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Power of a Point Theorem/Introductory Problem 1

448 bytes (67 words) - 15:15, 23 March 2020
If I have four boxes arranged in a $2 \times 2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes, using each digit exactly once, such that each box contains at most one digit? (I only have one of each digit, so one box
...he third, and one for the last. Thus, there are <math>4\cdot 3\cdot 2\cdot 1 = \boxed{24}</math> distinct ways to fill the boxes.

328 bytes (60 words) - 17:19, 12 June 2022
The perfect squares from $1$ through $2500,$ inclusive, are printed in a sequence of digits $1491625\ldots2500.$ How many digits are in the sequence?
...<math>3</math> perfect squares that only have <math>1</math> digit, <math>1^{2},</math> <math>2^{2},</math> and <math>3^{2}.</math> So we have a total of <math>1\times3+2\times6+3\times22+4\times19=\boxed{157}</math> digits.

939 bytes (140 words) - 17:27, 12 June 2022
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?
The table shows the percent of families in Mathville that have <math>0, 1, 2, 3</math> and <math>4</math> or more children. If there are a total of <

214 bytes (35 words) - 09:55, 1 August 2022
In triangle ABC, D be a point in BC, ∠BAD = 30 , ∠CAD = 90 , BD=1=AC, DC =

0 bytes (0 words) - 13:26, 6 February 2024

Page text matches

Pythagorean Theorem
=== Proof 1 === <math>(a+b)^2=c^2+4\left(\frac{1}{2}ab\right)\implies a^2+2ab+b^2=c^2+2ab\implies a^2 + b^2=c^2</math>. {{Ha

5 KB (886 words) - 13:51, 15 May 2024
2005 AMC 12A Problems/Problem 16
pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3); draw(Circle(O1,1));

2 KB (307 words) - 15:30, 30 March 2024
2016 AMC 10A Problems/Problem 2
<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textb ==Solution 1==

1 KB (190 words) - 10:58, 16 June 2023
2016 AMC 10A Problems/Problem 3
==Solution 1==

1 KB (176 words) - 10:58, 16 June 2023
2016 AMC 10A Problems/Problem 4
<math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf ==Solution 1==

2 KB (257 words) - 10:57, 16 June 2023
2016 AMC 10A Problems/Problem 5
A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box? == Solution 1==

1 KB (184 words) - 13:58, 22 August 2023
2016 AMC 10A Problems/Problem 6
..., replacing each occurrence of the digit <math>2</math> by the digit <math>1</math>. Star adds her numbers and Emilio adds his numbers. How much larger ...git and 2 appears 3 times as a units digit, the answer is <math>10\cdot 10+1\cdot 3=\boxed{\textbf{(D) }103.}</math>

967 bytes (143 words) - 03:18, 27 June 2023
2016 AMC 10A Problems/Problem 7
== Solution 1 ==

2 KB (268 words) - 18:19, 27 September 2023
2016 AMC 10A Problems/Problem 8
...eft(\frac{1}{2}\right)=30 , \Rightarrow (+40)=70 , \Rightarrow \left(\frac{1}{2}\right)=\boxed{\textbf{(C) }35}</cmath>

1 KB (169 words) - 14:59, 8 August 2021
2016 AMC 10A Problems/Problem 9
A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math == Solution 1==

2 KB (315 words) - 15:34, 18 June 2022
2016 AMC 10A Problems/Problem 10
...er rectangle is one foot wide, and each of the two shaded regions is <math>1</math> foot wide on all four sides. What is the length in feet of the inner filldraw(rectangle((1,1),(6,4)),gray(0.75));

2 KB (337 words) - 14:56, 25 June 2023
2016 AMC 10A Problems/Problem 11
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8)); label("$1$",(1/2,5),dir(90));

8 KB (1,016 words) - 00:17, 31 December 2023
2016 AMC 10A Problems/Problem 12
Three distinct integers are selected at random between <math>1</math> and <math>2016</math>, inclusive. Which of the following is a correc ...\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}</math>

