Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Search results

Create the page "Problem 6" on this wiki! See also the search results found.

Page title matches

Page text matches

  • Pythagorean Theorem
    <cmath>6-8-10 = (3-4-5)*2</cmath> * [[2006_AIME_I_Problems/Problem_1 | 2006 AIME I Problem 1]]
    5 KB (886 words) - 21:12, 22 January 2024
  • 2005 AMC 12A Problems/Problem 16
    == Problem == <math> (\mathrm {A}) \ 5 \qquad (\mathrm {B}) \ 6 \qquad (\mathrm {C})\ 8 \qquad (\mathrm {D}) \ 9 \qquad (\mathrm {E})\ 10 <
    2 KB (307 words) - 15:30, 30 March 2024
  • 2016 AMC 10A Problems/Problem 5
    == Problem == {{AMC10 box|year=2016|ab=A|num-b=4|num-a=6}}
    1 KB (184 words) - 13:58, 22 August 2023
  • 2016 AMC 10A Problems/Problem 7
    == Problem == ...using the mean in your reasoning, you can just take the mean of the other 6 numbers, and it'll solve it marginally faster. -[[User:Integralarefun|Integ
    2 KB (268 words) - 18:19, 27 September 2023
  • 2016 AMC 10A Problems/Problem 9
    == Problem == <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf
    2 KB (315 words) - 15:34, 18 June 2022
  • 2016 AMC 10A Problems/Problem 10
    == Problem == filldraw(rectangle((1,1),(6,4)),gray(0.75));
    2 KB (337 words) - 14:56, 25 June 2023
  • 2016 AMC 10A Problems/Problem 11
    == Problem == ...uad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8</math>
    8 KB (1,016 words) - 00:17, 31 December 2023
  • List of United States elementary school mathematics competitions
    * [[Noetic Learning Math Contest]] - semiannual problem solving contest for elementary and middle school students. [Grades 2-8] ...ills through [https://www.beestar.org/exercise/samples.jsp?adid=105 Weekly Problem Solving Contests] and [https://www.beestar.org/competition/?adid=105 Beesta
    4 KB (473 words) - 16:14, 1 May 2024
  • List of United States middle school mathematics competitions
    * [https://www.hardestmathproblem.org Hardest Math Problem] math contest for grades 5-8 with great prizes. * [[Noetic Learning Math Contest]]: a popular problem-solving contest for students in grades 2-8.
    7 KB (792 words) - 10:14, 23 April 2024
  • MathILy (serious mathematics infused with levity)
    Class meets for about 7 hours per day, in two shifts (morning and evening), 6 days per week. Each class has a Lead Instructor who is a mathematician with ...ctures and providing proofs. Classes include independent and collaborative problem solving as well as lots of laughter; in this way, students learn creative a
    5 KB (706 words) - 23:49, 29 January 2024
  • MATHCOUNTS
    ...cluding Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete be ...ics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.
    10 KB (1,497 words) - 11:42, 10 March 2024
  • ICTM Math Contest
    ...thematics offers two areas of math contests: Grade School (Grades 3, 4, 5, 6, 7, 8 + Algebra 1) and High School (Regional and State Finals). *2024 State FInals - Saturday 4/6/24
    8 KB (1,182 words) - 14:26, 3 April 2024
  • Math textbooks
    ...ks''' page is for compiling a list of [[textbook]]s for mathematics -- not problem books, contest books, or general interest books. See [[math books]] for mo * Getting Started is recommended for students grades 6 to 9.
    7 KB (901 words) - 14:11, 6 January 2022
  • William Lowell Putnam Mathematical Competition
    ...Problem A/B, 1/2</u>: 7<br><u>Problem A/B, 3/4</u>: 8<br><u>Problem A/B, 5/6</u>: 9}} ...chool olympiads are, although they include more advanced mathematics. Each problem is graded on a scale of 0 to 10. The top five scorers (or more if there are
    4 KB (623 words) - 13:11, 20 February 2024
  • Computer science books
    These '''Computer Science books''' are recommended by [[Art of Problem Solving]] administrators and members of the [http://www.artofproblemsolving .../ref_list_smcs.jsp?&mid=1500&div=9 Computer Science Reading for Grade 3-5, 6-8]
    2 KB (251 words) - 00:45, 17 November 2023
  • 2005 Alabama ARML TST Problems/Problem 7
