Create the page "No name" on this wiki! See also the search results found.

Page title matches

• Jugemu Jugemu Goko no Surikire Kaijarisuigyo no Suigyomatsu Unraimatsu Furaimatsu Ku Neru Tokoro ni Sumu Tokoro Yabura Koji no Bura Koji Paipo-paipo Paipo no ShuringanShuringan no Gurindai Gurindai no Ponpokopi no Ponpokona no Chokyumei no Chosuke.
  ...rindai no Ponpokopi no Ponpokona no Chokyumei no Chosuke. That is the full name of a legendary user named duskstream. It is also duskstream's WiFi password
  611 bytes (89 words) - 16:32, 12 July 2022
• Steve has one quarter, two nickels and three pennies. Assuming no items are free, for how many different-priced items could Steve individually pay for with exact change?
  Steve can use no quarters or one quarter, for two possibilities.
  605 bytes (93 words) - 23:18, 1 January 2024
• If no one shares an office, in how many ways can 3 people be assigned to 5 different offices? (Each person gets exactly one office).
  254 bytes (42 words) - 16:52, 28 February 2021
• The table shows the percent of families in Mathville that have $0, 1, 2, 3$ and $4$ or more children. If there are a total of $10,250$ families, how many are there with no children?
  ...there are a total of <math>10,250</math> families, how many are there with no children?
  214 bytes (35 words) - 09:55, 1 August 2022

Page text matches

• Pythagorean Theorem
  ...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
• Main Page
  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
• 2016 AMC 10A Problems/Problem 13
  ...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
• Mathematics scholarships
  ...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
• MATHCOUNTS
  {{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
• ICTM Math Contest
  *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
• Art of Problem Solving
  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
• Resources for mathematics competitions
  ...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]
  24 KB (3,269 words) - 00:43, 24 April 2024
• William Lowell Putnam Mathematical Competition
  {{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
• AMC historical results
  ...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
• Inequality
  ...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
• United States of America Mathematical Talent Search
  {{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
• AMC 8
  ...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
• American Invitational Mathematics Examination
  {{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
• American Mathematics Competitions
  ...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
• 2015 IMO Problems
  <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
• United States of America Mathematical Olympiad
  {{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
• Joining an ARML team
  ...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
• Mu Alpha Theta
  ...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
• Factoring
  ...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
• Pigeonhole Principle
  # 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
• Relatively prime
  ...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
• 1965 AHSME Problems/Problem 40
  ...>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
• Principle of Inclusion-Exclusion
  .... 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
• 2020 CIME II Problems/Problem 10
  ...+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
• Cauchy-Schwarz Inequality
  ...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
• Fermat's Little Theorem
  ...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
• Number theory/Advanced
  ...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
• 1959 IMO Problems/Problem 2
  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
• 1970 IMO Problems/Problem 6
  ...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
• Divisibility rules
  ...550</math>. Difference: <math>D = 49</math>. <math>A = 50 \cdot D</math>? No!
  8 KB (1,315 words) - 18:18, 2 March 2024
• Triangle
  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
• Constructive counting
  ...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
• Casework
  ...>. 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
• Prime number
  ...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
• Complementary counting
  ...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
• Integer
  ...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
• Mathematical Kangaroo
  ...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
• Pi
  ...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
• Fibonacci sequence
  # 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
• Fundamental Theorem of Calculus
  ...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
• Polygon
  ...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
• 2011 AMC 10B Problems/Problem 4
  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
• Complex number
  ...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
• Georgia ARML
  ...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
• Fermat's Last Theorem
  ...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
• Geometry
  *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
• Diophantine equation
  ...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
• 1985 IMO Problems/Problem 5
  ==Solution 3 (No Miquel's point)==
  3 KB (496 words) - 13:35, 18 January 2023
• Exponentiation
  ...if a equation has [[parentheses]] or the first one performed when there is no parentheses.
  5 KB (803 words) - 16:25, 10 August 2020
• Riemann Hypothesis
  * The zeta function has no zeros with real part between <math>\frac{1}{2}</math> and 1
  2 KB (425 words) - 12:01, 20 October 2016
• Fundamental Theorem of Algebra
  ...at <math>P(z)</math> is a complex polynomial of degree <math>n</math> with no complex roots; without loss of generality, suppose that <math>P</math> is [
  5 KB (832 words) - 14:22, 11 January 2024
• International Mathematical Olympiad
  {{Contest Info|name=IMO |region=International|type=Proof|difficulty=5.5 - 10|breakdown=<u>Probl There is no official team competition. Unofficially, however, the scores of each team
  3 KB (490 words) - 03:32, 23 July 2023
• Prime Number Theorem
  has no zeros on the line <math>\Re s = 1</math>, and from this deduced <math>\zeta(1+ri)</math> has no zeros was about 25 pages long; Hadamard's
  10 KB (1,729 words) - 19:52, 21 October 2023
• Mock AMC
  ...ular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but
  51 KB (6,175 words) - 20:58, 6 December 2023
• Permutation
  A permutation in which no object remains in the same place it started is called a [[derangement]].
  3 KB (422 words) - 11:01, 25 December 2020
• Riemann zeta function
  <math>\zeta(s)</math> has no zeros for <math>\Re s > 1</math>. Indeed, suppose this Since the [[gamma function]] has no zeros, it follows that these
  9 KB (1,547 words) - 03:04, 13 January 2021
• Modular arithmetic/Introduction
  ...>3</math>, or <math>4</math> in modulo 5, we give these integers their own name: the '''[[residue class]]es''' modulo 5. In general, for a natural number No, you cannot. Rewriting the question, we see that it asks us to find an inte
  15 KB (2,396 words) - 20:24, 21 February 2024
• Uncountable
  A [[set]] <math>S</math> is said to be '''uncountable''' if there is no [[injection]] <math>f:S\to\mathbb{N}</math>. Assuming the [[Axiom of choice
  2 KB (403 words) - 20:53, 13 October 2019
• Modular arithmetic/Intermediate
  <math>c_i</math> is a member of the set {1, 2, 3}. Since no <math>c_i</math> divides 7, 7 may be factored and <math>7\frac{k_2-k_1}{c_i
  14 KB (2,317 words) - 19:01, 29 October 2021
• Limit
  ...f(x)</math> to be to its limit value. Then the formal definition says that no matter how close we want to be (for ''any'' <math>\epsilon > 0</math>), we ...ghtarrow 0}\frac{1}{x}</math> does not exist, since, in fact, there exists no <math>\epsilon</math> for which there exists <math>\delta</math> satisfying
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems/Problem 13
  Since there is no obvious way to approach this problem, we start by experimenting with small
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2006 AIME I Problems/Problem 10
  ...e [[concave]] hexagonal region of the squares equally (as of yet, there is no substantiation that such would work, and definitely will not work in genera
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 8
  ...6-h^4</math> specifically must be in the form <math>n^2/4</math>. There is no restriction on h as long as it is a positive real number, so all we have to
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AMC 12B Problems
  ...ams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  <math>(1)</math> No two consecutive integers belong to <math>S</math>. ...th>S</math> contains <math>k</math> elements, then <math>S</math> contains no number less than <math>k</math>.
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2004 AMC 12A Problems
  ...at-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  ...rst <math>5</math> letters, no B's in the next <math>5</math> letters, and no C's in the last <math>5</math> letters?
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2001 AMC 12 Problems
  ...est such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2003 AMC 12B Problems
  ...is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. he walks at the rate of 5000 feet per hour while pushing t Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whos
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AMC 12B Problems
  ...umbers is <math>221</math>. None of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included amon ...nt vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 6
  ...ams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
  978 bytes (145 words) - 13:57, 4 December 2015
• 2006 AMC 12B Problems/Problem 7
  If there was no restriction, there would be 4!=24 ways to sit. However, only 2/4 of the peo
  1 KB (213 words) - 15:33, 9 April 2024
• 2006 AMC 12B Problems/Problem 9
  a. <math>c=2</math>. No such number exists.
  3 KB (409 words) - 17:10, 30 April 2024
• 2006 AMC 12B Problems/Problem 19
  ...8</math>, which is not divisible by <math>4</math>), which means there are no <math>2</math>, <math>4</math>, <math>6</math>, or <math>8</math> year olds
  4 KB (696 words) - 09:47, 10 August 2015
• 2006 AMC 12A Problems/Problem 14
  ...at <math>300p + 210g = 30(10p + 7g)</math> is divisible by <math>30</math> no matter what <math>p</math> and <math>g</math> are, so our answer must be di
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 20
  ...to visit <math>D</math> on the plane of <math>ABC</math> last, as there is no way in and out from <math>D</math>.
  5 KB (908 words) - 19:23, 22 September 2022
• 2006 AMC 12A Problems/Problem 25
  <math>(1)</math> No two consecutive [[integer]]s belong to <math>S</math>. ...</math> contains <math>k</math> [[element]]s, then <math>S</math> contains no number less than <math>k</math>.
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 25
  ...d as many solutions as possible, until it becomes intuitive that there are no more size of triangles left.
  4 KB (498 words) - 00:46, 4 August 2023
• 2005 AMC 12B Problems/Problem 3
  This was just a simple manipulation of the equation. No solving was needed!
  978 bytes (156 words) - 14:14, 14 December 2021
• 2005 AMC 12B Problems/Problem 4
  Here, only the <math>A</math>'s matter... No complicated stuff!
  1 KB (197 words) - 14:16, 14 December 2021
• 2005 AMC 12B Problems/Problem 11
  ==Solution 5 (Quick if no time)==
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 15
  ...umbers is <math>221</math>. None of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included amon
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 17
  As <math>a,b,c,d</math> must all be rational, and there are no powers of <math>3</math> or <math>7</math> in <math>10^{2005}</math>, <math
  1 KB (159 words) - 21:18, 21 December 2020
• 2005 AMC 12B Problems/Problem 23
  No other value of <math>a + b</math> is possible for all members of <math>S</m
  5 KB (786 words) - 16:49, 31 January 2023
• 2005 AMC 12B Problems/Problem 25
  ...nt vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex? ...choose the 4 vertices that have a cross section as a square, there exists no connecting diagonal.
  10 KB (1,840 words) - 21:35, 7 September 2023
• Root (polynomial)
  ...the interval <math>(r,s)</math>. In particular, if <math>P'(x)</math> has no roots in an interval <math>(a,b)</math>, then <math>P(x)</math> has at most No similar formula exists for quintics or polynomials of any higher degree. Al
  8 KB (1,427 words) - 21:37, 13 March 2022
• Vertical line test
  The vertical line test states that a [[relation]] is a [[function]] if no vertical [[line]] intersects the graph in more than one point.
  659 bytes (114 words) - 10:41, 27 April 2024
• Set
  ...tually imagine those numbers flowing freely inside those curly braces with no preference given to any of them. What matters is that these four numbers ar Since there are no reals such that the square of it is less than 0, that set is the empty set.
  11 KB (2,021 words) - 00:00, 17 July 2011
• 1991 AJHSME Problems
  ...finishing second and <math>1</math> point for finishing third. There are no ties. What is the smallest number of points that a student must earn in th
  17 KB (2,246 words) - 13:37, 19 February 2020
• 2022 USAJMO Problems/Problem 1
  Condition <math>(1)</math> holds if no consecutive terms in <math>a_i</math> are equivalent modulo <math>m</math>,
  4 KB (792 words) - 00:29, 13 April 2024
• 2005 AIME II Problems/Problem 2
  ...6 is <math>{6 \choose3}</math>. After that, the three remaining rolls have no more configurations.
  4 KB (628 words) - 11:28, 14 April 2024
• Empty set
  ...set</math> or <math>\varnothing</math>) is the (unique) [[set]] containing no elements. It is therefore a [[subset]] of every set.
  489 bytes (84 words) - 21:33, 27 February 2020
• Problem solving
  ...dn’t solve a single one. That’s right – I was 0-for-60+. I came away no longer confident that I was good at math. I assumed that most of the other ...I realized that formula sheets are not really math. Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling wo
  6 KB (1,039 words) - 17:43, 30 July 2018
• Research Science Institute
  ...na. (To this day, alumni of the program are known as "Rickoids".) RSI's name changed to Research Science Institute in 1987, following Admiral Rickover's ...m the United States. Admission to RSI is extremely competitive. There is no tuition or room-and-board charge to attend RSI. Students in all branches o
  2 KB (284 words) - 16:37, 6 July 2023
• 2005 AIME I Problems
  ...arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have. .... He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguis
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 4
  ...arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have. ...th>n(n + 7) = 14\cdot 21 = 294 = 17^2 + 5</math>, so this number works and no larger number can. Thus, the answer is <math>\boxed{294}</math>.