2 KB (297 words) - 14:54, 25 June 2023
2016 AMC 10A Problems/Problem 13
...n a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "ri <math>\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad

2 KB (402 words) - 14:54, 25 June 2023
List of United States middle school mathematics competitions
...tp://euclidlab.org/programs/archimedean-challenge/1 Archimedean Challenge #1] * [http://www.mindresearch.org/gameathon/ Game-a-thon] challenges students to design and build math games in this national co

7 KB (792 words) - 10:14, 23 April 2024
MathILy (serious mathematics infused with levity)
...duate or undergraduate mathematics students. The weeks break down into a 2-1-2 schedule: We start with two weeks of Root Class, which consists of a gall

5 KB (706 words) - 23:49, 29 January 2024
MATHCOUNTS
...rint: 1-1.5 (School/Chapter), 2-2.5 (State/National) Target: 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}

10 KB (1,506 words) - 21:31, 14 May 2024
Georgia mathematics competitions
* [[Archimedean Challenge #1]] * [[Archimedean Challenge #1]]

4 KB (565 words) - 13:24, 13 September 2019
ICTM Math Contest
...wo areas of math contests: Grade School (Grades 3, 4, 5, 6, 7, 8 + Algebra 1) and High School (Regional and State Finals). The contest is given in-school May 1-15. Each school sets its own testing date and time within these guidelines.

8 KB (1,182 words) - 14:26, 3 April 2024
Oklahoma mathematics competitions
*[[Math Kangaroo]] is available to students in grades 1-12. The competition occurs annually during March held at ORU. Students do

2 KB (279 words) - 19:41, 23 March 2017
Art of Problem Solving
...artofproblemsolving.com/store/item/aops-vol1 Art of Problem Solving Volume 1: the Basics]

5 KB (667 words) - 17:09, 3 July 2023
Resources for mathematics competitions
* [https://artofproblemsolving.com/news/articles/establishing-a-positive-culture Establishing a Positive Culture of Expectation in Math Educ * [https://artofproblemsolving.com/news/articles/how-to-write-a-solution How to Write a Math Solution] by [[Richard Rusczyk]] and [[user:MCr

16 KB (2,152 words) - 21:46, 6 May 2024
Math books
...zon.com/gp/product/B09PMLFHX2/ref=ox_sc_act_title_1?smid=ATVPDKIKX0DER&psc=1 Getting Started with Competition Math], a textbook meant for true beginners ...itu-Andreescu/dp/0817643265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]

24 KB (3,177 words) - 12:53, 20 February 2024
Richard Rusczyk
...fproblemsolving.com/store/item/aops-vol1 the Art of Problem Solving Volume 1: the Basics]''

2 KB (254 words) - 09:04, 25 January 2019
Math Olympiads for Elementary and Middle Schools
...Response|difficulty=1|breakdown=Division E: 1 Division M: 1}}

2 KB (215 words) - 02:54, 18 November 2020
William Lowell Putnam Mathematical Competition
...me=Putnam|region=USA|type=Proof|difficulty=7 - 9|breakdown=Problem A/B, 1/2: 7 Problem A/B, 3/4: 8 Problem A/B, 5/6: 9}} ...if the top three students tie, they are all awarded a rank of <math>\frac{1 + 2 + 3}{3}=2</math>.) Before 2019, schools were required to choose their t

4 KB (623 words) - 13:11, 20 February 2024
Physics competitions
Khan Academy: AP Physics 1/2

4 KB (506 words) - 11:46, 6 September 2023
Computer science books
...s/ASIN/0201485419/artofproblems-20 The Art of Computer Programming Volumes 1-3 (boxed set)] by [[D. E. Knuth]]

2 KB (251 words) - 00:45, 17 November 2023
Physics
...Thermodynamics states that the efficiency of heat engines must always be < 1.