    ==Problem== =\frac{2}{2}+\frac{4}{4}+\frac{6}{8}+\frac{8}{16}+\cdots\\
    1 KB (193 words) - 21:13, 18 May 2021
  • Inequality
    ...parentheses. For instance, <math>x \in [3,6)</math> means <math>3 \le x < 6</math>. The problem here is that we multiplied by <math>x+5</math> as one of the last steps. W
    12 KB (1,798 words) - 16:20, 14 March 2023
  • United States of America Mathematical Talent Search
    The USAMTS is administered by the [[Art of Problem Solving Foundation]] with support and sponsorship by the [[National Securit ...|difficulty=3-6|breakdown=<u>Problem 1-2</u>: 3-4<br><u>Problem 3-5</u>: 5-6}}
    4 KB (613 words) - 13:08, 18 July 2023
  • AMC 10
    ...administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC. ...iculty=1-3|breakdown=<u>Problem 1-5</u>: 1<br><u>Problem 6-20</u>: 2<br><u>Problem 21-25</u>: 3}}
    4 KB (574 words) - 15:28, 22 February 2024
  • AMC 12
    ...administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! ...ulty=2-4|breakdown=<u>Problem 1-10</u>: 2<br><u>Problem 11-20</u>: 3<br><u>Problem 21-25</u>: 4}}
    4 KB (520 words) - 12:11, 13 March 2024
  • American Invitational Mathematics Examination
    ...ministered by the [[Mathematical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! ...<u>Problem 6-10</u>: 4<br><u>Problem 10-12</u>: 5<br><u>Problem 12-15</u>: 6}}
    8 KB (1,057 words) - 12:02, 25 February 2024
  • 1954 AHSME Problems/Problem 46
    == Problem 46== draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2));
    3 KB (415 words) - 18:01, 24 May 2020
  • 2015 IMO Problems
    ==Problem 1== [[2015 IMO Problems/Problem 1|Solution]]
    4 KB (692 words) - 22:33, 15 February 2021
  • United States of America Mathematical Olympiad
    ...administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC and of the recent expansion ...9|breakdown=<u>Problem 1/4</u>: 7<br><u>Problem 2/5</u>: 8<br><u>Problem 3/6</u>: 9}}
    6 KB (869 words) - 12:52, 20 February 2024
  • American Regions Mathematics League
    {{WotWAnnounce|week=June 6-12}} The '''American Regions Math League''' (ARML) is a [[mathematical problem solving]] competition primarily for U.S. high school students.
    2 KB (267 words) - 17:06, 7 March 2020
  • Joining an ARML team
    ...tices held during the Spring Semester to determine the team each year. The 6 practices include 3 individual tests to help determine the team and some le ...[[MAML]] (Maine Association of Math Leagues) Meets. Training includes the problem set "Pete's Fabulous 42."
    21 KB (3,500 words) - 18:41, 23 April 2024
  • Simon's Favorite Factoring Trick
    ...This yields <math>x(y+5)+6(y+5)=60</math>. Now, we can factor as <math>(x+6)(y+5)=60</math>. ...ecause this keeps showing up in number theory problems. Let's look at this problem below:
    7 KB (1,107 words) - 07:35, 26 March 2024
  • Mathematical problem solving
    ...oblem solving]] involves using all the tools at one's disposal to attack a problem in a new way. <math>\frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots</math><br>
    2 KB (314 words) - 06:45, 1 May 2014
  • Geometric mean
    The geometric mean of the numbers 6, 4, 1 and 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>. * [[1966 AHSME Problems/Problem 3]]
    2 KB (282 words) - 22:04, 11 July 2008
  • Principle of Inclusion-Exclusion
    ...d only once. In particular, memorizing a formula for PIE is a bad idea for problem solving. ==== Problem ====
    9 KB (1,703 words) - 07:25, 24 March 2024
  • Factorial
    * <math>3! = 6</math> * <math>6! = 720</math>
    10 KB (809 words) - 16:40, 17 March 2024
  • 2006 AIME II
    ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[2006 AIME II Problems/Problem 1]]
    1 KB (133 words) - 12:32, 22 March 2011
  • Fermat's Little Theorem
    int n = 6; ...),red+linewidth(2));match(2,3,1); </asy>For <math>p=2,3</math> and <math>a=6,4</math>, respectively.</center>
    16 KB (2,658 words) - 16:02, 8 May 2024
  • Parabola
    f.p=fontsize(6); f.p=fontsize(6);
    3 KB (551 words) - 16:22, 13 September 2023