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  .... He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguis ...e a string of tails followed by a string of heads, since after the first H no more tails can appear. The first H can occur in a maximum of eight times di
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 9
  The unit cube at the center of our large cube has no exterior faces, so all of its orientations work.
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 13
  # There are no right angle turns in the particle's path. ...h>3</math> <math>U</math>'s, and <math>2</math> <math>D</math>'s such that no two <math>R</math>'s or <math>U</math>'s are adjacent. The <math>D</math>'s
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME II Problems/Problem 14
  == Solution 9 (Short and no IQ Required Altogether-Bash) ==
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  ...</math> operations, it follows that <math>x_{10} \ge 20</math>, so that we no longer have to worry about reaching <math>1</math> again. ...>10</math> <math>A</math>s must be preceded by a <math>B</math>. There are no other restrictions on the remaining seven characters. Letting <math>\square
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 13
  ...</math> and <math>19</math> are [[relatively prime]], this means there are no duplicate roots. Thus, <math>a_1, a_2, a_3, a_4</math> and <math>a_5</math
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 2
  ...irst term is negative and last is positive for each set, it has absolutely no effect on the end result! This is a great strategy that can help significa
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 8
  * the points <math> P_1, P_2,\ldots, P_n </math> are coplanar and no three of them are collinear, There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed sta
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  * the points <math> P_1, P_2,\ldots, P_n </math> are coplanar and no three of them are collinear, There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed sta
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 14
  ...th>. However, for <math>9k \ge 963 \Longrightarrow n \le 37</math>, we can no longer apply this approach. ...ath>b = 11,9,10,11 + 12p</math> for integers <math>p</math>) that their is no way to satisfy the inequality <math>11b + 111c < 1000</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 6
  == Solution 4 (No Thinking Required) ==
  6 KB (950 words) - 14:18, 15 January 2024
• 2019 AIME I Problems/Problem 2
  ...obability that <math>B</math> and <math>J</math> differ by <math>1</math> (no zero, because the two numbers are distinct). There are <math>20 \cdot 19 = ...th>2</math> distinct elements from a <math>20</math> element set such that no <math>2</math> elements are adjacent. Using the well-known formula <math>\d
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ...likely. Let <math>\frac m n</math> in lowest terms be the probability that no two birch trees are next to one another. Find <math>m+n</math>.
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  ...math>S</math> be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of <math>S</math> have the same sum. What is the large
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  ..."simple" if the addition <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of simple ordered pairs of non-negative integers
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...ion, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  ...}</math> be a subset of <math>\{1,2,3^{}_{},\ldots,1989\}</math> such that no two members of <math>S^{}_{}</math> differ by <math>4^{}_{}</math> or <math
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  ...umber of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that
  6 KB (870 words) - 10:14, 19 June 2021
• 1991 AIME Problems
  Expanding <math>(1+0.2)^{1000}_{}</math> by the binomial theorem and doing no further manipulation gives ...f radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent t
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  ...nsecutive integers in <math>\{1000,1001,1002^{}_{},\ldots,2000\}</math> is no carrying required when the two integers are added?
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1994 AIME Problems
  ...in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty.
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1996 AIME Problems
  ...am has a <math>50\%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  ...th>10</math> and <math>11</math>. An equilateral triangle is drawn so that no point of the triangle lies outside <math>ABCD</math>. The maximum possible
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  .... Let <math>D_{40}</math> be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos
  7 KB (1,084 words) - 02:01, 28 November 2023
• 1999 AIME Problems
  Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are ...th>50 \%</math> chance of winning any game it plays. The probability that no two teams win the same number of games is <math>m/n,</math> where <math>m_{
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  Find the least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two posit
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  ...ath>x^{2001}+\left(\frac 12-x\right)^{2001}=0</math>, given that there are no multiple roots. ...same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are pos
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2003 AIME I Problems
  ...region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the gre
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  ...ith <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizontal or vertical sides, and <math>\overline{AB}</math> and <math>\ove
  6 KB (947 words) - 21:11, 19 February 2019
• 2002 AIME II Problems
  ...ch the equation <math>\left\lfloor\frac{2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math>. (The notation <math>\lfloor x\rfloor< ...of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</math> is divide
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 10
  ...se complementary counting. We will split up into <math>3</math> cases: (1) no number is repeated, (2) <math>2</math> numbers are repeated, and <math>2</m
  5 KB (855 words) - 20:26, 14 January 2023
• 1983 AIME Problems/Problem 15
  ...which is non-negative, so the root outside of <math>(-3,3)</math> must be no less than <math>3</math>. By considering the graph of <math>y=f(x)</math>, === Solution 7 (No Trig) ===
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 10
  ...+4(23)-3=119</math>. So Mary scored at least <math>119</math>. To see that no result other than <math>23</math> right/<math>3</math> wrong produces <math
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1984 AIME Problems/Problem 11
  ...ly. Let <math>\frac m n</math> in lowest terms be the [[probability]] that no two birch trees are next to one another. Find <math>m+n</math>. First notice that there is no difference between the maple trees and the oak trees; we have only two type
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 12
  satisfies the conditions and has no other roots.
  3 KB (588 words) - 14:37, 22 July 2020
• 1984 AIME Problems/Problem 14
  ...e possible ways of splitting <math>38</math> into two addends, we see that no pair of odd composites add to <math>38</math>. Therefore, <math>\boxed{038} ...ite (except for 5 itself). This means that we will have 15, 25, 35, etc... no matter what. What it also means is that if we look at the end digit, if 15
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1984 AIME Problems/Problem 15
  ...ing this solution, I highly recommend never following this unless you have no idea what to do with an hour of your time. Even so, learning the above solu
  6 KB (1,050 words) - 18:07, 16 January 2024
• 1995 IMO Problems/Problem 2
  === Solution 8 (fast Titu's Lemma no substitutions) ===
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 13
  ...+ 1)</math> to get <math>\gcd(n^2 - 200n, 2n + 1)</math>. Now we know that no matter what, <math>n</math> is relatively prime to <math>2n + 1</math>. The
  4 KB (671 words) - 20:04, 6 March 2024
• 1986 AIME Problems/Problem 12
  ...math>S</math> be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of <math>S</math> have the same sum. What is the large ...61}</math>, with <math>S=\{ 15,14,13,11,8\}</math>. We must now prove that no such set has sum greater than 61. Suppose such a set <math>S</math> existed
  2 KB (364 words) - 19:41, 1 September 2020
• 1986 AIME Problems/Problem 10
  ...mely <math>m \equiv 358</math> mod <math>666</math>. We see that there are no other 3-digit integers that are <math>358</math> mod <math>666</math>, so <
  3 KB (565 words) - 16:51, 1 October 2023
• 1986 AIME Problems/Problem 7
  ...base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Therefore, we can recast this into
  5 KB (866 words) - 00:00, 22 December 2022
• 1987 AIME Problems/Problem 12
  Because if its equal to, then there is no integer in between the two values. - resources
  4 KB (673 words) - 19:48, 28 December 2023
• 1987 AIME Problems/Problem 1
  ...ple" if the [[addition]] <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of simple ordered pairs of non-negative integers Since no carrying over is allowed, the range of possible values of any digit of <mat
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 15
  ...ion, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.) <math>\textbf{Case 1:}</math> Since letter 9 arrived before lunch, no further letters will arrive, and the number of possible orders is simply th
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 13
  ...ath>c_{i}</math> is some coefficient. However, since <math>F(x)</math> has no <math>x</math> term, it must be true that <math>c_{15} = 1</math>. Again, try to only use Engineer's Induction when you have no other options. A rigorous proof is usually not needed, but when you have ex
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1989 AIME Problems/Problem 13
  ...h>S</math> be a [[subset]] of <math>\{1,2,3,\ldots,1989\}</math> such that no two members of <math>S</math> differ by <math>4</math> or <math>7</math>. W ...choose at most 5 numbers from <math>\{1, 2, \ldots , 11\}</math> such that no two numbers have a difference of <math>4</math> or <math>7</math>. We take
  2 KB (274 words) - 04:07, 17 December 2023
• 1989 AIME Problems/Problem 10
  Since no additional information is given, we can assume that triangle ABC is right w
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 4
  <math>m = 5</math> gives no solution for k <math>m = 10</math> gives no solution for k
  3 KB (552 words) - 12:41, 3 March 2024
• 1990 AIME Problems/Problem 13
  ...ements in <math>T</math> and 3817 sections that have 1 or 2 elements. And, no matter how many elements a section has, each section contains exactly one e
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 9
  ...math>5</math> tails occur. The heads must fall between the tails such that no two heads fall between the same tails, and must fall in the positions label
  3 KB (425 words) - 19:31, 30 July 2021
• 1990 AIME Problems/Problem 6
  ...umber of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that
  2 KB (325 words) - 13:16, 26 June 2022
• 1990 AIME Problems/Problem 3
  ...18 then the highest it can be is 118. However, this gives an equation with no solution. The second largest possibility in this case is <math>s=59</math>,
  3 KB (516 words) - 19:18, 16 April 2024
• 2006 AMC 10B Problems/Problem 7
  ...c{-3}{1-\frac{4}{3}}}=\sqrt{\frac{-3}{\frac{-1}{3}}}=\sqrt{9}=3</math>. As no other option choice fits, <math>\boxed{\textbf{(A)}-x}</math> is the correc
  1 KB (179 words) - 10:33, 19 August 2022
• 2006 AMC 10B Problems/Problem 9
  ...ath>386</math> calories in <math>100</math> grams of sugar. Water contains no calories. How many calories are in <math>200</math> grams of her lemonade?
  1 KB (140 words) - 00:54, 2 April 2023
• 1991 AIME Problems/Problem 11
  ...radius]] 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is [[tangent
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 3
  Expanding <math>(1+0.2)^{1000}_{}</math> by the binomial theorem and doing no further manipulation gives
  5 KB (865 words) - 12:13, 21 May 2020
• 1991 AIME Problems/Problem 1
  ...of <math>16</math> can sum greater than <math>32</math>, and so there are no integral solutions for <math>(x,y)</math>. The solution is <math>5^2 + 11^2
  4 KB (628 words) - 22:05, 7 June 2021
• 1992 AIME Problems/Problem 6
  ...of consecutive integers in <math>\{1000,1001,1002,\ldots,2000\}</math> is no carrying required when the two integers are added? ...\textbf{Hundreds} & \textbf{Tens} & \textbf{Ones} & \textbf{Conditions for No Carrying} & \boldsymbol{\#}\textbf{ of Ordered Triples} \\ [0.5ex]
  3 KB (455 words) - 02:03, 10 July 2021
• 2006 AMC 10B Problems/Problem 22
  The first pair violates <math>N>1</math> and the third and fourth pairs have no positive integer solutions for <math>B</math> and <math>J</math>.
  2 KB (394 words) - 00:51, 25 November 2023
• 2006 AMC 10B Problems/Problem 25
  ...h> accounts for the 5 and one factor of 2. This leaves <math>X</math>, but no single digit number contains all the prime factors besides 5 and 2 of the c
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 10
  ...=2.</math> Trying other cases of <math>P</math> and <math>T</math> yields no solutions. Therefore, <math>T=2, P=2</math> and after solving for <math>t,
  4 KB (716 words) - 20:50, 17 April 2022
• 1993 AIME Problems/Problem 8
  We can perform casework based on the number of overlapping elements. If no elements overlap, there is <math>\binom60=1</math> way to choose the overla ...math>6\rightarrow6</math> we just have <math>\binom{6}{6}</math> (there is no double counted case since ABCDEF, ABCDEF is only counted once).
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 7
  ...th> is a dimension of the box but <math>x_2,x_3</math> aren’t, there are no possibilities (same for <math>x_5</math>).
  5 KB (772 words) - 09:04, 7 January 2022
• 1993 AIME Problems/Problem 1
  Now that there are no restrictions we proceed to find that there are <math>8</math> choices for t Now that there are no restrictions we proceed to find that there are <math>8</math> choices for t
  3 KB (440 words) - 21:20, 22 July 2021
• 1994 AIME Problems/Problem 9
  ...in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty.