9 KB (1,355 words) - 07:29, 29 September 2021
2005 Alabama ARML TST Problems/Problem 7
...c{n^2}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{2n}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)</math> \sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)=x\\

1 KB (193 words) - 21:13, 18 May 2021
Mathematics
==Overview=={{asy image|<math>1\,2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up 

6 KB (902 words) - 12:53, 3 September 2019
AMC historical results
*Distinguished Honor Roll: Awarded to top 1% of scorers on each AMC 8, 10 and 12 respectively. *AIME qualifiers: 960 (1.5%)

17 KB (1,921 words) - 20:53, 10 May 2024
Inequality
...>b</math> if <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive. ...<b</math> if <math>a</math> is smaller than <math>b</math>, that is, <math>a-b</math> is negative.

12 KB (1,798 words) - 16:20, 14 March 2023
United States of America Mathematical Talent Search
...Info|name=USAMTS|region=USA|type=Proof|difficulty=3-6|breakdown=Problem 1-2: 3-4 Problem 3-5: 5-6}}

4 KB (613 words) - 13:08, 18 July 2023
AMC 8
...ty=1 - 1.5|breakdown=Problems 1 - 12: 1 Problems 13 - 25: 1.5}} The AMC 8 is usually administered on the third week of January. There is a 1-week window for students to take the test.

4 KB (558 words) - 22:25, 28 April 2024
AMC 10
...gion=USA|type=Multiple Choice|difficulty=1-3|breakdown=Problem 1-5: 1 Problem 6-20: 2 Problem 21-25: 3}} ...incorrect questions are worth 0 points, and unanswered questions are worth 1.5 points, to give a total score out of 150 points. From 2002 to 2006, unan

4 KB (574 words) - 15:28, 22 February 2024
AMC 12
...AMC 12|region=USA|type=Multiple Choice|difficulty=2-4|breakdown=Problem 1-10: 2 Problem 11-20: 3 Problem 21-25: 4}} ..., incorrect answers are worth 0 points, and unanswered questions are worth 1.5 points, to give a total score out of 150 points. From 2002 to 2006, the

4 KB (520 words) - 12:11, 13 March 2024
1955 AHSME Problems/Problem 38
...c}{3}+d+16=a+b+c+d</math>. This, with some algebra, means that <math>\frac{1}{3}(a+b+c)=8</math>. <math>d</math> must be <math>\boxed{\textbf{(B)} 21}</

1 KB (200 words) - 23:35, 28 August 2020
American Invitational Mathematics Examination
...ame=AIME|region=USA|type=Free Response|difficulty=3-6|breakdown=Problem 1-5: 3 Problem 6-10: 4 Problem 10-12: 5 Problem The AIME is a 15 question, 3 hour exam<math>^1</math> taken by high scorers on the [[AMC 10]], [[AMC 12]], and [[USAMTS]]

8 KB (1,057 words) - 12:02, 25 February 2024
1954 AHSME Problems/Problem 46
draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); label("$\frac{1}{2}$",(.5,.25));

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
...Info|name=USAMO|region=USA|type=Proof|difficulty=7-9|breakdown=Problem 1/4: 7 Problem 2/5: 8 Problem 3/6: 9}}

6 KB (869 words) - 12:52, 20 February 2024
American Regions Mathematics League
...dividual: 4 (Problem 6/8), 6 (Problem 10) Team: 3.5 (Problem 1-5), 5 (Problem 6-10)}}

2 KB (267 words) - 17:06, 7 March 2020
Joining an ARML team
...The three teams are the Reals, Naturals, and Primes. Two compete in A and 1 in B... Michigan was also awarded the best shirt design at ARML Iowa 2015.. ...ille, etc.) on the MO ARML team. (eg. the 2008 team fielded 6 from KS, and 1 from IL)

21 KB (3,500 words) - 18:41, 23 April 2024
Chemistry scholarships
....htm American Electroplaters and Surface Finishers Society Scholarship] of 1,500 dollars for undergraduate and graduate students in chemistry and some t

1 KB (182 words) - 22:00, 4 February 2017
Engineering scholarships
...ican Electroplaters and Surface Finishers Society Scholarship] of <dollar/>1,500 for undergraduate and graduate students in chemistry and some types of ....com/2020-usa-cargo-trailer-scholarship USA Cargo Trailer Scholarship] of $1,500 for students pursuing engineering.

