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• 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>.
  4 KB (597 words) - 18:39, 9 May 2024
• 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> NO SOLUTION HERE! PLEASE TRY AGAIN NEXT TIME
  30 KB (4,794 words) - 23:00, 8 May 2024
• 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
• 1990 USAMO Problems/Problem 1
  ...re, in the set <math> \{0, 1, \ldots, 9\}^6 </math> (hypercube), there are no collinear points (parallel to the coordinate axes).
  4 KB (565 words) - 02:21, 25 May 2024
• 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) - 03:05, 19 May 2024
• 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 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
  5 KB (806 words) - 17:13, 18 May 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

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