  3 KB (589 words) - 14:18, 21 July 2019
• 1995 AIME Problems/Problem 15
  Think of the problem as a sequence of <tt>H</tt>'s and <tt>T</tt>'s. No two <tt>T</tt>'s can occur in a row, so the sequence is blocks of <math>1</
  6 KB (979 words) - 13:20, 11 April 2022
• 1996 AIME Problems/Problem 14
  ...uts by the diagonal is therefore <math>a+b+c</math>, if we can ensure that no more than <math>1</math> positive integer is present in the x, y, or z coor
  5 KB (923 words) - 21:21, 22 September 2023
• 1996 AIME Problems/Problem 6
  ...am has a <math>50\%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament No more than 1 team can win or lose all games, so at most one team can win all
  3 KB (461 words) - 01:00, 19 June 2019
• 1996 AIME Problems/Problem 3
  Notice that the coefficients in the problem statement have no effect on how many unique terms there will be in the expansion. Therefore t
  3 KB (515 words) - 04:29, 27 November 2023
• 1997 AIME Problems/Problem 15
  ...0</math> and <math>11</math>. An [[equilateral triangle]] is drawn so that no point of the triangle lies outside <math>ABCD</math>. The maximum possible
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 12
  Because there are no other special numbers other than <math>19</math> and <math>97</math>, take
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 8
  ...mn, and the second column is determined. For the third and fourth columns, no two numbers can be in the same row (to make the sum of each row 0), so agai
  4 KB (638 words) - 16:41, 22 January 2024
• 1997 AIME Problems/Problem 7
  ==Solution 3 (No Coordinates)==
  4 KB (617 words) - 18:47, 17 July 2022
• 1997 AIME Problems/Problem 4
  NOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been fo
  2 KB (354 words) - 22:33, 2 February 2021
• 1998 AIME Problems/Problem 15
  .... Let <math>D_{40}</math> be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos ...n to another, and a proper sequence is represented as a path that retraces no segment. Each time that such a path reaches a non-terminal vertex, it must
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 2
  ...h>y</math> both must be equal or less than 30. The inequalities given have no complicated qualities. We can recompile them by understanding: Two times th
  6 KB (913 words) - 16:34, 6 August 2020
• 1999 AIME Problems/Problem 13
  ...0 \%</math> chance of winning any game it plays. The [[probability]] that no two teams win the same number of games is <math>\frac mn,</math> where <mat ...mind, we see that there are a total of <math>40!</math> outcomes in which no two teams win the same number of games. Further, note that these are all th
  2 KB (329 words) - 01:38, 6 October 2015
• 1999 AIME Problems/Problem 10
  Ten [[point]]s in the plane are given, with no three [[collinear]]. Four distinct [[segment]]s joining pairs of these poi
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 7
  *Case of no 2's:
  3 KB (475 words) - 13:33, 4 July 2016
• 2000 AIME I Problems/Problem 15
  ...each Round 5, where the number of cards that remain (<math>125</math>) is no longer a multiple of 2. Therefore, when we remove all the cards with indice
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 5
  If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle of
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2000 AIME I Problems/Problem 1
  Find the least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two posit
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 15
  ...] so that each face contains a different number. The [[probability]] that no two consecutive numbers, where <math>8</math> and <math>1</math> are consid ...o A uniquely determine a configuration that satisfies the conditions, i.e. no two vertices have consecutive numbers. Thus, the number of desired configur
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 14
  ...same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are poss ...ath>1</math>'s such that there are no two consecutive <math>1</math>'s and no three consecutive <math>0</math>'s.
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2014 USAJMO Problems
  ...which <math>A</math> cannot win in a finite number of moves, or prove that no such minimum value exists.
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 4
  ==Solution 2 (no trig)==
  3 KB (534 words) - 03:22, 23 January 2023
• 2001 AIME I Problems/Problem 3
  ...ath>x^{2001}+\left(\frac 12-x\right)^{2001}=0</math>, given that there are no multiple roots. ...has a <math>2000^{\text{th}}</math> degree polynomial. Note that there are no multiple roots. Thus, if <math>\frac{1}{2} - x</math> is a root, <math>x</m
  2 KB (335 words) - 18:38, 9 February 2023
• 2002 AIME I Problems/Problem 10
  == Solution 4 No Trig ==
  4 KB (643 words) - 22:44, 8 August 2023
• 2002 AIME I Problems/Problem 9
  Again, note that <math>h,t,u \neq 1</math>. The three conditions state that no picket number <math>n</math> may satisfy any two of the conditions: <math>n The three conditions state that no picket number <math>n</math> may satisfy any two of the conditions: <math>n
  4 KB (749 words) - 19:44, 25 April 2024
• 2002 AIME I Problems/Problem 1
  ...last letter/number for there to be a palindrome. Thus, the probability of no palindrome is <cmath>\frac{25}{26}\cdot \frac{9}{10}=\frac{45}{52}</cmath>
  3 KB (369 words) - 23:36, 6 January 2024
• 2003 AIME I Problems/Problem 2
  ...region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The [[ratio]] of the total area of the
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 14
  ==Solution 4 (No Trig)==
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2003 AIME II Problems/Problem 2
  Let <math>N</math> be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when <math>N</math> We want a number with no digits repeating, so we can only use the digits <math>0-9</math> once in co
  1,013 bytes (162 words) - 09:00, 11 July 2023
• 2002 AIME II Problems/Problem 12
  ...a_5\leq .4</math>. Therefore we can have two shots made, one shot made, or no shots made in group A. '''Case 1: Group A contains no made shots.'''
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2002 AIME II Problems/Problem 9
  ...of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</math> is divide
  3 KB (404 words) - 23:07, 4 May 2024
• 2002 AIME II Problems/Problem 8
  ...ch the equation <math>\left\lfloor\frac{2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math>. (The notation <math>\lfloor x\rfloor< ...ch the equation <math>\left\lfloor\frac{2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math> is <math>\boxed{049}</math>.
  6 KB (908 words) - 14:22, 14 July 2023
• 2002 AIME II Problems/Problem 7
  Thus, there are no restrictions on <math>k</math> in <math>\pmod{3}</math>.
  3 KB (403 words) - 12:10, 9 September 2023
• 2002 AIME II Problems/Problem 6
  ...(n-2)A + (n+2)B</math> or <math>1 = n(A+B)+ 2(B-A)</math>. Since there is no n term on the left hand side, <math> A+B=0</math> and by inspection <math>1
  2 KB (330 words) - 05:56, 23 August 2022
• 2002 AIME II Problems/Problem 5
  <math>0>1 \cdot 3^m</math> which has no solution for non-negative integers m <math>30 > 32 \cdot 3^m</math> which has no solution for non-negative integers m
  3 KB (515 words) - 14:46, 14 February 2021
• 2001 AIME II Problems/Problem 10
  To show that no other possibilities work, suppose <math>j-i \equiv a \pmod{6},\ 1 \le a \le
  4 KB (549 words) - 23:16, 19 January 2024
• 2001 AIME II Problems/Problem 9
  ...<math>2^9=512</math> ways to paint the <math>3 \times 3</math> square with no restrictions, so there are <math>512-95=417</math> ways to paint the square ...it square is blue--The blue lies on the center of the bigger square, makes no 2*2 grid <math>9-1=8</math> cases
  8 KB (1,207 words) - 20:04, 5 September 2023
• 2000 AIME II Problems/Problem 14
  ...ath>16!</math> place followed by <math>0</math>’s. To add these numbers, no carrying will be necessary, because there is only one non-zero value for ea
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2000 AIME II Problems/Problem 13
  ...e come to the conclusion that <math>y=\frac{-1\pm i\sqrt{3}}{2}</math>, so no real solution exists for <math>x</math>. Thus our solution is correct.
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 11
  ...ith <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizontal or vertical sides, and <math>\overline{AB}</math> and <math>\ove
  4 KB (750 words) - 22:55, 5 February 2024
• 2005 AMC 10B Problems/Problem 25
  ...from <math>1</math> to <math>100</math>, inclusive, has the property that no two elements of <math>B</math> sum to <math>125</math>. What is the maximum ==Solution 2 (If you have no time)==
  3 KB (517 words) - 19:15, 15 October 2023
• 2006 USAMO Problems/Problem 3
  integer coefficients, suppose further that no prime divides all the
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  ...n only be made from a number that is divisible by <math>2^M</math> (and by no higher power of 2). Thus we must have <math>M < i_0</math>, since otherwise
  7 KB (1,280 words) - 17:23, 26 March 2016
• Diagonal
  [[Triangle]]s have no diagonals, while [[convex polygon|convex]] [[quadrilateral]]s have two inte ...trahedra]] have no space or face diagonals. [[Octahedron|Octahedra]] have no face diagonals but have 3 space diagonals. [[Cube (geometry) | Cube]]s hav
  2 KB (374 words) - 00:37, 25 January 2015
• Ring
  However, there is no multiplicative identity, so this is not a proper ring.
  6 KB (994 words) - 06:16, 8 April 2015
• Holomorphic function
  by the previous paragraph, there is no holomorphic extension of <math>f</math>
  9 KB (1,537 words) - 21:04, 26 July 2017
• Young Scholars Program
  ...extensions. Students do a significant amount of research. The program is no longer free.
  463 bytes (79 words) - 19:14, 20 February 2016
• Convex polygon
  ...s on the [[perimeter]] of the polygon are connected by a [[line segment]], no point on that segment will be outside the polygon. For example, every [[re
  948 bytes (147 words) - 22:10, 27 February 2020
• 2006 AMC 10B Problems
  ...rams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...from <math>1</math> to <math>100</math>, inclusive, has the property that no two elements of <math>B</math> sum to <math>125</math>. What is the maximum
  12 KB (1,874 words) - 21:20, 23 December 2020
• Platonic solid
  ...fit anywhere from three to five equilateral triangles around a point, but no more. This adds three, so we have a total of five. We have exhausted all ca
  8 KB (1,168 words) - 22:48, 19 February 2022
• Furman University Wylie Mathematics Tournament
  ...//math.furman.edu/tournament/tournament.html Furman website] but there are no solutions to the earlier ones. Solutions are posted below:
  2 KB (365 words) - 21:21, 18 March 2017
• University of South Carolina High School Math Contest/1993 Exam/Problem 11
  ...boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card? ...uivalently, one could say that the only [[permutation]]s of 4 objects with no fixed points are those with [[cycle notation]] <math>(abcd)</math> and <mat
  2 KB (334 words) - 16:27, 25 October 2023
• University of South Carolina High School Math Contest/1993 Exam/Problem 25
  ...ngent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.)
  1 KB (172 words) - 00:11, 16 February 2016
• 1977 Canadian MO Problems/Problem 1
  ...ath>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> ...us, <math>b^{2} + b + 1</math> is not a perfect square, and thus there is no <math>b</math> that satisfies <math>4f(a) = f(b)</math>.
  1 KB (245 words) - 10:04, 28 September 2023
• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label as the card? ...by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 2
  ...d to place 12 indistinguishable balls into 7 distinguishable boxes so that no box contains more than 9 balls. There are <math>{12 + 7 - 1 \choose 7 - 1}
  1 KB (188 words) - 15:53, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 5
  then <math>f(n)=y</math>. If there is no such <math>y</math>, then <math>f(n)=0</math>. If <math>p=11</math>, find t
  2 KB (340 words) - 15:52, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...se variables, which does uniquely determine these variables - but there is no obvious way of computing them. We will show a different solution.
  3 KB (568 words) - 15:50, 3 April 2012
• Mock AIME 1 2006-2007/Problems
  then <math>f(n)=y</math>. If there is no such <math>y</math>, then <math>f(n)=0</math>. If <math>p=11</math>, find t
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 8
  ...= 1</math> nor <math>x_3 = 2</math> is a solution, so in this case we have no solutions. ...\cdot 3\cdot 4 = 144</math> so <math>x_3^2(x_3 + 2) = 12</math>, which has no solutions in positive integers
  3 KB (470 words) - 00:33, 10 August 2019
• Mock AIME 2 2006-2007 Problems/Problem 9
  ...a,b,</math> and <math>d</math> are relatively prime and <math>c</math> has no perfect square divisors excluding <math>1,</math> find <math>a+b+c+d.</math
  2 KB (358 words) - 23:22, 3 May 2014
• Strict inequality
  ...ath> (less than). That is, a strict inequality is an inequality which has no [[equality condition]]s.
  827 bytes (127 words) - 11:53, 2 September 2020
• Mock AIME 2 2006-2007 Problems/Problem 15
  For the second level, no red square can have the same row and column location as any of the red squa For the third level, no red square can have the same row and column location as any of the red squa
  4 KB (739 words) - 17:04, 24 November 2023
• Mock AIME 2 2006-2007 Problems
  ...a,b,</math> and <math>d</math> are relatively prime and <math>c</math> has no perfect square divisors excluding <math>1,</math> find <math>a+b+c+d.</math
  5 KB (848 words) - 23:49, 25 February 2017
• Base numbers/Conversion
  ...For example, <math>3\cdot 3\neq 9</math> (in base 7). For one, there is no 9 in base 7. Second, we need to go back to the definition of multiplicatio
  7 KB (1,177 words) - 15:56, 18 April 2020
• IMO Problems and Solutions
  [[1980 IMO | There was no IMO in 1980]]
  35 KB (4,009 words) - 20:25, 21 February 2024
• 1977 Canadian MO Problems
  ...ath>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math>
  3 KB (560 words) - 19:23, 10 March 2015
• 1970 IMO Problems/Problem 4
  ...not work because only one of the numbers is a multiple of 5. So there are no such sets. ...e cannot partition the original set into two sets of equal product. Thus, no such <math>n</math> exist.