4 KB (511 words) - 14:57, 16 July 2020
Princeton University Mathematics Competition
...participate in the Finals round, and they have to solve three problems in 1 hour.

2 KB (295 words) - 23:19, 5 January 2019
Mu Alpha Theta
*[[Algebra]] 1 *-1 for each incorrect question

4 KB (632 words) - 17:09, 11 October 2020
AoPS Online School
* [[AoPS Online School/Prealgebra 1 | Prealgebra 1]] — [https://artofproblemsolving.com/school/course/catalog/prealgebra1 De .../Introduction to Algebra A | Introduction to Algebra A]] (formerly Algebra 1) — [https://artofproblemsolving.com/school/course/catalog/algebra-a Detai

8 KB (965 words) - 03:41, 17 September 2020
Power Mean Inequality
...th> positive real weights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in\mathb \prod_{i=1}^n a_i^{w_i} &\text{if } t=0 \\

3 KB (606 words) - 23:59, 1 July 2022
Wilson's Theorem
...rem''' states that if [[integer ]]<math>p > 1</math> , then <math>(p-1)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prim ...p-1)! + 1</math>. Therefore <math>p</math> does not divide <math>(p-1)! + 1</math>.

4 KB (639 words) - 01:53, 2 February 2023
Trivial Inequality
...lutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>. *Show that <math>\sum_{k=1}^{n}a_k^2 \geq a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1</math>. [[Inequality_Introductory_Problem_2|Solution]]

3 KB (560 words) - 22:51, 13 January 2024
Arithmetic mean
For example, if I wanted to find the average of the numbers 3, 1, 4, 1, and 5, I would compute: <center><math> \frac{3+1+4+1+5}{5} = \frac{14}{5}.</math></center>

699 bytes (110 words) - 12:44, 20 September 2015
Completing the square
What is the value of <math>x</math> if <math>x=1+\dfrac{1}{x}</math>

2 KB (422 words) - 16:20, 5 March 2023
Heron's Formula
<cmath>[ABC]=\frac{ab}{2}\sqrt{1-\cos^2 C}.</cmath> <math>[ABC]=\frac{ab}{2}\sqrt{1-\left(\frac{a^2+b^2-c^2}{2ab}\right)^2}.</math>

4 KB (675 words) - 00:05, 22 January 2024
Polynomial
* <math>x^{-1}+2+3x+x^2</math> * <math>x^{1/3}=\sqrt[3]{x}</math>

6 KB (1,100 words) - 01:44, 17 January 2024
Simon's Favorite Factoring Trick
...h> are integer constants, and the coefficient of xy must be 1(If it is not 1, then divide the coefficient off of the equation.). According to Simon's Fa ...y)</math> that are solutions to the equation <math>\frac{4}{x}+\frac{5}{y}=1</math>. (2021 CEMC Galois #4b)

7 KB (1,107 words) - 07:35, 26 March 2024
University of South Carolina High School Math Contest
The test is scored as 5 points for every correct response, 1 point for a blank response, and 0 points for an incorrect response.

972 bytes (141 words) - 11:12, 30 September 2018
Factoring
<cmath>a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})</cmath> ...this creates the difference of squares factorization, <cmath>a^2-b^2=(a+b)(a-b)</cmath>

3 KB (532 words) - 22:00, 13 January 2024
Proof by contradiction
...e|primes]] <math>p_1, p_2,\ldots, p_n</math>. Let <math>N=p_1p_2p_3...p_n+1</math>. <math>N</math> is not divisible by any of the known primes since i

2 KB (374 words) - 14:01, 21 August 2022
Mathematical problem solving
...e sum of the [[series]] <math>\frac11 + \frac14 + \frac19 + \cdots + \frac{1}{n^2} + \cdots</math> ...{\pi}\right)\left(1+\frac{x}{\pi}\right)\left(1-\frac{x}{2\pi}\right)\left(1+\frac{x}{2\pi}\right)\cdots</math> 

