problems/Problem 7
  ...tions, so we try <math>125</math>: <cmath>3n + 2 = 125</cmath> <cmath>3n = 123</cmath> <cmath>n = \boxed{041}</cmath> <math>25</math> - that gives a non-integer <math>n</math>.
  2 KB (340 words) - 00:26, 9 January 2023
• 1989 AJHSME Problems
  <math>\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 <math>\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\
  13 KB (1,765 words) - 11:55, 22 November 2023
• 1988 AJHSME Problems/Problem 3
  <math>\text{(A)}\ .1 \qquad \text{(B)}\ .123 \qquad \text{(C)}\ .2 \qquad \text{(D)}\ .3 \qquad \text{(E)}\ .6</math> {{AJHSME box|year=1988|num-b=2|num-a=4}}
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• 1989 AJHSME Problems/Problem 14
  <math>\text{(A)}\ 58 \qquad \text{(B)}\ 123 \qquad \text{(C)}\ 149 \qquad \text{(D)}\ 171 \qquad \text{(E)}\ 176</math> ...math>a</math> is three digit and <math>b</math> is two digit, we set <math>a=245</math> and <math>b=96</math>. Their difference is <math>149\rightarrow
  887 bytes (126 words) - 11:49, 30 July 2023
• 2010 AIME I Problems/Problem 4
  Jackie and Phil have two fair coins and a third coin that comes up heads with [[probability]] <math>\frac47</math>. J ...is <math>3+10x+11x^2+4x^3</math>. (<math>ax^n</math> means there are <math>a</math> ways to get <math>n</math> heads, eg there are <math>10</math> ways
  3 KB (470 words) - 22:15, 27 August 2023
• 2011 AIME I Problems/Problem 3
  .../math> the <math>y</math>-axis. In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is ...>L</math> has slope <math>\frac{5}{12}</math> and contains the point <math>A=(24,-1)</math>, we may write the point-slope equation for <math>L</math> as
  4 KB (699 words) - 16:01, 13 April 2024
• 2003 AMC 10B Problems
  <math>\textbf{(A)} -1 \qquad\textbf{(B)} -\frac{2}{3} \qquad\textbf{(C)} \frac{2}{3} \qquad\ ...<math> \ </math><math>1</math> more than a pink pill, and Al's pills cost a total of <math> \ </math><math>546</math> for the two weeks. How much does
  15 KB (2,166 words) - 21:17, 16 February 2021
• 2003 AMC 10B Problems/Problem 25
  <math>\textbf{(A) } 27 \qquad\textbf{(B) } 30 \qquad\textbf{(C) } 33 \qquad\textbf{(D) } 81 ...three). However certain numbers can be added to make the sum of the digits a multiple of three.
  6 KB (882 words) - 11:45, 12 November 2023
• 1996 AJHSME Problems/Problem 14
  <math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qq ...e the three numbers on the right to <math>123</math>, the first number has a maximum value of <math>6</math>. This is the maximum for the intersection
  2 KB (254 words) - 00:24, 5 July 2013
• 1997 AHSME Problems
  If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </m
  17 KB (2,590 words) - 13:38, 19 February 2020
• 1997 AHSME Problems/Problem 14
  ...h>, and <math>1997</math> were <math>39</math>, <math>60</math>, and <math>123</math>, respectively, then the population in <math>1996</math> was <math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 84\qquad\textbf{(C)}\ 87\qquad\textbf{(D)}\ 90\qqu
  1 KB (217 words) - 14:13, 5 July 2013
• 2003 AMC 10B Problems/Problem 24
  ...d\textbf{(C)}\ 0\qquad\textbf{(D)}\ \frac{27}{20}\qquad\textbf{(E)}\ \frac{123}{40} </math> ...{x}{y}-2y=\frac{9\cdot 5}{8\cdot 3}+\frac{6}{5}=\boxed{\textbf{(E)}\ \frac{123}{40}}</cmath>
  4 KB (779 words) - 16:16, 12 March 2024
• Primitive Pythagorean Triple
  27 120 123 [9 - 40 - 41] 123 164 205 [3 - 4 - 5]
  55 KB (3,565 words) - 11:05, 21 May 2020
• 1964 AHSME Problems/Problem 3
  When a positive integer <math>x</math> is divided by a positive integer <math>y</math>, the quotient is <math>u</math> and the rem <math>\textbf{(A)}\ 0 \qquad
  2 KB (296 words) - 06:04, 31 December 2023
• 1991 AHSME Problems/Problem 26
  .../math>, for <math>k = 1,2,...,n</math>. For example, <math>321</math> is a cute <math>3</math>-digit integer because <math>1</math> divides <math>3</m <math>\text{(A) } 0\quad
  4 KB (604 words) - 03:40, 24 February 2018
• 1974 AHSME Problems
  <math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1974 AHSME Problems/Problem 26
  <math> \mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \ } 123 \qquad \mathrm{(D) \ } 30 \qquad \mathrm{(E) \ }\text{none of these} </m ...th> 1 </math> and <math> 30^4 </math>, so our final answer is <math> 125-2=123, \boxed{\text{C}} </math>}$
  895 bytes (116 words) - 02:27, 4 May 2024
• 1974 AHSME Answer Key
  #A #A
  2 KB (233 words) - 06:08, 20 January 2023
• 1980 USAMO Problems/Problem 2
  Find the maximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals. If <math>n = 3</math>, there will be one ascending triplet (123). Let's only consider the ascending order for now.