  1 KB (250 words) - 03:31, 2 January 2023
• 2006 Romanian NMO Problems
  * no student lost all <math>4</math> matches.
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 4
  * no student lost all <math>4</math> matches.
  2 KB (325 words) - 15:53, 11 December 2011
• Pierre de Fermat
  ...icial. He, however, is most famous for being an amateur mathematician. His name is attached to several results in number theory, though he worked in many o ...he Franciscan monastery in Grandselve and then the University of Toulouse. No record shows that he was particularly adept with numbers.
  5 KB (860 words) - 17:10, 21 March 2023
• 1969 Canadian MO Problems/Problem 4
  ...h>PD,PE,PF</math> are drawn to the three sides of the triangle. Show that, no matter where <math>P</math> is chosen, <math>\frac{PD+PE+PF}{AB+BC+CA}=\fra
  1 KB (215 words) - 00:23, 9 October 2020
• 2005 Canadian MO Problems
  * Prove that there are no integer <math>n</math> and Pythagorean triple <math>(a,b,c)</math> satisfyi * Show that for no value of <math>n</math> can four such points in <math>S</math> (and corresp
  2 KB (436 words) - 11:45, 26 December 2018
• 2005 Canadian MO Problems/Problem 3
  * Show that for no value of <math>n</math> can four such points in <math>S</math> (and corresp ...th> is the center), where <math>OQ_i < 1 - \frac{\sqrt{3}}{2},</math> then no point can be chosen on the circle which is closer to one of the <math>Q_i</
  2 KB (460 words) - 13:35, 9 June 2011
• IMO Shortlist Problems
  * There was no IMO in [[1980 IMO | 1980]].
  2 KB (177 words) - 09:19, 23 June 2023
• P versus NP
  ...th> the class of decision problems (those whose answer is either "yes" or "no," as opposed to other classes such as counting problems) that can be solved A ''decision problem'' is a problem that admits a yes or no answer (as opposed to an optimization problem, such as "What is the length
  6 KB (1,104 words) - 15:11, 25 October 2017
• Congruent (geometry)
  ...laced exactly on top of the other, with all parts lining up perfectly with no parts on either figure left over. In plain language, two objects are congr ...of the same or of another straight line <math>a'</math> having, likewise, no point other than <math>B'</math> in common. Then, if <math>\overline{AB}\c
  10 KB (1,655 words) - 21:43, 24 March 2022
• 2005 AMC 10A Problems/Problem 18
  ...s wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B w
  2 KB (260 words) - 17:42, 7 July 2023
• 2005 AMC 10A Problems/Problem 25
  ==Solution 2 (no trig)==
  6 KB (854 words) - 20:25, 24 July 2022
• Yang-Mills Theory
  However, the quantum Yang-Mills theory (no quarks) with a non-abelian gauge group is an exception. It has a property k
  2 KB (363 words) - 15:31, 1 December 2015
• Zero (constant)
  '''Zero''', or 0, is the name traditionally given to the additive [[identity]] in number systems such as ...h>x</math>. So anything times its reciprocal has to be 1. Therefore, 0 has no reciprocal because anything times 0 is 0.
  2 KB (371 words) - 12:55, 21 June 2023
• Perfect cube
  ...ath>5^3 = 125</math>. However, 121 is not a perfect cube because there is no integer <math>n</math> such that <math>n^3 = 121</math>.
  704 bytes (91 words) - 14:12, 24 August 2023
• Obtuse triangle
  ...if two of its [[altitude]]s lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the trian
  1 KB (193 words) - 14:20, 12 June 2022
• Linear congruence
  ...on. For instance, the congruence equation <math>2x\equiv3\pmod8</math> has no solutions. A solution is guaranteed [[iff]] <math>a</math> is [[relatively * if <math>d</math> does not divide <math>b</math>, there will be no solution
  1 KB (225 words) - 15:52, 3 August 2022
• Divisibility rules/Rule for 2 and powers of 2 proof
  ...this is a base-10 expression rather than a product. If <math>N</math> has no more than <math>n</math> digits, then the last <math>n</math> digits of <ma
  2 KB (406 words) - 16:46, 3 May 2020
• Power set
  ...ent, ... , or n elements. There are <math>\binom{n}{0}</math> ways to have no elements, <math>\binom{n}{1}</math> ways to have one element, ... , and <ma
  4 KB (757 words) - 11:44, 8 March 2018
• AwesomeMath
  ...kend for each 12-week semester. Students choose their classes and there is no application requirement to participate. Benefits to students include:
  1 KB (195 words) - 11:19, 30 November 2023
• 1970 Canadian MO Problems
  b) Show that there is no integer such that the deletion of the first digit produces a result that is show that there is no integer <math>k</math> such that <math>f(k)=8</math>.
  4 KB (604 words) - 04:32, 8 October 2014
• Perfect number
  ...greater than <math>10^{500}</math>. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.
  1 KB (193 words) - 19:09, 3 March 2022
• Angle
  ...ways by choosing different points along the side of the angle. If there is no ambiguity, this notation can be shortened to simply <math>\angle B</math>.
  3 KB (568 words) - 20:21, 21 June 2019
• Derangement
  ...on]] with no [[fixed point]]s. That is, a derangement of a [[set]] leaves no [[element]] in its original place. For example, the derangements of <math>
  3 KB (473 words) - 12:57, 20 February 2024
• Reduced fraction
  ...c a b</math> where <math>a</math> and <math>b</math> are [[integer]]s with no common [[divisor]]s (that is, which are [[relatively prime]]).
  1 KB (225 words) - 13:22, 7 March 2021
• 1970 IMO Problems
  ...onsider all possible triangles having these point as vertices. Prove that no more than <math>70 \%</math> of these triangles are acute-angled.
  3 KB (558 words) - 00:17, 10 December 2022
• Mock MathCounts
  ...ar in the months leading up to the actual MATHCOUNTS competition. There is no guarantee that community members will make Mock MATHCOUNTS in any given yea
  26 KB (3,265 words) - 21:34, 20 March 2024
• Proper divisor
  ...<math>n</math> other than <math>n</math> itself. Thus <math>1</math> has no proper divisors, [[prime number]]s have exactly one proper divisor <math>1<
  436 bytes (63 words) - 16:27, 10 May 2021
• Distributive property
  Also note that there is no particular reason that distributivity should be one-way, as it is with conv
  2 KB (267 words) - 10:13, 23 July 2020
• Rockdale Mathematics Competition
  No calculators are permitted on any part of the competition.
  2 KB (267 words) - 21:50, 6 March 2016
• 2007 iTest Problems
  \text{(L) There is no such value of } r\qquad</math> ...e all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math
  33 KB (5,177 words) - 21:05, 4 February 2023
• Qq808casino
  Online casino gaming is no longer a new phenomenon. Ever since the first one was introduced, its popul
  2 KB (276 words) - 03:46, 9 December 2019
• 1981 IMO Problems/Problem 4
  ...n another [[multiple]] of <math>p_i^{e_i}</math>. In particular, there is no set of cardinality 3 satisfying our conditions, because each number greater ...reasons. However, for <math>n = 4 </math>, this does not work, and indeed no set works other than <math>\{ 3,4,5,6 \} </math>. To prove this, we simply
  3 KB (516 words) - 09:43, 28 March 2012
• Duke Math Meet
  ...d into teams and complete problems successively), and a tiebreaker round. No calculators are permitted during the competition.
  1 KB (174 words) - 09:48, 10 November 2012
• Dimension
  A 0-dimensional object is a [[point]], which has no [[length]], height, or [[depth]]. ...mensional object is a [[line]], or [[line segment]], which has length, but no other characteristics.
  1 KB (167 words) - 16:58, 19 February 2024
• Least upper bound
  ...uch that <math>s \leq M</math> for every <math>s \in S</math> and there is no <math>m < M</math> with this same property. ...] of <math>S</math>. If <math>M \not\in S</math>, then <math>S</math> has no maximum.
  1,011 bytes (177 words) - 14:08, 5 March 2022
• Plane
  ...imension]]al object that is infinitely [[length | long]] and wide, but has no [[thick]]ness and zero curvature.
  920 bytes (148 words) - 13:52, 2 October 2023
• 2005 AMC 10A Problems
  ...s wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B w ...and four sides of length <math>\frac{\sqrt{2}}{2}</math>, arranged so that no two consecutive sides have the same length. What is the area of the octago
  14 KB (2,026 words) - 11:45, 12 July 2021
• Expression
  ...arithmetic expression, while <math>7 \times + 43</math> is not. There are no equal signs in expressions.
  2 KB (239 words) - 14:42, 11 March 2023
• 2005 AMC 10A Problems/Problem 20
  ...and four sides of length <math>\frac{\sqrt{2}}{2}</math>, arranged so that no two consecutive sides have the same length. What is the area of the octago
  3 KB (414 words) - 10:25, 15 May 2023
• Euler line
  ...<math>\triangle A'B'C',</math> so point <math>K</math> lies on line <math>NO'.</math>
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2004 AMC 10A Problems/Problem 4
  ...= x - 2</math>, so we must solve <math>x - 1 = x- 2</math>, which also has no solutions. Finally, if <math>1 \leq x \leq 2</math>, then <math>|x - 1| =
  2 KB (289 words) - 18:01, 16 January 2021
• Common Multiplication
  ...mber of negative numbers, the result is negative. However, if you multiply no 0's, any amount of positive numbers, and an even number of negative numbers
  2 KB (250 words) - 18:38, 30 December 2010
• 2004 AMC 12A Problems/Problem 4
  ...t-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters? ...<math>24/6 = 4</math> of Bertha's daughters, so the number of women having no daughters is <math>30 - 4 = \boxed{26}</math>.
  2 KB (232 words) - 13:56, 19 January 2021
• 1961 IMO Problems/Problem 1
  ...th> (which, in this case, implies <math>y,z = 0 </math>). Hence there are no positive solutions when <math>a = 0 </math>.
  5 KB (916 words) - 18:15, 26 March 2024
• Cyprus Mathematical Olympiad
  ...ts, each wrong answer subtracts 1 point and no answer gives no points (but no points are lost). This means there is a penalty for wrong answers. Each set
  2 KB (366 words) - 14:26, 4 September 2017
• Pick's Theorem
  ...minimum distance between two distinct lattice points is <math>1</math>. If no two lattice points have distance <math>1</math>, then by <math>\frac{1}{2}b
  2 KB (301 words) - 13:08, 20 February 2024
• Fermat point
  ...alled the Torricelli point) of a triangle <math>\triangle ABC</math> (with no angle more than <math>120^{\circ}</math> is a point <math>P</math> which ha
  4 KB (769 words) - 16:07, 29 December 2019
• 1985 IMO Problems/Problem 2
  ...(ii) may only imply that 0 has the same color as <math>k</math>, which put no restrictions on the colors of the other residues.
  1 KB (267 words) - 00:11, 12 July 2020
• 2004 AMC 10A Problems
  ...t-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
  15 KB (2,092 words) - 20:32, 15 April 2024
• Pythagorean triple
  A Pythagorean triple is called ''primitive'' if its three members have no common [[divisor]]s, so that they are [[relatively prime]]. Some triples l ...> and <math>(m^2+n^2) - (m^2 - n^2) = 2n^2</math>, which are integers with no common factor greater than 2. <math>\blacksquare</math>
  4 KB (684 words) - 16:45, 1 August 2020
• 2003 AMC 12A Problems/Problem 7
  By the [[triangle inequality]], no side may have a length greater than the semiperimeter, which is <math>\frac
  1 KB (189 words) - 21:45, 19 July 2023
• 2003 AMC 12A Problems/Problem 8
  Clearly, there are no positive factors of <math>60</math> between <math>7</math> and <math>\sqrt{
  2 KB (326 words) - 15:40, 19 August 2023
• Cyprus Seniors Provincial Problems and Solutions
  '''Note:''' Until 2003 there was no Provincial competition for 1st grade students. There was only [[Seniors Pan
  1 KB (122 words) - 15:11, 4 September 2017
• Cyprus Juniors Provincial Problems and Solutions
  '''Note:''' Until 2003 there was no Provincial competition for junior high-school (Gymnasium) students. There w
  1 KB (107 words) - 15:00, 4 September 2017
• 1987 IMO Problems/Problem 1
  ...to be careful, because now, what about for <math>k=n-1</math>? We see that no matter how we choose <math>n-1</math> fixed points, we always have to put t
  3 KB (604 words) - 18:54, 10 May 2023
• 1987 IMO Problems/Problem 4
  Prove that there is no function <math>f </math> from the set of non-negative integers into itself
  5 KB (923 words) - 19:51, 21 January 2024
• 1987 IMO Problems
  Prove that there is no function <math>f </math> from the set of non-negative integers into itself
  3 KB (459 words) - 14:24, 17 September 2023
• 2003 AMC 12A Problems/Problem 25
  <math>a>0</math>: A semi-hyperbola above or on the x-axis. Therefore, no positive value of <math>a</math> allows the domain and range to be the same ...wnwards. Now, either the entire graph of the parabola is below the x-axis (no roots), or some part including the vertex sticks out above it. The graph of
  9 KB (1,606 words) - 11:34, 10 July 2020
• Minkowski Inequality
  As the latter can be iterated, there is no loss of generality by putting <math>m=2</math>.
