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• 2004 AIME I Problems/Problem 1
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi A brute-force solution to this question is fairly quick, but we'll try someth
  2 KB (374 words) - 14:53, 27 December 2019
• 1983 AIME Problems/Problem 10
  ...<math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-digit number beginning with <math>1</math> that has exactly Consider a sequence of <math>4</math> digits instead of a <math>4</math>-digit number. Only looking at the sequences which have one d
  5 KB (855 words) - 20:26, 14 January 2023
• Base numbers/Conversion
  The next natural question is: how do we convert a number from another base into base 10? For example, what does <math>4201_5 ...der for us than converting from other bases to base 10. This shouldn't be a surprise, though. We work in base 10 ''all the time'' so we are naturally
  7 KB (1,177 words) - 15:56, 18 April 2020
• 2008 iTest Problems
  ...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 48
  A repunit is a natural number whose digits are all <math>1</math>. For instance, <cmath>1,11,111,1111, \ldots</cmath>
  988 bytes (145 words) - 17:14, 14 July 2018
• 1997 AHSME Problems/Problem 30
  <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qqu ...begin with a <math>1</math>, and if you switch digits twice, you will have a <math>1</math> at the end. Thus, we are only concerned with the <math>49</
  10 KB (1,552 words) - 21:01, 5 October 2023
• Primitive Pythagorean Triple
  220 1089 1111 [20 - 99 - 101] Here is a more complete list up to 10,000 [with thanks to Tom Wallett] (2.8Mb)
  55 KB (3,565 words) - 11:05, 21 May 2020
• 2017 AMC 8 Problems
  <math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo
  12 KB (1,771 words) - 21:13, 20 January 2024
• 1994 AHSME Problems/Problem 21
  <math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qqu In the first case, <math>N = 1111 = 101 \cdot 11</math> so this is a counter example.
  1 KB (204 words) - 14:11, 28 May 2021
• 1978 AHSME Problems
  <math>\textbf{(A) }-1\qquad If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
  15 KB (2,432 words) - 01:06, 22 February 2024
• 2016 AMC 12A Problems/Problem 25
  ...st perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then w <math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\
  8 KB (1,224 words) - 18:28, 16 November 2022
• 2017 AMC 8 Problems/Problem 7
  ...igits taken in the same order. Which of the following numbers must also be a factor of <math>Z</math>? ...xtbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111</math>
  3 KB (451 words) - 18:22, 21 January 2024
• 2018 AMC 10A Problems/Problem 25
  ...s is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> su <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) }
  6 KB (1,001 words) - 16:50, 21 January 2024
• 2008 iTest Problems/Problem 98
  ...<math>AB=7</math>, <math>BC=9</math>, <math>CD=15</math>, and there exists a circle, lying inside the quadrilateral and having center <math>I</math>, th ...pposite sides of <math>QMPN</math> have equal length, <math>QMPN</math> is a [[parallelogram]]. Thus, the diagonals bisect each other, and <math>QI = IP
  3 KB (586 words) - 23:47, 8 January 2019
• 2019 AIME I Problems/Problem 1
  ...10-321=789</math>. This method will remove three <math>1</math>'s, and add a <math>7</math>, <math>8</math> and <math>9</math>. Therefore, the sum of th A similar and simpler way to consider the initial manipulations is to observe
  3 KB (433 words) - 07:57, 9 February 2023
• 1978 AHSME Problems/Problem 4
  ...ath>d = 1000</math>, then <math>(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d)</math> is equal to <math>\textbf{(A) }1111\qquad
  549 bytes (86 words) - 12:00, 13 February 2021
• 2021 JMPSC Accuracy Problems/Problem 15
  ...ath>1(100)+10(99)+100(98)+1000(97)+ \cdots + 10^{100}</math>, so factoring a <math>10</math> we have <math>1(10)+99+10(98)+ \cdots + 10^{99}</math>. Now
  2 KB (304 words) - 01:19, 12 July 2021
• 2022 AIME I Problems/Problem 13
  ...orm <math>0.\overline{abcd},</math> where at least one of the digits <math>a,</math> <math>b,</math> <math>c,</math> or <math>d</math> is nonzero. Let < Make cases by factors of <math>x</math>. (A venn diagram of cases would be nice here.)
  7 KB (1,207 words) - 21:53, 2 January 2024
• 1957 AHSME Problems/Problem 19
  <math>\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 10011\qquad \textbf{(D)}\ ...wo every single place to the left. For example, <math>1111_2</math> (<math>1111</math> in base <math>2</math>) would equate to <math>1 * 2^3 + 1 * 2^2 + 1
  1 KB (135 words) - 09:58, 29 March 2021
• 2021 GMC 12B
  When <math>30\%</math> of <math>x</math> is a positive perfect square integer, what is <math>min(x)</math> such that <mat <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~12 \qquad\textbf{(C)} ~30 \qquad\textbf{(D)} ~36 \
  11 KB (1,691 words) - 18:56, 25 April 2022

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