  2 KB (405 words) - 19:04, 9 August 2023
• 2003 AMC 12B Problems/Problem 8
  ...ath>x</math>. For example, <math>\clubsuit(8)=8</math> and <math>\clubsuit(123)=1+2+3=6</math>. For how many two-digit values of <math>x</math> is <math>\ <math>\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 9 \qqu
  1 KB (159 words) - 15:38, 30 March 2020
• 1957 AHSME Problems
  ...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2015 AIME II Problems/Problem 10
  Call a permutation <math>a_1, a_2, \ldots, a_n</math> of the integers <math>1, 2, When inserting an integer <math>n</math> into a string with <math>n - 1</math> integers, we notice that the integer <math>n
  2 KB (245 words) - 11:43, 20 December 2021
• Gmaas
  ...Plenipotentiary GMAAS cannot get rid of a language that's been around for a <math>999999999999999999999999999999999999999999^{9999999999999999999999999 ...ke and reciprocated the fee," you must pay <math>1,000,000</math> to solve a problem using the Games theorem. You wouldn't waste that much money to solv
  69 KB (11,805 words) - 20:49, 18 December 2019
• 2017 AMC 10B Problems/Problem 23
  <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\ ...1+12+ \cdots 43 + 44 = \frac{44 \cdot 45}2 = 22 \cdot 45</math>, which has a remainder of 0 mod 9. Solving these modular congruence using the [[Chinese
  3 KB (485 words) - 16:39, 29 September 2023
• Reaper Archives
  |- <!-- Start of a new row --> |- <!-- Start of a new row -->
  187 KB (10,824 words) - 18:27, 3 February 2022
• 2018 AIME II Problems/Problem 13
  Misha rolls a standard, fair six-sided die until she rolls <math>1-2-3</math> in that ord Call the probability you win on a certain toss <math>f_n</math>, where <math>n</math> is the toss number.
  11 KB (1,860 words) - 13:12, 24 January 2024
• 2004 Indonesia MO Problems
  A trough, if filled with cold water tap, will be full in 14 minutes. To empty ...th> and <math> c</math>. Show that all these tangent points are located on a circle.
  3 KB (478 words) - 13:45, 28 July 2018
• 2004 Indonesia MO Problems/Problem 5
  Given a system of equations: ...49x_6 + 64x_7 = 12\\9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 = 123\end{array}\right.</math>
  1 KB (171 words) - 18:14, 28 July 2018
• 2006 SMT/General Problems/Problem 13
  ...that no term "dominates" the sum, we see that the solution is <math>\boxed{123-45-67+89=100}</math>
  514 bytes (84 words) - 18:34, 14 January 2020
• GMAAS
  ...entiary Almighty Gmaas cannot get rid of a language that's been around for a <math>999999999999999999999999999999999999999999^{9999999999999999999999999
  85 KB (13,971 words) - 18:02, 16 May 2024
• Sieve of Sundaram
  The '''Sieve of Sundaram''' is a sieve for a range of '''odd''' [[prime number|prime numbers]].The sieve starts out list 121&122&123&124&125&126&127&128&129&130&131&132&133&134&135\\
  5 KB (580 words) - 18:10, 27 February 2020
• 2020 AIME II Problems/Problem 4
  ...m<180</math>, will transform <math>\triangle ABC</math> to <math>\triangle A'B'C'</math>. Find <math>m+x+y</math>. ...is done about <math>(x,y)</math>. Looking between <math>A</math> and <math>A'</math>, <math>x+y=18</math>. Thus <math>90+18=\boxed{108}</math>.