  1 KB (240 words) - 16:49, 29 December 2021
• 2006 Seniors Pancyprian/2nd grade/Problem 4
  A quadrilateral <math>ABCD</math>, that has no parallel sides, is inscribed in a circle, its sides <math>DA</math>, <math>
  856 bytes (129 words) - 00:10, 20 February 2020
• Georgia Tech High School Math Competition
  There are currently no distinction between junior varsity and varsity (though this distinction exi ...on multiple choice test. Each question is worth three points, and there is no deduction for incorrect answers. The top quintile of contestants from this
  3 KB (475 words) - 21:51, 31 December 2013
• ELMO
  ...of most of the returning MOPpers as well as the Black group, create a new name for the ELMO based on these initials. Names in past years have included "Er ...s fair. Throughout all this, the ELMO room is kept securely locked so that no one not involved in ELMO coordination can see any potential test problems,
  5 KB (773 words) - 19:16, 17 June 2022
• Asymptote: Advanced Configuration
  ...with a button that says '''Refresh FNDB'''. Click it to refresh your file name database. MikTeX should now see your Asymptote style files. ...asymptote.sty package to find the end of the Asymptote code; there can be no whitespaces or other characters on the line containing <tt>\end{asy}</tt>.
  12 KB (1,931 words) - 13:53, 26 January 2020
• Asymptote: Macros and Packages
  ...In addition, you want it to simply return the pair <math>(a,b)</math> if no value of <math>r</math> is specified, so you want <math>r</math> to default ...ine a function with no output simply put <tt>void</tt> before the function name. This is the general format for a function definition.
  3 KB (515 words) - 00:43, 24 April 2019
• Pigeonhole Principle/Solutions
  ...th> to <math>n-2</math> friends for the remaining friends not to also have no friends. By pigeonhole again, this leaves at least <math>1</math> other per Therefore, no two points within the same region can have a distance of <math>2</math> or
  10 KB (1,617 words) - 01:34, 26 October 2021
• 2006 Canadian MO Problems/Problem 4
  ...th>2n+1</math> teams, where two teams play exactly one match and there are no ties. We say that the teams <math>X</math>, <math>Y</math>, and <math>Z</ma
  604 bytes (100 words) - 18:20, 28 November 2023
• Canada/USA Mathcamp
  ...eturning students, and including both domestic and international students. No student is turned away for lack of funds, and financial aid requests are no
  4 KB (680 words) - 23:22, 12 April 2021
• Greatest lower bound
  ...uch that <math>m \leq s</math> for every <math>s \in S</math> and there is no <math>M > m</math> with this same property. ...] of <math>S</math>. If <math>m \not\in S</math>, then <math>S</math> has no minimum.
  597 bytes (109 words) - 13:55, 5 March 2022
• Juniors Pancyprian Problems and Solutions
  '''Note:''' Until 2003 there was no Pancyprian competition for 1st and 2nd grade. As of 2004 each grade (1st, 2
  755 bytes (72 words) - 15:00, 4 September 2017
• Reducible fraction
  A fraction is no longer reducible or [[irreducible fraction|irreducible]] when the [[numerat
  622 bytes (83 words) - 22:24, 5 January 2024
• 2005 PMWC Problems/Problem I1
  The length of the resulting number is the same no matter what, so to maximize the number we want to invoke the [[greedy algor
  1 KB (176 words) - 16:57, 28 November 2018
• 2002 USAMO Problems/Problem 2
  ...triangle <math>T </math> whose side lengths are all positive integers with no common divisor and determine those integers.
  2 KB (298 words) - 22:32, 6 April 2016
• Mock AIME 3 Pre 2005 Problems/Problem 5
  Case <math>5</math>: The word contains no <math>O</math>s.
  5 KB (795 words) - 16:03, 17 October 2021
• Mock AIME 3 Pre 2005 Problems/Problem 8
  ...<math>\pm</math>'s all the <math>b_i</math>'s I've listed are positive so no restrictions are imposed here. There are restrictions imposed by <math>b_6<
  3 KB (520 words) - 12:55, 11 January 2019
• IMO problems statistics
  A * in front of a country name means that this country no longer exists, since it split or merged into other countries.
  2 KB (184 words) - 09:51, 23 July 2021
• 2007 AIME I Problems/Problem 5
  === Solution 5(similar to solution 3 but faster solution if you have no time) ===
  7 KB (1,076 words) - 00:10, 29 November 2023
• 2007 AIME I Problems/Problem 10
  ...e column with no balls in the top half must all be filled in, so there are no choices here. In the column with two balls already, we can choose any of t ...has columns <math>1</math> and <math>2</math> shaded. Note how if there is no complement to this, then all the other five rows must have at least one squ
  13 KB (2,328 words) - 00:12, 29 November 2023
• LaTeX:Downloads
  LaTeX can also be used Online! No need to download. '''Overleaf''' is one of the most popular online LaTeX ed
  8 KB (1,318 words) - 01:16, 28 February 2022
• LaTeX:Basics
  ...produces <math>\sqrt[n]{x}</math>. The [n] does not have to be included if no root is needed, so you can just use \sqrt{x} to produce <math>\sqrt{x}</mat ...or click File, then Save) as 'Hello' (don't include the quote marks in the name) in a folder of your choice.
  6 KB (1,112 words) - 08:03, 22 August 2023
• LaTeX:Math
  ...r click File, then Save) as 'mymath' (don't include the quote marks in the name) in a folder of your choice. The file will appear in your folder as 'mymath ..., the align command requires that you use the package amsmath (and there's no reason to ''not'' use this package). Second, the * after align prevents li
  8 KB (1,323 words) - 15:09, 3 December 2020
• 2007 AIME II Problems/Problem 1
  ...aracters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the fou
  1 KB (180 words) - 14:01, 3 December 2023
• 2007 AIME II Problems/Problem 5
  In the top row, there clearly are no squares that can be formed. In the second row, we see that the line <math>y
  4 KB (559 words) - 00:38, 3 January 2023
• 2007 AIME II Problems/Problem 6
  ...igit is less than <math>9</math>). There are <math>4</math> options to add no matter what(try some examples if you want) so the recursion is <math>S_n=4S
  2 KB (338 words) - 15:30, 7 August 2022
• 2007 AIME II Problems/Problem 8
  :(ii) no [[segment]]s of drawn lines lie inside the rectangle.
  3 KB (399 words) - 21:17, 24 February 2021
• 2007 AIME II Problems/Problem 11
  ...e revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a [[distance]] <math>x</math> from where
  2 KB (407 words) - 16:31, 29 February 2020
• 2007 AIME II Problems/Problem 14
  ...of finite degree) and <math>f(0)</math> is nonzero, <math>f(x)</math> has no real roots. ...re is no reason that <math>f(-2)\neq1</math>. Since <math>f(x)</math> has no real roots, the degree must be even. Consider <math>g(x)= f(x)/f(-x)</math
  7 KB (1,335 words) - 17:44, 25 January 2022
• 2007 AIME II Problems
  ...en from the four letters in AIME and the four digits in <math>2007</math>. No character may appear in a sequence more times than it appears among the fou :(ii) no [[segment]]s of drawn lines lie inside the rectangle.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2005 USAMO Problems
  ...all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>.
  4 KB (609 words) - 09:24, 14 May 2021
• 2007 AMC 12A Problems
  Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many [[subset]]s o
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2003 USAMO Problems/Problem 1
  ...h <math>10^{n-1}</math> are odd, and since <math>a<10^n</math>, there are no other digits.
  4 KB (736 words) - 22:17, 3 March 2023
• 2003 USAMO Problems/Problem 3
  ...erms <math>c_0, \ldots, c_{j-1} </math> are all less than <math>j </math>, no other terms that precede <math>c_{k} </math> can be unequal to <math>c_k </
  3 KB (636 words) - 13:39, 4 July 2013
• 2007 USAMO Problems/Problem 2
  ...e parallel to the gridlines in the circle.) That is a contradiction. Hence no such tiling exists. ...ap any member of <math>\mathcal{F}</math>. Then <math>D(O,r)</math> covers no grid point. Take the disc <math>D(O,r)</math> to be maximal in the sense th
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 4
  ...l connected since you can go around the other way in the loop. Now we have no loops. Each vertex can have at most 4 edges coming out of it. For each poin ...ting of a central cell and four arms with <math>s - 1</math> cells apiece. No connected figure with at least <math>s</math> cells can be removed without
  10 KB (1,878 words) - 14:56, 30 June 2021
• BMO
  ...nited Kingdom, the abbreviation usually refers to the Balkan Olympiad when no further explanation is given.
  306 bytes (42 words) - 23:45, 10 November 2015
• 2007 BMO Problems/Problem 3
  Now we prove that no <math>n \ge 5 </math> is a solution. ...math> \sqrt{3 + \sqrt{2 + \sqrt{4}}} </math> is irrational. Thus there is no solution for <math>n=4 </math> and the only solutions are <math>n=1, 3 </ma
  4 KB (682 words) - 10:53, 13 January 2016
• 2004 IMO Shortlist Problems/N2
  We now claim that for <math>a = 2^k </math>, the equation has no non-canonical solutions. ...But since <math> \frac{\psi(x_1)}{x_1} </math> is an integer and there are no factors in the denominator common to the odd prime which divides <math>p </
  6 KB (1,007 words) - 09:10, 29 August 2011
• 2005 IMO Shortlist Problems/C2
  ...e only finitely many blossoms, there exists at least one blossom which has no blossoms as indirect successors. We remove this blossom, one of its buds,
  3 KB (538 words) - 09:12, 29 August 2011
• Order of operations
  Afterward, there are no multiplication and division, so we can move on to addition and subtraction.
  2 KB (271 words) - 13:19, 5 March 2022
• LaTeX:Pictures
  ...article, working with graphics and pictures without Asymptote in LaTeX is no easy feat. ...he link to the image and choose "Save link as...") Save the image with the name 'myimage.png'. Make sure you pay attention to what folder you are saving th
  9 KB (1,454 words) - 16:52, 19 February 2024
• LaTeX:Layout
  ...eXnicCenter and trying out each of the examples as you go. It takes almost no time at all to just copy-paste, compile, and view the results. ...ines in your source file, there is not only no new paragraph, but there is no line break, either. To end one paragraph and start another it is not enough
  30 KB (5,171 words) - 10:16, 4 April 2021
• Chicken McNugget Theorem
  ...</math> for all <math>k \in \mathbb{Z}</math>. We now prove that there are no others. Suppose <math>(x_{1},y_{1})</math> and <math>(x_{2},y_{2})</math> a ...th>7</math>. What is the largest integer <math>n</math> such that there is no way to buy exactly <math>n</math> nuggets? Can you Generalize ? (Source: Th
  17 KB (2,748 words) - 19:22, 24 February 2024
• Zermelo-Fraenkel Axioms
  ...alled the [[empty set]] and denoted <math>\emptyset</math>) which contains no elements. This makes sure no set contains itself, thus avoiding certain paradoxical situations. <br/>
  4 KB (732 words) - 20:49, 13 October 2019
• Parity
  No matter what Jack chooses, the number of numbers with the same parity is fou
  4 KB (694 words) - 22:00, 12 January 2024
• 2007 AMC 12A Problems/Problem 14
  ...f the product of any four is at least <math> |(-3)(-1)(1)(3)|=9</math>, so no factor can have an absolute value greater than <math>5</math>. Thus the fac
  2 KB (278 words) - 02:10, 16 February 2021
• 2007 AMC 12A Problems/Problem 22
  ...1995 + 3u = 2007 \Longrightarrow u = 4</math>. This exceeds our bounds, so no solution here. ...ming these, we have <math>n+S(n)+S(S(n)) \equiv 3 \pmod 9</math>. Clearly, no integers of this form will satisfy the condition because <math>2007</math>
  15 KB (2,558 words) - 19:33, 4 February 2024
• 2007 AMC 12A Problems/Problem 25
  Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many [[subset]]s o ...| | | o | corresponds to the arrangment o o o | o |. Notice that there is no longer any restriction on consectutive numbers. Therefore, we can easily pl
  9 KB (1,461 words) - 23:07, 27 January 2024
• 2005 AMC 12A Problems/Problem 18
  ...0</math>, which is a fact we must somehow utilize. Since there seems to be no easy way to directly calculate the number of "prime-looking" numbers, we ca
  2 KB (277 words) - 18:15, 25 November 2020
• 2005 AMC 12A Problems/Problem 21
  ...nt of <math>a</math> becomes huge, and since <math>a \ge 2</math> there is no way we can satisfy the second condition. Hence we have two ordered triples
  991 bytes (153 words) - 21:20, 3 July 2013
• 2006 AIME II Problems/Problem 8
  ...follows: Pick the colors to the immediate right of a divider. If there is no color to the immediate right of a divider, then that color is 6.
