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• 2004 AIME II Problems/Problem 6
  ...hird monkeys have at the end of the process are in the [[ratio]] <math> 3: 2: 1, </math>what is the least possible total for the number of bananas? == Solution 2 ==
  6 KB (950 words) - 14:18, 15 January 2024
• 2006 AMC 10B Problems/Problem 7
  ...hrm{(B) \ } x\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \sqrt{\frac{x}{2}}\qquad \mathrm{(E) \ } x\sqrt{-1} </math> ...sqrt{\frac{x}{\frac{x-(x-1)}{x}}} = \sqrt{\frac{x}{\frac{1}{x}}} = \sqrt{x^2} = |x|</math>
  1 KB (179 words) - 10:33, 19 August 2022
• 2007 USAMO Problems/Problem 5
  ...math> has at least two more prime factors than does <math>x + 1</math>. To confirm that <math>(x^7 + 1)/(x + 1)</math> is composite, observe that &= (x + 1)^6 - \frac{7x(x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)}{x + 1} \\
  2 KB (240 words) - 09:47, 7 August 2014
• Proofs of AM-GM
  ...ction in which one proves a result for <math>2</math>, all powers of <math>2</math>, and then that <math>n</math> implies <math>n-1</math>. ...2 = 4xy</cmath> <cmath>\frac{(x+y)^2}{4} \geq xy</cmath> <cmath>\frac{x+y}{2} \geq \sqrt{xy},</cmath> with equality if and only if <math>x=y</math>, jus
  12 KB (2,171 words) - 07:55, 11 May 2023
• 2003 JBMO Problems/Problem 1
  A &= 4(10^{2n-1} + 10^{2n-2} \cdots 10^1 + 10^0) \\ B &= 8(10^{n-1} + 10^{n-2} \cdots 10^1 + 10^0) \\
  1,002 bytes (145 words) - 00:14, 28 August 2018
• 2001 AMC 12 Problems/Problem 24
  ...math>. Point <math>D</math> is on <math>\overline{BC}</math> so that <math>2\cdot BD=CD</math> and <math>\angle DAB=15^\circ</math>. Find <math>\angle A == Solution 2 ==
  8 KB (1,316 words) - 22:48, 7 March 2024
• 2011 AMC 12A Problems/Problem 19
  ...number of players given elite status is equal to <math>2^{1+\lfloor \log_{2} (N-1) \rfloor}-N</math>. Suppose that <math>19</math> players are given el ...ing, we get <math>\lfloor\log_{2}(N-1)\rfloor = \log_{2} \left(\frac{N+19}{2}\right)</math>.
  4 KB (563 words) - 11:12, 3 December 2023
• 2011 AMC 10A Problems/Problem 19
  .... Now, we find the percent increase from <math>22^2=484</math> to <math>28^2=784</math>. Since the increase is <math>300</math>, the percent increase is == Solution 2 ==
  3 KB (545 words) - 20:54, 21 August 2023
• 2011 AIME I Problems/Problem 7
  ...s having <math>\frac{1}{m}</math> <math>m-1</math> times, <math>\frac{1}{m^2}</math> <math>m-1</math> times, <math>\ldots</math>. To make other sequence ==Solution 2==
  10 KB (1,673 words) - 19:27, 8 April 2024
• 1997 AHSME Problems/Problem 27
  Let <math>f(x) = x</math> for <math>x \in \{1, 2, 3, 4, 5, 6, 7, 8\}</math>. Plugging in <math>x=5</math> into the initial Continuing, <math>f(10) = f(6) - f(2) = 4</math>, <math>f(11) = f(7) - f(3) = 4</math>, and <math>f(12) = f(8) -
  3 KB (456 words) - 14:14, 5 July 2013
• 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? ...rac{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>
  3 KB (578 words) - 00:47, 20 March 2024
• 1950 AHSME Problems/Problem 17
  <cmath> \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline\end{tabular} </cmath> ...-5x^{2}\qquad\textbf{(C)}\ y=100-5x-5x^{2}\qquad\\ \textbf{(D)}\ y=20-x-x^{2}\qquad\textbf{(E)}\ \text{None of these} </math>
  843 bytes (134 words) - 00:51, 12 October 2020
• 2003 AMC 12B Problems/Problem 9
  ...function for which <math>f(6) - f(2) = 12.</math> What is <math>f(12) - f(2)?</math> <cmath>m = \frac{f(6) - f(2)}{\Delta x} = \frac{12}{6 - 2} = 3</cmath>
  818 bytes (128 words) - 19:05, 7 July 2020
• 2006 iTest Problems/Problem 28
  &= (10^2 + 1)(10^4 - 10^2 + 1)(10^3 - 1)(10^3 + 1) ...ctor of the original number can not be greater than <math>2006</math>), we confirm that <math>9901</math> is the largest prime factor of <math>999999999999</m
  2 KB (222 words) - 21:20, 2 December 2018
• 1969 AHSME Problems/Problem 29
  Plug in <math>t = 2</math> and test each expression (if the relation works, then both sides are ...xponent of both <math>x^y</math> and <math>y^x</math> are the same, we can confirm that <math>\boxed{\textbf{(C) } y^x = x^y}</math>.
  1 KB (258 words) - 18:49, 21 June 2018
• 2007 iTest Problems/Problem 56
  In the binary expansion of <math>\dfrac{2^{2007}-1}{2^{225}-1}</math>, how many of the first <math>10,000</math> digits to the ri ...math>2^{2007} - 1</math> has <math>2007</math> ones while the number <math>2^{225} - 1</math> has <math>225</math> ones.