  • Euler's Totient Theorem
    ...ast two digits of <math> 7^{81}-3^{81} </math>. ([[Euler's Totient Theorem Problem 1 Solution|Solution]]) ...ast two digits of <math> 3^{3^{3^{3}}} </math>. ([[Euler's Totient Theorem Problem 2 Solution|Solution]])
    3 KB (542 words) - 17:45, 21 March 2023
  • Number theory/Advanced
    ...uited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]]. ...es <math>G_4</math> and <math>G_6</math> are modular forms of weight 4 and 6 respectively.
    5 KB (849 words) - 16:14, 18 May 2021
  • Harmonic mean
    ...instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we would be trying to calculate <math>\frac 3{\frac 13 + \frac 16 * [[2002 AMC 12A Problems/Problem 11]]
    1 KB (196 words) - 00:49, 6 January 2021
  • Divisibility rules
    ...if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. ...s at the end of it to be divisible by 1,000,000 because <math>1,000,000=10^6</math>.
    8 KB (1,315 words) - 18:18, 2 March 2024
  • Induction
    Other, odder inductions are possible. If a problem asks you to prove something for all integers greater than 3, you can use <m ...ernational Mathematics Olympiad | IMO]]. A good example of an upper-level problem that can be solved with induction is [http://www.artofproblemsolving.com/Fo
    5 KB (768 words) - 20:45, 1 September 2022
  • Overcounting
    An example of a classic problem is as follows: ...hem twice. A number that is divisible by both 2 and 3 must be divisible by 6, and there are 16 such numbers. Thus, there are <math>50+33-16=\boxed{67}</
    4 KB (635 words) - 12:19, 2 January 2022
  • Constructive counting
    This is a problem where constructive counting is not the simplest way to proceed. This next e ...wer is <math>8 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 8 \cdot 7^6 = 941,192</math>, as desired. <math>\square</math>
    12 KB (1,896 words) - 23:55, 27 December 2023
  • Euclidean algorithm
    Similarly, <math>42 \equiv 6 \pmod{9}</math>, so <math>\gcd(42,9) = \gcd(9,6)</math>. <br/> Continuing, <math>9 \equiv 3 \pmod{6}</math>, so <math>\gcd(9,6) = \gcd(6,3)</math>. <br/>
    6 KB (924 words) - 21:50, 8 May 2022
  • Generating function
    ...multiply the functions together, getting <math>1+3x+6x^2+8x^3+8x^4+6x^5+3x^6+x^7</math>. We want the number of ways to choose 4 eggs, so we just need to
    4 KB (659 words) - 12:54, 7 March 2022
  • Binomial Theorem
    ...th>\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \dots</cmath> for all <math>x</math>. ...[[polynomial]]s with [[binomial coefficient]]s. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>, one could factor
    5 KB (935 words) - 13:11, 20 February 2024
  • Function
    ...function: <math>f(x) = x^2 + 6</math>. The function <math>g(x) = \sqrt{x-6}</math> has the property that <math>f(g(x)) = x</math>. Therefore, <math>g ([[2006 AMC 10A Problems/Problem 2|Source]])
    10 KB (1,761 words) - 03:16, 12 May 2023
  • Complementary counting
    ...ng may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ''[[2006 AMC 10A Problems/Problem 21 | 2006 AMC 10A Problem 21]]: How many four-digit positive integers have at least one digit that is
    8 KB (1,192 words) - 17:20, 16 June 2023
  • Bijection
    [[2008 AMC 12B Problems/Problem 22]] [[2001 AIME I Problems/Problem 6]]
    1,016 bytes (141 words) - 03:39, 29 November 2021
  • Combinatorial identity
    .../www.artofproblemsolving.com/Forum/viewtopic.php?p=394407#394407 1986 AIME Problem 11] ...lving.com/Forum/resources.php?c=182&cid=45&year=2000&p=385891 2000 AIME II Problem 7]
    12 KB (1,993 words) - 23:49, 19 April 2024
  • Circle
    ...- 3)^2 + (y + 6)^2 = 25</math> represents the circle with center <math>(3,-6)</math> and radius 5 units. ([[2006 AMC 12A Problems/Problem 16|Source]])
    9 KB (1,581 words) - 18:59, 9 May 2024
  • Base numbers
    ...umber can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six <math>10^0</math>'s, the second d
    4 KB (547 words) - 17:23, 30 December 2020

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php/Special:Search"