  4 KB (695 words) - 10:37, 4 November 2023
• 2006 AIME II Problems/Problem 5
  ...carried out exactly the same steps and found exactly the same probability no matter which face it was. We could have labelled <math>A(6)</math> as <math
  5 KB (712 words) - 12:10, 5 November 2023
• 2006 AIME II Problems/Problem 15
  The area of the triangle must be the same no matter how we measure it; therefore <math>x\cdot h_x = y\cdot h_y = z \cdot
  4 KB (725 words) - 17:18, 27 June 2021
• 2006 AIME II Problems/Problem 4
  ...wo distinct sets that are automatically ordered. For this reason, there is no need to multiply by two to count doubles or treat as a permutation. ...sets are distinct because one side has 12 and the other does not. There is no need to multiply by two.
  2 KB (384 words) - 14:12, 20 April 2024
• 2006 AIME II Problems/Problem 7
  ...e thus <math>9^2</math> cases that we overlooked, where <math>a</math> had no zero digits, but <math>1000 - a</math> did. Adding up the cases with <math>a \in [1, 999]</math> with no zero digits and removing the cases with <math>1000 - a</math> with zero dig
  7 KB (1,114 words) - 03:41, 12 September 2021
• 2006 AIME II Problems/Problem 10
  ...soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a <math> 50\% </math> chance of winning each game
  6 KB (983 words) - 13:42, 8 December 2021
• 2006 AIME II Problems
  ...soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a <math> 50\% </math> chance of winning each game
  8 KB (1,350 words) - 12:00, 4 December 2022
• 2005 PMWC Problems
  ...<math>x</math> is <math>455</math> and the numerator and denominator have no common factor except <math>1</math>, how many possible values are there for
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  ...heet. Each student noticed five A’s. No student saw all the grades and no student saw her or his own grade. What is the minimum number of students w Every year there is at least one Friday the thirteenth, but no year has more than three. This year there are exactly three : in February,
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  ...umbers less than <math>1000</math> contain at least one <math>2</math> but no <math>3</math>? ...the largest number of candies below <math>4000</math> in each bag so that no candies are left?
  11 KB (1,713 words) - 22:47, 13 July 2023
• 1971 Canadian MO Problems
  .../math> and <math>p(1)</math> are both odd, show that <math>p(x)</math> has no integral roots.
  3 KB (519 words) - 08:58, 13 September 2012
• 1999 USAMO Problems/Problem 5
  ...emaining (since from the first moves we have constructed at least one, and no one has played between them otherwise the game will have ended), and the fi
  2 KB (433 words) - 13:35, 4 July 2013
• Main Page/Suggestions
  During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.
  2 KB (284 words) - 17:05, 5 August 2020
• 1960 IMO Problems/Problem 1
  When <math>q=1</math>, we get that <math>a</math> is not an integer. There is no <math>N</math> for this case. When <math>q=2</math>, we get that <math>a</math> is not an integer. There is no <math>N</math> for this case.
  4 KB (751 words) - 05:01, 17 August 2022
• 1960 IMO Problems/Problem 2
  ...th>x < 45/8</math>. But if <math>x < -1/2</math> then the inequality makes no sense, since <math>\sqrt{2x + 1}</math> is imaginary. So the original inequ
  2 KB (239 words) - 05:51, 7 April 2024
• 1960 IMO Problems/Problem 6
  ...dratic equation <math>3 \tan^4 \theta - 3 \tan^2 \theta + 1 = 0</math> has no real roots for <math>\tan \theta</math>, which means that <math>k \neq 1</m
  7 KB (1,214 words) - 18:49, 29 January 2018
• Asymptote: CSE5
  ...h function; a function with a full name and a function with an abbreviated name. ...k (<math>1</math> by default). Can be set to <math>0</math>, in which case no arcs will be drawn, <math>1</math> through <math>7</math>, in which case <m
  4 KB (607 words) - 15:09, 30 June 2020
• Analysis
  ...variety of topics which appear quite different to each other, so there is no way to define exactly what is "analysis" and what is not. However, one can
  2 KB (352 words) - 18:12, 8 February 2015
• Zorn's Lemma
  ...set. Evidently, <math>a</math> is an extreme point of <math>M</math>, as no element of <math>M</math> is less than <math>a</math>, so every element les ...an <math>x</math>. Then <math>f</math> is an increasing function, but for no <math>x \in A</math> does <math>x = f(x)</math>, which contradicts the Bour
  9 KB (1,669 words) - 19:02, 1 August 2018
• 2004 AMC 12A Problems/Problem 11
  ...s is <math>25+25+25+5</math>. Thus, having three quarters, one nickel, and no dimes <math>\boxed{\mathrm{(A)}\ 0}.</math>
  2 KB (310 words) - 17:39, 31 December 2022
• IDEAMATH
  ...up their solutions. The solutions must then be submitted by email or mail no later than the indicated postmark deadline. There is also an early applicat
  4 KB (648 words) - 03:13, 15 February 2021
• 2004 AMC 12A Problems/Problem 23
  ...should be zero (in fact, that quantity is <math>P(1)</math>, and there is no evidence that <math>1</math> is a root of <math>P(x)</math>).
  3 KB (517 words) - 14:13, 5 September 2021
• 2005 Alabama ARML TST Problems/Problem 8
  Therefore, there are absolutely no solutions to the above equation.
  1 KB (250 words) - 00:38, 28 October 2015
• Work
  ...is perpendicular to the direction in which the object is moving, there is no work done. The most common uses of this fact are in that gravity does not d
  1 KB (192 words) - 00:46, 16 July 2018
• Mass
  *The [[imperial system]] has no measure for mass, as at the time it was popularized, the distinction betwee
  1 KB (188 words) - 22:44, 10 October 2013
• 1976 USAMO Problems/Problem 5
  ...ves <math>(A(1), B(1), C(1)) = (0,0,0)</math> as a solution, and there are no others because the system of equations is linear. Hence, <math>A(1) = 0</ma
  3 KB (572 words) - 17:14, 16 August 2015
• Graph (graph theory)
  ...simple, loopless'' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For grap ...<math>v</math> is ''isolated'' if <math>d(v) = 0</math>, i.e. if there are no edges incident to <math>v</math>.
  8 KB (1,428 words) - 10:26, 27 August 2020
• 2000 AMC 12 Problems/Problem 25
  Notice that once the outer 4 vertices are colored, no matter how the inner 4 vertices are colored, the resulting graphs are disti ...tices. Permutation is used because the coloring of the inner vertices have no restrictions. In total that is <math>\binom{8}{4} \cdot 4! = \boxed{\textbf
  7 KB (998 words) - 21:04, 23 December 2023
• 2007 AMC 10A Problems/Problem 12
  ...least one tourist, we subtract the <math>2</math> cases where a guide has no tourist. Thus the answer is <math>2^6 - 2 = \boxed{62}\ \mathrm{(D)}</math>
  3 KB (411 words) - 18:07, 14 March 2023
• 1995 AHSME Problems
  There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1995 AHSME Problems/Problem 24
  There exist positive integers <math>A,B</math> and <math>C</math>, with no [[greatest common divisor|common factor]] greater than <math>1</math>, such
  977 bytes (142 words) - 14:05, 5 July 2013
• 1997 PMWC Problems
  'D': No, either A or B is telling the truth. 'E': No, 'D', that's not true.
  15 KB (2,057 words) - 19:13, 10 March 2015
• Euclid's Elements
  *A point is that which has no part.
  12 KB (2,094 words) - 15:42, 1 December 2015
• Algebraic manipulation
  ...zero. However, after multiplying both sides by <math>x-5</math>, there is no denominator that eliminates <math>x = 5</math> as a possible solution.
  4 KB (562 words) - 18:49, 8 November 2020
• 1959 AHSME Problems
  ...}\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} </math> ...and only if a,b,c are each relatively prime in pairs} \text{ (This means: no two have a common factor greater than 1.)}</math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  <math>\text{(A) no value of } n \quad \text{(B) one value of } n \quad \text{(C) two values of
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1966 AHSME Problems/Problem 5
  ...= 5</math>, which isn't in the [[domain]] of the equation. Thus there are no values of <math>x \Rightarrow \mathrm{(A)}</math>.
  752 bytes (116 words) - 12:39, 5 July 2013
• 1991 USAMO Problems/Problem 1
  ...rrow \sqrt{3} < \frac{y}{x} < 2</math>. For <math>x \le 3</math> there are no integer solutions. For <math>x = 4</math>, we have <math>y = 7</math> that
  5 KB (755 words) - 08:58, 6 May 2023
• 2002 AMC 12B Problems/Problem 4
  == Solution 2 (no limits)==
  2 KB (321 words) - 10:54, 9 August 2022
• Construction
  No other tools are allowed in a construction. However, the two basic tools alo
  3 KB (443 words) - 20:52, 28 August 2014
• 2007 AMC 8 Problems
  or "4", with no repeats. Three of these pieces are drawn, one at a time without
  12 KB (1,800 words) - 20:01, 8 May 2023
• 2008 iTest Problems/Problem 97
  ...of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If <math>m</math> leaves a remainder
  3 KB (497 words) - 13:29, 20 October 2019
• 2003 AMC 12B Problems/Problem 24
  ...lution. We already know of one such solution, so we need to make sure that no other solution appears. The last two inequalities contradict each other, thus there are no <math>a,b,c</math> that would satisfy both of them.
  7 KB (1,183 words) - 11:47, 15 February 2016
• 1994 USAMO Problems
  Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> ...e of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modi
  2 KB (391 words) - 07:58, 19 July 2016
• 2008 AMC 10A Problems
  ...re <math>B</math> offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <math>\textdollar15</math> by buying the computer at
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 2
  ...othesis, so this is a contradiction. Therefore <math>P_{k+1}(x)</math> has no double roots. This proves that that the roots of <math>P_{k+1}(x)</math> ar ...ath>. As we can choose the range <math>0\leq\theta\leq\pi</math> to ensure no duplications, we get that, upon rearranging, <math>0\leq k\leq2^{n-1}-\frac
  3 KB (596 words) - 16:19, 28 July 2015
• Partition (combinatorics)
  ...{n}</math> is [[even integer | even]], despite the presence of a formula! No simpler formula is known, and the existence of such a formula is doubtful. The ''empty partition'' (with no parts) is the unique partition of <math>0</math>, so <math>P(0) = 1</math>.
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2008 AMC 12A Problems
  ...re <math>B</math> offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <math> \textdollar 15</math> by buying the computer a
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 6
  ...re <math>B</math> offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <math>\$15</math> by buying the computer at store <ma
  2 KB (240 words) - 19:53, 4 June 2021
• Elliptical geometry
  ...'' is a term used to refer to [[geometry | geometries]] in which there are no parallel lines and the third axiom of order, which states that, of three po
  685 bytes (113 words) - 12:22, 20 November 2012
• 2007 AMC 12B Problems/Problem 24
  ...iple of <math>a</math>, and because <math>a</math> and <math>b</math> have no common factors greater than 1, it follows that 14 is divisible by <math>a</
  9 KB (1,522 words) - 22:46, 12 May 2022
• 2007 AMC 12B Problems/Problem 17
  Note that if <math>a</math> is positive, then, the equation will have no solutions for <math>b</math>. This becomes more obvious by noting that at <
  1 KB (242 words) - 12:41, 19 June 2023
• 2008 AMC 10B Problems
  ...xtdollar3</math> each, and carnations cost <math>\textdollar2</math> each. No other flowers are to be used. How many different bouquets could be purchase ...hat is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is <math>0</math>.)
