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  • ...e [[Primitive Pythagorean Triple]]s, those in which the three numbers have no common [[divisor]], are most interesting. A few of them are: There are no more cases as the hypotenuse has to be greater than the leg.
    5 KB (886 words) - 21:12, 22 January 2024
  • During AMC 10/12 testing week, the AoPS Wiki is in read-only mode. No edits can be made.
    2 KB (258 words) - 14:05, 15 November 2023
  • ...t Edie and Dee were originally in seats 3 and 4. If this were so, there is no possible position for which Bea can move 2 seats to the right. The same app ==Solution 3 (no casework)==
    2 KB (402 words) - 14:54, 25 June 2023
  • ...du/%7Ecalculus/prize2005.html website] (NOTE: website from 2005, currently no plans for future competitions) ...ams.org/employment/mimoscow.html website] (NOTE: was not offered in 2010 - no longer funded)
    3 KB (409 words) - 15:02, 28 January 2022
  • {{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u> ...d the last 10 problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.
    10 KB (1,497 words) - 11:42, 10 March 2024
  • *No calculator ...2 people in 12th grade, no more than 3 people in 11th grade or above, and no more than 4 people in 10th grade or above
    8 KB (1,182 words) - 14:26, 3 April 2024
  • The accomplishments of every student are unique, and there is no way to measure that success. However, we try to record and celebrate [[AoP
    5 KB (667 words) - 17:09, 3 July 2023
  • ...et-Union math competitions 1961-1986] - Many problems, no solutions. [Site no longer exists. Site has been replaced by a web capture] ...ympiad Math Madness] - Stacks of challenging problems, no solutions. [Site no longer exists. Site has been replaced by a web capture]
    16 KB (2,152 words) - 21:46, 6 May 2024
  • {{Contest Info|name=Putnam|region=USA|type=Proof|difficulty=7 - 9|breakdown=<u>Problem A/B, 1/2 ...average of the range of ranks that would have been attained had there been no tie -- that is, if the top three students tie, they are all awarded a rank
    4 KB (623 words) - 13:11, 20 February 2024
  • ...IME would remain after the new year. Thus there are two "2021 AMC 10/12s", no "2021 AMC 8", and one “2021 AIME”. All future AMC contests will follow
    17 KB (1,921 words) - 13:00, 28 April 2024
  • ...tor, resulting in a positive fraction and thus satisfying the inequality. No value between <math>x=-5</math> and <math>x=\frac{3}{2}</math> (except <mat
    12 KB (1,798 words) - 16:20, 14 March 2023
  • {{Contest Info|name=USAMTS|region=USA|type=Proof|difficulty=3-6|breakdown=<u>Problem 1-2</u>: 3 ...TS) was initiated in the fall of 1989 through a regular column by the same name in Consortium, a quarterly newsletter published by COMAP.
    4 KB (613 words) - 13:08, 18 July 2023
  • ...eir school to take the more challenging [[AMC 10]] exam. However, there is no requirement for the AMC 10 besides the fact that you have to be 10<math>^\t {{Contest Info|name=AMC 8|region=USA|type=Multiple Choice|difficulty=1 - 1.5|breakdown=<u>Probl
    4 KB (558 words) - 22:25, 28 April 2024
  • {{Contest Info|name=AIME|region=USA|type=Free Response|difficulty=3-6|breakdown=<u>Problem 1-5< ...0 to 999, inclusive, making guessing almost futile. Wrong answers receive no credit, while correct answers receive one point of credit, making the maxim
    8 KB (1,057 words) - 12:02, 25 February 2024
  • ...s vary widely in difficulty. All three of the tests are designed such that no background in [[calculus]], [[analysis]], or any other higher mathematics i
    5 KB (696 words) - 03:47, 24 December 2019
  • <math>\mathcal{S}</math>, there is no point <math>P</math> in <math>\mathcal{S}</math> such that
    4 KB (692 words) - 22:33, 15 February 2021
  • {{Contest Info|name=USAMO|region=USA|type=Proof|difficulty=7-9|breakdown=<u>Problem 1/4</u>: 7< ...da. Qualification is based on [[AMC 10]], [[AMC 12]], and [[AIME]] scores. No calculators are allowed on this two day, 9 hour exam (two 4.5 hour sessions
    6 KB (869 words) - 12:52, 20 February 2024
  • ...s team were regular attendees at SDMC. Also, since the 2007 team contained no seniors, the organizers for the 2008 team are not planning on extending inv The coach is Dr. Kent Merryfield, a professor at CSULB. His AoPS user name is Kent Merryfield. Please contact him for further information.
    21 KB (3,500 words) - 18:41, 23 April 2024
  • ...into regionals, invitationals, states, and national competitions. There is no need to progress through the levels- you simply register and you can partic
    4 KB (632 words) - 17:09, 11 October 2020
  • ...even]], <math>(-b)^n + b^n = 2b^n</math>, rather than 0, so this gives us no useful information.
    3 KB (532 words) - 22:00, 13 January 2024
  • # Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to r ...n</math> such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee. ([[Pigeonhole Princi
    11 KB (1,985 words) - 21:03, 5 August 2023
  • ...n) = 1</math>. Equivalently, <math>m</math> and <math>n</math> must have no [[prime]] divisors in common. The positive integers <math>m</math> and <ma
    2 KB (245 words) - 15:51, 25 February 2020
  • ...>x \neq 10</math> and <math>x < -9</math> or <math>x > 3</math>, there are no solutions for <math>x</math>. ...e force and checking each value individually, we can assert that there are no such solutions for <math>x</math>, leaving us with only <math>1</math> solu
    3 KB (571 words) - 00:42, 22 October 2021
  • .... Compute the number of ways they can arrange themselves in line such that no three consecutive people are in increasing order of height, from front to b
    9 KB (1,703 words) - 07:25, 24 March 2024
  • ...+b</math>; hence <math>a+b</math> is a multiple of <math>c</math> which is no more than <math>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c\}< Case IV) <math>a+b=4c\Rightarrow (4a-1)(4b-1)=65</math> for which there are no solutions.