  10 KB (1,542 words) - 13:29, 19 January 2024
• 2021 AMC 10A Problems/Problem 20
  <math>\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \q ...id sequence if and only if <math>(6-x_1,6-x_2,6-x_3,6-x_4,6-x_5)</math> is a valid sequence. We enumerate the valid sequences that start with <math>1,2,
  14 KB (1,940 words) - 08:44, 3 October 2023
• 2021 MECC Mock AMC 10
  <math>\textbf{(A)} ~31 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~33 \qquad\textbf{(D)} ~34 ...sible ordered pair of positive integers <math>(a,b)</math> such that <math>a\%b=25</math>.
  12 KB (1,915 words) - 17:38, 29 April 2021
• 1957 AHSME Problems/Problem 19
  ...base of the decimal number system is ten, meaning, for example, that <math>123 = 1\cdot 10^2 + 2\cdot 10 + 3</math>. In the binary system, which has base <math>\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 10011\qquad \textbf{(D)}\
  1 KB (135 words) - 09:58, 29 March 2021
• 2021 GMC 12
  <math>\textbf{(A) } 6 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \ <math>\textbf{(A) } 20 \qquad\textbf{(B) } 24 \qquad\textbf{(C) } 28 \qquad\textbf{(D) } 32
  13 KB (2,097 words) - 17:38, 29 April 2021
• 2021 Fall AMC 12B Problems/Problem 18
  This sequence tends to a limit; call it <math>L</math>. What is the least value of <math>k</math> su <math>\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401</math>
  5 KB (733 words) - 10:36, 5 November 2022
• 2021 Fall AMC 12B Problems
  <math>\textbf{(A)}\: 10{,}000\qquad\textbf{(B)} \: 10{,}010\qquad\textbf{(C)} \: 10{,}110\qq <math>\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10
  14 KB (2,191 words) - 19:57, 12 November 2023
• 2022 AMC 10B Problems/Problem 21
  Let <math>P(x)</math> be a polynomial with rational coefficients such that when <math>P(x)</math> is d is <math>2x+1</math>. There is a unique polynomial of least degree with these two properties. What is the su
  7 KB (1,219 words) - 23:07, 19 September 2023
• Pell's equation (simple solutions)
  ...uation of the form <math>x^2 – Dy^2 = 1,</math> where <math>D</math> is a given positive nonsquare integer, and integer solutions are sought for <ma a) to construct a recurrent sequence <math>\{x_{i+1}, y_{i+1} \} = f({x_i, y_i});</math>
  15 KB (2,351 words) - 08:28, 20 April 2023
• Know the Facts: Gmaas
  ...e contributors section. WARNING: Security details will be checking once in a while, so please edit this page appropriately! ..., though his godly power can turn him into anything, he just prefers to be a cat. As you read on, you will discover more things about him then you will
  88 KB (14,928 words) - 13:54, 29 April 2024
• 2023 IOQM/Problem 1
  Let <math>n</math> be a [[positive integer]] such that <math>1 \leq n \leq 1000</math>. Let <math>M ...4n + 1}, \sqrt{4n + 2}, \ldots, \sqrt{4n + 1000}\right\}</math>. Let <math>a = \max\{M_n : 1 \leq n \leq 1000\}</math>, and <math>b = \min\{M_n : 1 \leq
  8 KB (670 words) - 22:07, 27 April 2024
• Mock AIME 6 2006-2007 Problems/Problem 5
  Then as a function of d, At <math>k=3</math>, <math>10k+93=123\;and\;86+k^2=95</math>, <math>123>95</math>.
  6 KB (974 words) - 23:02, 24 November 2023
• 2024 AMC 8 Problems/Problem 15
  <math>\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{ ...ath>. All the digits are different, so <math>FLY+BUG</math> would be <math>123+984</math>, which is <math>1107</math>. So, the answer is <math>\boxed{\tex
  4 KB (559 words) - 01:46, 18 April 2024
• 2025 AMC 8 Problems/Problem 11
  <math>\textbf{(A)}\ 122 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 124 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 126<
  249 bytes (34 words) - 07:06, 23 February 2024