  12 KB (1,838 words) - 16:52, 7 October 2022
• 2008 AMC 12B Problems
  ...th>3</math> dollars each, and carnations cost <math>2</math> dollars each. No other flowers are to be used. How many different bouquets could be purchase ...th 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let <math>R</math> be the region formed by the union of t
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2005 Alabama ARML TST Problems/Problem 13
  ...th> gets bigger, then that expression gets smaller, so there is absolutely no way that <math>n=p_1^5</math>. So the second case is true.
  2 KB (279 words) - 19:14, 10 March 2015
• 2006 Alabama ARML TST Problems
  ...rs <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> with no common factor greater than 1, such that ...nuing consecutively as he writes. When he stops, he realizes that there is no set of 5 composite integers among the ones he wrote such that each pair of
  5 KB (769 words) - 20:56, 24 March 2015
• 2008 AMC 12B Problems/Problem 5
  ...th>3</math> dollars each, and carnations cost <math>2</math> dollars each. No other flowers are to be used. How many different bouquets could be purchase The class could send <math>25</math> carnations and no roses, <math>22</math> carnations and <math>2</math> roses, <math>19</math>
  885 bytes (131 words) - 13:47, 15 February 2021
• 2008 AMC 12B Problems/Problem 21
  In other words, the two chosen <math>x</math>-coordinates must differ by no more than <math>\sqrt{3}</math>. To find this probability, we divide the pr
  6 KB (1,008 words) - 11:46, 24 December 2020
• 2008 AMC 12B Problems/Problem 25
  Note: Once <math>DY</math> is found, there is no need to do the trig. Notice that the hexagon consists of two trapezoids, <m
  12 KB (2,015 words) - 20:54, 9 October 2022
• 2008 AMC 12B Problems/Problem 22
  ...roblem, this means the number of distinct ways to order the cars such that no two spaces are adjacent is exactly the number of ways to pick 4 spots out o ...that 4 spaces must be left empty, and we care about the arrangements where no empty spaces are next to one another. Notice that this is simply the number
  4 KB (653 words) - 11:06, 15 October 2022
• 2008 AIME II Problems
  ...d is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math>. No point of circle <math>Q</math> lies outside of <math>\bigtriangleup\overlin ...ts using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when <math>
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 13
  ...</math> are now completely arbitrary because the original equations impose no more restrictions on them. Hence, for the final equation to hold for all po
  8 KB (1,218 words) - 00:07, 11 April 2024
• 2008 AIME II Problems/Problem 10
  ...)</math> or <math>(2,3)</math>. However, from <math>(0,3)</math>, there is no way to move <math>\sqrt{9}</math> away, so we discard it as a possibility.
  4 KB (569 words) - 09:44, 25 November 2019
• 2008 AIME II Problems/Problem 11
  ...d is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math>. No point of circle <math>Q</math> lies outside of <math>\triangle ABC</math>.
  6 KB (1,065 words) - 20:12, 9 August 2022
• 2008 AIME II Problems/Problem 12
  ...ts using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when <math> ...cond pole could start with a green, since the original string assumed that no greens could be consecutive. We solve this problem by introducing an extra
  10 KB (1,550 words) - 12:58, 15 July 2023
• Ultrafilter
  ...math>\mathcal{F}</math> is an ultrafilter on <math>X</math>, then there is no filter <math>\mathcal{F'}</math> on <math>X</math> such that <math>\mathcal ...rafilter <math>\mathcal{F}</math> is nontrivial if and only if it contains no [[finite]] element.
  9 KB (1,685 words) - 20:28, 13 October 2019
• 2005 USAMO Problems/Problem 2
  has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>. It suffices to show that there are no solutions to this system in the integers mod 19. We note that <math>152 =
  7 KB (1,053 words) - 10:38, 12 August 2015
• 1989 APMO Problems
  has no solutions in integers except <math>a=b=c=n=0</math>.
  2 KB (340 words) - 10:27, 11 April 2016
• 2005 USAMO Problems/Problem 1
  ...all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. No such circular arrangement exists for <math>n=pq</math>, where <math>p</math
  4 KB (650 words) - 13:40, 4 July 2013
• 1990 APMO Problems
  1. No one in a subset knows all the others in the subset,
  2 KB (364 words) - 23:07, 12 September 2016
• 1997 PMWC Problems/Problem T7
  ...traint. (If you think a cuboid can have more than 8 corners, then you have no business messing with this page. If you do, you are either very bad at math
  2 KB (283 words) - 21:47, 13 July 2023
• 0.999...
  ...limit definitions to resolve this problem. If you think about it, there is no <math>n</math> such that <math>a_n=1</math>. ...such a limit seems to be a real number. But how do we know that there are no 'gaps' so it could possibly not be a real number. But lets prove it.
  3 KB (577 words) - 20:04, 4 February 2023
• 2008 AMC 10B Problems/Problem 8
  ...e hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchase
  2 KB (401 words) - 20:32, 26 July 2022
• 2008 AMC 10B Problems/Problem 16
  ...hat is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)
  2 KB (357 words) - 22:58, 4 October 2023
• 2008 AMC 10B Problems/Problem 21
  ...rried couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seatin
  2 KB (262 words) - 22:37, 6 November 2021
• 2008 AMC 10B Problems/Problem 22
  ...blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? ...beads first. If these two colors are ordered first, we must make sure that no neighboring beads are the same color, or only one pair of neighboring beads
  4 KB (663 words) - 13:49, 7 June 2021
• 2008 USAMO Problems/Problem 1
  ...teger <math>m</math>, <math>P(mp_1p_2\cdots p_k)</math> is an integer with no prime factors, which must equal <math>1</math> or <math>-1</math>. However,
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 USAMO Problems/Problem 2
  ...e{ABC}</math>. Note that since <math>AP \perp PO</math> and <math>AN \perp NO</math>, then <math>A</math>, <math>N</math>, <math>O</math> and <math>P</ma ...means <math>NM \perp PO</math>. It follows analogously that <math>PM \perp NO</math>. This means that <math>M</math> is the orthocenter of triangle <math
  20 KB (3,565 words) - 11:54, 1 May 2024
• 2008 USAMO Problems/Problem 3
  ...f black dots (which occurs when both endpoints are red). Thus there can be no more than <math>m + l</math> red dots than black dots in <math>S_n</math>. ...jacent to each other, so both are adjacent to at least three points. Then, no two points adjacent to at least three others are travelled consecutively.
  9 KB (1,585 words) - 01:00, 14 August 2014
• 2008 USAMO Problems/Problem 6
  ...th>n\geq 2</math>. The case in which some participant, <math>P</math>, has no friends is trivial. In this case, <math>P</math> can eat in either of the t
  13 KB (2,414 words) - 14:37, 11 July 2016
• Free magma
  ...is as general as possible&mdash;a magma generated from an initial set with no constraints or relations.
  4 KB (887 words) - 13:19, 6 July 2016
• 2008 Mock ARML 2 Problems
  ...ohn removes blocks one at a time, removing only blocks that currently have no blocks on top of them. Find the number of ways (order matters) in which Joh
  4 KB (582 words) - 21:57, 8 May 2019
• Sylow Theorems
  ...P' \cap H</math> is a <math>p</math>-subgroup of <math>H</math>, it can be no larger than <math>P</math>. <math>\blacksquare</math>
  11 KB (2,071 words) - 12:25, 9 April 2019
• Relativity
  ...n achieve. Objects moving at ''c'' would then move to their destination in no time since time slows down to a stop.
  2 KB (226 words) - 19:19, 16 August 2019
• KGS math club
  ...possible, such that any integer value from 1 to 100 cents can be paid with no more than two coins. ...s always possible to color the vertices black and white in such a way that no vertex is connected to more than two vertices of the same color as itself;
  22 KB (3,358 words) - 15:17, 18 July 2017
• KGS math club/solution 2 1
  Answer: No.
  2 KB (393 words) - 12:54, 11 August 2023
• 1998 AHSME Problems
  ...ntersect when <math>k</math> satisfies <math>a \le k \le b</math>, and for no other values of <math>k</math>. Find <math>b-a</math>.
  15 KB (2,222 words) - 10:40, 11 August 2020
• 1995 AHSME Problems/Problem 13
  ...would be possible to change the <math>2</math> to either a <math>5</math> (no carry) or a <math>6</math> (carry) to create a correct statement.
  1 KB (223 words) - 13:59, 5 July 2013
• 2001 IMO Shortlist Problems/C1
  <math>n < 3</math>: Then there are no subsequences, because there are only two possible values for each element, Now we show no higher <math>m</math> is allowed. Let <math>A_i=\{k\in A\mid a_k\equiv i\pm
  4 KB (703 words) - 12:45, 27 November 2017
• 2001 IMO Shortlist Problems/C2
  ...h means there are <math>n!</math> distinct sums <math>S(a)</math>. Because no two of them are congruent modulo <math>n!</math>, we have that for some <ma
  2 KB (364 words) - 08:55, 31 August 2011
• 2001 IMO Shortlist Problems/C3
  ...ove that there are two or fewer people at the party whose departure leaves no 3-clique remaining. ...here exists only one 3-clique, remove anyone in that clique. (If there are no 3-cliques, we are done!) Otherwise, consider the following cases:
  3 KB (479 words) - 15:01, 23 November 2017
• 2001 IMO Shortlist Problems/C7
  ...s called a <i>final configuration</i>. For each <math>n</math>, show that, no matter what choices are made at each stage, the final configuration obtaine At no point during this process can there be three or more columns with the same
  3 KB (567 words) - 16:49, 5 January 2021
• 2001 IMO Shortlist Problems/N1
  Prove that there is no positive integer <math>n</math> such that, for <math>k = 1,2,\ldots,9</math ...igit can't be 4. This contradicts with how <math>n</math> is defined, thus no such value of <math>n</math> exists.
  3 KB (440 words) - 08:14, 1 April 2022
• 2001 IMO Shortlist Problems/N4
  ...ring all <math>a_0\geq\sqrt{p}</math> and taking their inverse. As long as no more than half of the values in that range can have the required property,
  3 KB (454 words) - 22:27, 16 October 2020
• Millennium Problems
  ...phic to the ''n'' sphere, or, more simply, that only a ''n'' manifold with no holes was simply connected. All cases of this conjecture had been proven up ...theory relating to whether decision problems (problems admitting a yes or no answer) whose solutions can be ''verified'' in polynomial time (as a functi
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1997 PMWC Problems/Problem T5
  'D': No, either A or B is telling the truth. 'E': No, 'D', that's not true.
  1 KB (263 words) - 22:58, 13 April 2013
• 1984 IMO Problems/Problem 4
  ...ath>, and with all conditions satisfied. We shall prove that there exists no point <math>T</math> on <math>CD</math> such that <math>T</math> is a midpo
  4 KB (750 words) - 23:49, 29 January 2021
• 1988 IMO Problems/Problem 4
  ...r_i</math> such that <math>f(r_i)=\frac{5}{4}</math>. There are obviously no other such <math>r_i</math> since <math>f(x)=\frac{5}{4}</math> yields a po
  3 KB (518 words) - 11:36, 30 January 2021
• 1990 AHSME Problems/Problem 29
  ...he integers from <math>34</math> to <math>100</math> is allowed as long as no integer between <math>11</math> and <math>33</math> inclusive is within the
  2 KB (285 words) - 19:25, 25 September 2020
• 2008 IMO Problems/Problem 5
  ...y, let <math>\cal{M}</math> denote the set of those sequences that contain no numbers from <math>B</math> and each of the numbers in <math>A</math> an od
  4 KB (677 words) - 01:10, 19 November 2023
• Integral domain
  * and it contains no [[zero divisor|zero divisors]] (i.e. there are no nonzero <math>x,y\in R</math> such that <math>xy = 0</math>).
  826 bytes (125 words) - 17:43, 16 March 2012
• 2000 USAMO Problems
  holds for all real numbers <math>x</math> and <math>y</math>. Prove that no very convex function exists.
  3 KB (495 words) - 19:02, 18 April 2014
• 2000 USAMO Problems/Problem 1
  holds for all real numbers <math>x</math> and <math>y</math>. Prove that no very convex function exists. ...on becomes vertical, which contradicts the definition of a function. Hence no very convex function exists.
  3 KB (523 words) - 00:45, 9 July 2017
• 2000 USAMO Problems/Problem 4
  ...r column, form a colored right triangle, giving us a contradiction. Hence, no filled row or column may exist. ...ach of the <math>1000</math> rows has <math>1</math> black square, leaving no place for the other <math>999</math>, contradiction. Hence <math>n = \boxed
  2 KB (382 words) - 13:37, 4 July 2013
• 2001 USAMO Problems/Problem 1
  ...th> colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justificati Again, no balls may appear <math>\ge 4</math> times, but by Pigeonhole, one ball must
  5 KB (841 words) - 17:19, 5 May 2022
• 2008 iTest Problems
  ...417</math> after six games, and the team is <math>6-0</math> (six wins and no losses). They are off to their best start in years. ...ntains the same number. How many jelly beans are in each bag? (Assume that no jelly bean gets put inside more than one bag.)