  3 KB (464 words) - 16:44, 19 August 2018
• 2015 AMC 12A Problems/Problem 20
  The area of <math>T</math> is <math>\dfrac{1}{2} \cdot 8 \cdot 3 = 12</math> and the perimeter is 18. ...area of <math>T'</math> is <math>\dfrac{1}{2} b \sqrt{a^2 - (\dfrac{b}{2})^2}</math> and the perimeter is <math>2a + b</math>.
  5 KB (912 words) - 22:32, 7 June 2021
• 2016 AMC 8 Problems/Problem 22
  draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0)); fill((3,0)--(2,4)--(1.5,3)--cycle, black);
  8 KB (1,157 words) - 16:12, 18 January 2024
• 2008 iTest Problems/Problem 52
  Note that <math>48^2 + 55^2 = 73^2</math>, so the sides are part of a right triangle. That means the altitude <cmath>\frac{48 \cdot 55}{\frac{73^2 + 48 \cdot 55}{73}}</cmath>
  2 KB (252 words) - 13:37, 13 July 2018
• 1960 AHSME Problems/Problem 25
  ...largest integer which divides all possible numbers of the form <math>m^2-n^2</math> is: <math>\textbf{(A)}\ 2\qquad
  1 KB (252 words) - 01:21, 18 June 2018
• Basic Programming With Python
  In the second iteration, i = 2, so Python prints 2. '''Find <math>\sum_{n=1}^{50} 2^{n}.</math>'''
  33 KB (5,277 words) - 22:14, 3 June 2023
• 1973 AHSME Problems/Problem 16
  ...the polygon. The number of interior angles in the polygon is <math>180(n-2)</math>. We know that the sum of all but one of them is <math>2190^{\circ} <cmath>180(n-2) > 2190</cmath>
  1 KB (184 words) - 14:00, 20 February 2020
• 1973 AHSME Problems/Problem 15
  ...ath>. By HL Congruency and CPCTC, <math>\angle AOC = \angle BOC = \theta /2</math>. <cmath>\cos (\theta /2) = \frac{3}{R}</cmath>
  2 KB (306 words) - 14:00, 20 February 2020
• 2000 JBMO Problems/Problem 4
  ...mber of games that a boy won is <math>\tfrac{7}{12} \cdot \tfrac{3n(3n-1)}{2} = \tfrac{7n(3n-1)}{8}.</math> ...is at least one boy playing is <math>\tfrac{3n(3n-1)}{2} - \tfrac{n(n-1)}{2}.</math> This means we can write and solve the following inequality.
  2 KB (328 words) - 15:50, 19 August 2018
• 2008 iTest Problems/Problem 69
  ...cdots,</math> where the first term is 1, the next 2 terms are congruent to 2 modulo 3, the next 3 terms are congruent to 3 modulo 3, and so on.''' ...tion, the last number of each grouping is term number <math>\tfrac{n(n+1)}{2}.</math>
  1 KB (190 words) - 10:19, 22 August 2018
• 1999 JBMO Problems/Problem 2
  ...egative integer <math>n</math> we define <math>A_n = 2^{3n}+3^{6n+2}+5^{6n+2}</math>. Find the greatest common divisor of the numbers <math>A_0,A_1,\ldo Note that <math>A_0 = 2^0 + 3^2 + 5^2 = 35</math>, so the [[GCD]] must be a factor of 35. The prime factorizatio
  2 KB (226 words) - 13:30, 25 August 2018
• 2019 AMC 10A Problems/Problem 23
  ...the number <math>1</math>, then Todd must say the next two numbers (<math>2</math> and <math>3</math>), then Tucker must say the next three numbers (< 1 & 1 & 2-3 & 4-6 \\
  6 KB (1,052 words) - 17:48, 6 November 2021
• 2006 iTest Problems/Problem U7
  ...6^2</math>, and applying difference of squares yields <math>(y-x)(y+x)=696^2</math>. Because <math>x</math> and <math>y</math> have the same parity (du Let <math>y+x = 2b</math> abd <math>y-x = 2a</math>; then <math>ab = 348^2</math>. Additionally,
  4 KB (674 words) - 16:03, 25 February 2021
• 2019 AMC 8 Problems/Problem 24
  ...ath>D</math> divides side <math>\overline{AC}</math> so that <math>AD:DC=1:2</math>. Let <math>E</math> be the midpoint of <math>\overline{BD}</math> an A = (1.2,1.7);
  27 KB (4,256 words) - 19:30, 17 January 2024
• 2020 AMC 10B Problems/Problem 18
  ==Solution 2== ...e. Now, in order to get <math>3</math> balls of each color, he needs <math>2</math> more of both <math>R</math> and <math>B</math>.
  11 KB (1,928 words) - 22:40, 12 November 2023
• 2021 Fall AMC 12B Problems/Problem 20
  ...How many different ways are there to construct the <math>2 \times 2 \times 2</math> cube using these smaller cubes? (Two constructions are considered th ...cube. Each edge of the planar graph represents a shared face between <math>2</math> neighboring unit cubes. Each face of the planar graph represents a f
  10 KB (1,641 words) - 22:17, 26 September 2023
• 2022 AMC 12A Problems/Problem 21
  <math>\textbf{(A)} \, x^2 -x + 1 \qquad\textbf{(B)} \, x^2 + x + 1 \qquad\textbf{(C)} \, x^4 + 1 \qquad\textbf{(D)} \, x^6 - x^3 + 1 ...<math>1011</math> and <math>3033</math> are <math>3*337</math> and <math>3^2*337</math>, respectively.
  8 KB (1,325 words) - 02:14, 11 April 2024
• 2022 AMC 10B Problems/Problem 10
  ...egers is <math>2</math> greater than their median, and the median is <math>2</math> greater than their arithmetic mean. What is the least possible value ...> be the median. It follows that the two largest integers are both <math>M+2.</math>
  3 KB (517 words) - 13:07, 11 November 2023