    2 KB (332 words) - 09:37, 30 December 2021
  • ...triangle <math>T </math> whose side lengths are all positive integers with no common divisor and determine those integers.
    13 KB (2,048 words) - 15:28, 22 February 2024
  • ...es of the theorem by <math>a</math>. The restated form is nice because we no longer need to restrict ourselves to integers <math>{a}</math> not divisibl
    16 KB (2,675 words) - 10:57, 7 March 2024
  • ...written as an expansion in the parameter <math>q=e^{2\pi i z}</math> with no negative exponents:
    5 KB (849 words) - 16:14, 18 May 2021
  • This tells there that there is no solution for (b), since we must have <math>A^2 \ge 2</math> ...h>x \in \left[\frac{1}{2}, 1\right]</math> is a solution for (a); there is no solution for (b); there is one solution for (c), which is <math>x=\frac{3}{
    3 KB (466 words) - 12:04, 12 April 2024
  • ...onsider all possible triangles having these point as vertices. Prove that no more than <math>70 \%</math> of these triangles are acute-angled.
    1 KB (234 words) - 19:26, 14 July 2017
  • ...550</math>. Difference: <math>D = 49</math>. <math>A = 50 \cdot D</math>? No!
    8 KB (1,315 words) - 18:18, 2 March 2024
  • A '''scalene''' triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the Hinge Theorem).
    4 KB (628 words) - 17:17, 17 May 2018
  • ...that each entry is between <math>2</math> and <math>9</math> inclusive and no two consecutive entries are equal?'' ...luding <math>2</math> and <math>9</math>; the key restriction here is that no two boxes right next to each other can have the same number.
    12 KB (1,896 words) - 23:55, 27 December 2023
  • ...>. But the question mandates that <math>y</math> is positive, so there are no solutions when <math>x \geq 3</math>. Thus, there are <math>68+53=121</math
    5 KB (709 words) - 10:28, 19 February 2024
  • ...6</math>; <math>2^{2^6} + 1 = 274177\cdot 67280421310721</math>. In fact, no other Fermat primes have been found to date.
    6 KB (985 words) - 12:38, 25 February 2024
  • ...use a complementary approach. The total number of positive integers, with no restrictions, is <math>99</math> integers. What we don't want are the multi ...git positive integers. What we don't want are the four-digit integers with no digit that is a two or three, Using a [[Constructive counting | constructiv
    8 KB (1,192 words) - 17:20, 16 June 2023
  • ...depending on whether <math>a</math> can be divided by <math>b</math> with no remainder.
    2 KB (296 words) - 15:04, 5 August 2022
  • ...he test is worth 4 points, and the remaining problems are worth 5 points. No penalty or partial credit is given to unanswered or incorrectly answered qu ...nutes is given to solve all problems, each with exactly 5 answer choices. No problems require the use of a calculator, and calculators of all types are
    6 KB (936 words) - 15:38, 22 February 2024
  • ...lementary Chebyshev's estimate <math>D_n\le n^{\sqrt n}\cdot 4^n</math> is no longer sufficient).
    8 KB (1,469 words) - 21:11, 16 September 2022
  • # Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to r
    6 KB (957 words) - 23:49, 7 March 2024
  • ...me <math>{t}</math> is <math>e^{t^2}</math> meters/second. (Yes, probably no object really moves this way, but just pretend.) ''Approximately'' how far
    11 KB (2,082 words) - 15:23, 2 January 2022
  • ...ve vertices. Most frequently, one deals with ''simple polygons'' in which no two edges are allowed to intersect. (In fact, the adjective "simple" is al
    2 KB (372 words) - 19:04, 30 May 2015
  • Since there are no restrictions on cost paid besides <math>A<B</math>, we can use an example w
    1 KB (249 words) - 13:05, 24 January 2024
  • ...e square of a [[real number]] cannot be [[negative]], so this equation has no solutions in the real numbers. However, it is possible to define a number,
    5 KB (860 words) - 15:36, 10 December 2023
  • ...RML/Welcome.html Georgia ARML page] - note that the author of this page is no longer, and thus this page is perpetually stuck in 2006. For instance, GA A
    1 KB (186 words) - 11:52, 4 June 2012
  • ...ve [[integers]] <math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>.
    3 KB (453 words) - 11:13, 9 June 2023
  • *A [[point]] is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said t ...ne]] satisfying a [[linear]] [[function]]. It has length and position, but no other properties. It is one-dimensional. A [[line segment]] means a finite
    3 KB (393 words) - 07:59, 25 September 2020
  • ...owever, <math>17</math> will never be a multiple of <math>3</math>, hence, no solutions exist. An equation of form <math>x^4+y^4=z^2</math> has no [[integer]] solutions, as follows:
    9 KB (1,434 words) - 13:10, 20 February 2024
  • ==Solution 3 (No Miquel's point)==
    3 KB (496 words) - 13:35, 18 January 2023
  • ...if a equation has [[parentheses]] or the first one performed when there is no parentheses.
    5 KB (803 words) - 16:25, 10 August 2020

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