  71 KB (11,749 words) - 01:31, 2 November 2023
• Asymptote: Macintosh
  ...see "x.xx", replace it with the Asymptote version number (e.g. If the file name was asymptote-2.86.src.tgz, you would replace x.xx with 2.86). If you get an error at the./configure step, stating that there is no acceptable C compiler found in $PATH, a solution is to download and install
  5 KB (732 words) - 00:47, 13 December 2023
• 2007 iTest Problems/Problem 9
  ...etric mean. But by the [[AM-GM inequality]], this is impossible. Therefore no such pairs <math>(m, n)</math> exist, and the answer is <math>0\Rightarrow\
  966 bytes (152 words) - 20:24, 10 January 2019
• 1991 IMO Problems/Problem 4
  ...that vertex will be sorrounded by edges whose greates common factor is 1, no matter how the other edges sorrounding it are labelled. Notice that from o
  4 KB (668 words) - 17:45, 30 January 2021
• 1993 IMO Problems/Problem 4
  ...s clear that since <math>a</math> is the longest side of <math>ABC</math>, no two points within <math>ABC</math> have distance exceeding <math>a</math>.
  4 KB (854 words) - 11:30, 21 November 2023
• 2007 AMC 12B Problems/Problem 25
  Because there can be no movement in the <math>z</math> direction, the k unit vector must be zero. A
  3 KB (470 words) - 19:46, 17 July 2023
• 1973 USAMO Problems/Problem 4
  ...ath>t^3-3t^2+3t-1=(t-1)^3</math>. Thus <math>x=y=z=1</math>, and there are no other solutions.
  5 KB (888 words) - 08:18, 22 April 2024
• 1976 USAMO Problems/Problem 3
  ...we must do this over and over again, and from infinite descent, there are no non-zero solutions when <math>a^2\equiv 0\bmod{4}</math>. ...math>c^2\equiv 3\bmod{4}</math>, which is an impossibility. Thus there are no non-zero solutions when <math>a^2\equiv 1\bmod{4}</math>.
  1 KB (214 words) - 00:00, 10 October 2020
• 1979 USAMO Problems/Problem 1
  ...14 modulo 16. But <math>1599\equiv 15\bmod{16}</math>, and thus there are no integral solutions to the given Diophantine equation. ...th>. Since <math>E_{16}</math> is less than <math>F_{16}</math>, there are no integral solutions for this equation.
  1 KB (183 words) - 04:45, 21 January 2023
• 1976 USAMO Problems/Problem 1
  ...nd two black squares. From this observation it is immediately obvious that no two columns agree on two cells of the same color.
  4 KB (660 words) - 19:21, 23 November 2016
• 1973 USAMO Problems/Problem 2
  Prove that, except for the "1", there is no term which occurs in both sequences.
  2 KB (342 words) - 18:54, 3 July 2013
• 1972 USAMO Problems/Problem 3
  The probability that there is no 5 is <math>\left( \frac{8}{9}\right)^n</math>. The probability that there is no 2 is <math>\left( \frac{5}{9}\right)^n</math>.
  1 KB (177 words) - 19:11, 3 June 2023
• 2001 IMO Problems/Problem 5
  ...r to the image in Solution 1 just rename Point X as P and Point Y as Q And no need of construction
  4 KB (705 words) - 07:41, 26 February 2024
• 2001 IMO Problems/Problem 6
  ...ce <math>KL+MN>KM+LN</math> if <math>KL+MN</math> is prime, then there are no common factors between the two. So, in order to have <cmath>(KM+LN)\mid (K
  1 KB (273 words) - 00:23, 19 November 2023
• 1999 AHSME Problems/Problem 2
  ...le A+\angle B+\angle C=180</math>, <math>\angle A=60^{\circ}</math>. Since no other angles are above <math>180^{\circ}</math>, all equilateral triangles
  2 KB (259 words) - 14:28, 13 February 2019
• For The Win
  ...lving.com/community/c10h296585p1607539 Unfortunately, the old FTW forum is no longer accessible, which makes it harder to find old information on FTW.] It was likely created or co-created by the no longer active [https://artofproblemsolving.com/community/user/2 Valentin Vo
  2 KB (282 words) - 02:51, 24 June 2023
• Absolute zero
  ...able of cooling the temperature to a very close approximation to 0 Kelvin, no existing technology allows any object to arrive at absolute zero.
  753 bytes (108 words) - 15:35, 5 April 2024
• Displacement
  There are no names for higher derivatives or integrals of displacement.
  883 bytes (111 words) - 19:37, 6 March 2024
• 2002 USAMO Problems/Problem 6
  ...le the plane by tiles containing one block for every five stamps such that no more blocks can be chosen. Two such tilings are shown below with one tile o ...eing torn out, and it is impossible to use one stamp in the center and use no other stamps in the cross.) In addition, each block not lying along an edge
  5 KB (940 words) - 17:33, 16 July 2014
• 2002 USAMO Problems
  ...triangle <math>T </math> whose side lengths are all positive integers with no common divisor and determine those integers.
  3 KB (486 words) - 09:21, 14 May 2021
• 2001 USAMO Problems
  ...th> colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justificati
  3 KB (446 words) - 09:56, 20 July 2016
• 2002 AMC 10B Problems
  ...ath> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2002 AMC 12A Problems/Problem 7
  ...Then we realize that the ratio has to be with square numbers, as there are no square roots. Therefore, it is <math>(A)</math>
  3 KB (492 words) - 14:46, 31 January 2024
• 1973 USAMO Problems
  Prove that, except for the "1", there is no term which occurs in both sequences.
  2 KB (273 words) - 18:53, 3 July 2013
• 1974 USAMO Problems/Problem 4
  ...her and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: By the pigeonhole principle, there may be no more than 4 games; once the 4th game is finished, someone will have won two
  6 KB (1,037 words) - 23:42, 11 May 2018
• 1974 USAMO Problems
  ...her and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:
  3 KB (427 words) - 18:55, 3 July 2013
• 1963 IMO Problems/Problem 6
  ...s very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second We are given that no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so in order
  2 KB (249 words) - 14:28, 31 July 2016
• 1963 IMO Problems
  ...s very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second
  2 KB (342 words) - 21:16, 20 August 2020
• 1975 USAMO Problems/Problem 5
  ...ction. Our base case is naturally when <math>n = 3</math>, as there can be no less than <math>3</math> cards in the deck. The only way to turn up the sec
  4 KB (723 words) - 02:30, 30 December 2017
• 2000 AMC 10 Problems
  ...on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1998 AHSME Problems/Problem 23
  ...ntersect when <math>k</math> satisfies <math>a \le k \le b</math>, and for no other values of <math>k</math>. Find <math>b-a</math>.
  1 KB (193 words) - 09:12, 2 December 2018
• 1999 AHSME Problems/Problem 28
  To minimize <math>x_1^3 + \cdots + x_n^3</math>, there are no <math>2</math>s and maximize the number of <math>-1</math>s.
  4 KB (607 words) - 12:25, 3 October 2023
• 2000 AMC 10 Problems/Problem 13
  ...on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?
  2 KB (308 words) - 21:32, 27 July 2022
• 2008 AMC 12B Problems/Problem 20
  ...arbage truck meet. It is clear that they met at these times, and will meet no more. Thus the answer is <math>\boxed{\textbf{(B)}}</math>.
  8 KB (1,241 words) - 17:55, 11 August 2023
• 1985 AJHSME Problems/Problem 4
  Obviously, there are no formulas to find the area of such a messed up shape, but we do recognize so
  3 KB (490 words) - 10:52, 24 June 2023
• 1985 AJHSME Problems/Problem 7
  ...s one, and then shifts the new row over a little. Since the first row had no black squares, the number of black squares in any row is one less than the
  3 KB (584 words) - 09:23, 10 January 2023
• 1986 AJHSME Problems
  ...of the following statements about the whole number <math>(\text{o}^2+\text{no})</math> is always true?
  14 KB (2,054 words) - 15:41, 8 August 2020
• 1986 AJHSME Problems/Problem 7
  No... of course you're not supposed to know what the square root of 8 is, or t
  2 KB (256 words) - 21:11, 3 July 2013
• 1986 AJHSME Problems/Problem 14
  ...than <math>1200</math>, <math>b = 1200</math>. Since <math>a</math> can be no less than <math>200</math>, <math>a = 200</math>. <math>\frac{1200}{200} =
  789 bytes (124 words) - 21:17, 3 July 2013
• 1986 AJHSME Problems/Problem 17
  ...of the following statements about the whole number <math>(\text{o}^2+\text{no})</math> is always true? .../math> is odd. If <math>\text{n}</math> is odd, then obviously <math>\text{no}</math> will be odd as well, since <math>\text{o}</math> is odd, and the pr
  3 KB (457 words) - 15:02, 4 April 2021
• 1986 AJHSME Problems/Problem 21
  * <math>C</math> would cause the figure to not be foldable at all - NO * <math>G</math> is the same case as <math>C</math> - NO
  2 KB (346 words) - 14:18, 28 September 2020
• 2004 AMC 10B Problems/Problem 2
  ...The complement of having at least one <math>7</math> as a digit is having no <math>7</math>s as a digit.
  1 KB (208 words) - 19:10, 24 January 2015
• Distinguishability
  Case 3: No quarters
  6 KB (1,071 words) - 12:54, 2 March 2024
• 2002 AMC 10B Problems/Problem 12
  ...ath> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?
  1 KB (198 words) - 13:49, 6 June 2023
• 2002 AMC 10B Problems/Problem 13
  .../math>. For there to be infinite solutions for <math>y</math> and there is no <math>x</math>, we simply find a value of <math>x</math> such that the equa
  1 KB (218 words) - 13:12, 30 November 2019
• 2004 AMC 10B Problems/Problem 25
  ...ave a <math>\sqrt 3</math> and not a <math>\sqrt 2</math> in them (There's no <math>45^{\circ}</math> angle either). So, our answer is <math>\boxed{\math
  4 KB (703 words) - 22:37, 2 November 2022
• 2004 AMC 12B Problems/Problem 14
  ...uch greater? Well, if we let the intersection of <math>AB</math> and <math>NO</math> be <math>X</math>, we can approximate that <math>\triangle NXK</math
  7 KB (1,083 words) - 22:41, 23 November 2020
• 2009 AMC 10A Problems
  ...count as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <math>\textdollar 6.00
  14 KB (2,130 words) - 11:32, 7 November 2021
• 2009 AMC 12A Problems/Problem 24
  Now we <math>\log</math> it a third time... and we're stuck 😐. There is no pretty formula for taking the <math>\log</math> of two things added togethe ...imes we can keep taking the <math>\log</math> of this expression, until we no longer can (once you reach a negative value, you can't take the log of it a
  10 KB (1,595 words) - 22:13, 20 September 2021
• 1971 IMO Problems/Problem 6
  ...s of columns - let the size of this subset be <math>k</math>. (If there is no such subset, <math>S\ge n^2</math>.)
  6 KB (1,192 words) - 14:14, 29 January 2021
• 2009 AMC 12A Problems/Problem 17
  ...s that the series are different, <math>r_1 \ne r_2</math>, and so there is no division by zero error.
  3 KB (453 words) - 10:22, 6 October 2023
• 2009 AMC 12A Problems/Problem 12
  ...to <math>94a + 4b - 5c = 0</math>. Clearly for <math>a>0</math> there are no solutions, hence <math>a=0</math> and we get the equation <math>4b=5c</math
  3 KB (481 words) - 20:06, 17 December 2017
• 2009 AMC 10A Problems/Problem 8
  ...count as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <math>\$6.00</math>, i
  1 KB (186 words) - 16:53, 28 July 2023
• 2009 AMC 10A Problems/Problem 24
  ...chosen for vertex A, so vertex C can be placed anywhere and the plane will no matter what be in the cube. Therefore, the probability of choosing a valid
  5 KB (795 words) - 20:25, 31 October 2022
• 2009 AMC 10A Problems/Problem 22
  Let second case(No Two Same Numbers on One Die) = <math>B</math>
  4 KB (721 words) - 13:48, 1 November 2023
• 2009 AMC 10A Problems/Problem 18
  * <math>36\%</math>: swimming no, soccer yes * <math>6\%</math>: swimming yes, soccer no
  3 KB (402 words) - 10:29, 2 August 2021

View (previous 500 | next 500) (20 | 50 | 100 | 250 | 500)