2 KB (314 words) - 06:45, 1 May 2014
Pigeonhole Principle
In [[combinatorics]], the '''pigeonhole principle''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one hole must ...ontain at most one ball implies that <math>b_r \leq 1</math> for all <math>1 \leq r \leq n</math>, so <cmath>b_1 + b_1 + \cdots + b_n \geq n.</cmath> Ho

11 KB (1,985 words) - 21:03, 5 August 2023
Quadratic equation
Next, we factor out our common terms to get <math>x(x-1)-2(x-1)=0</math>. ...<math>(x-1)(x-2)=0</math>. By the zero-product property, either <math> (x-1) </math> or <math> (x-2) </math> equals zero.

2 KB (264 words) - 12:04, 15 July 2021
Relatively prime
...r than 1. That is, their [[greatest common divisor]] is <math>\gcd(m, n) = 1</math>. Equivalently, <math>m</math> and <math>n</math> must have no [[pri ...r difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</math>.

2 KB (245 words) - 15:51, 25 February 2020
1965 AHSME Problems/Problem 40
\textbf{(D) }\ 1 \qquad We can then try <math>(x^2+3x+1)=x^4+6x^3+11x^2+6x+1</math>.

3 KB (571 words) - 00:42, 22 October 2021
Trigonometric substitution
...>\int \sqrt{a^2+x^2}\,dx</math>, we make use of the identity <math>\tan^2x+1=\sec^2x</math>. Set <math>x=a\tan\theta</math> and the radical will go awa Making use of the identity <math>\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>.

1 KB (173 words) - 18:42, 30 May 2021
Geometric mean
...eometric 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>. MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W);

2 KB (282 words) - 22:04, 11 July 2008
Principle of Inclusion-Exclusion
...gain, <math>|A\cap C|</math> would be putting five guys in order, so <math>1!\binom{6}{5}=6</math>. <math>|A\cap D|</math> is just choosing <math>3</mat ...>, <math>|A\cap C\cap D|</math> is again ordering everybody which is <math>1</math>, and <math>|B\cap C\cap D|</math> is the same as <math>|A\cap B\cap

9 KB (1,703 words) - 07:25, 24 March 2024
Googol
Googol is a huge number. It has a 1 followed by 100 zeroes or <math>10^{100}.</math>

84 bytes (14 words) - 10:46, 25 October 2020
Combination
* <math>\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r}</math> * <math>\binom{n}{r}=\frac{n}{r}\binom{n-1}{r-1}</math>

4 KB (615 words) - 11:43, 21 May 2021
2020 CIME II Problems/Problem 10
...row (a-1)(b-1)=2</math> from whence we have <math>(a,b,c)\in\{(2,3,1),(3,2,1)\}</math>. ...b\geq 6c</math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Rightarrow c=1</math>; a contradiction since <math>c\geq 2</math>.

2 KB (332 words) - 09:37, 30 December 2021
Cauchy-Schwarz Inequality
...a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with th ...a^2b^2</math>. The inequality then follows from <math> |\cos\theta | \le 1 </math>, with equality when one of <math> \mathbf{a,b} </math> is a multipl

13 KB (2,048 words) - 15:28, 22 February 2024
Factorial
...[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. * <math>0! = 1</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))

10 KB (809 words) - 16:40, 17 March 2024
Discriminant
...ng about the roots of a given [[polynomial]] <math>p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of degree <math>n</math> with all the coefficients being re ...integer such that <math>k\geq\frac{n-2}{4}</math>, then there are <math>2k+1</math> pairs of complex conjugate roots and <math>n-4k+2</math> real roots.

4 KB (734 words) - 19:19, 10 October 2023
Menelaus' theorem
<cmath>\frac{PB}{CP} \cdot \frac{QC}{QA} \cdot \frac{AR}{RB} = 1.</cmath> ...;label("C",C,(1,0));label("P",P,(0,-1));label("Q",Q,(1,0));label("R",R,(-1,1));

5 KB (804 words) - 03:01, 12 June 2023
ARML historical results
| [[South Carolina ARML]] (South Carolina #1)

19 KB (2,632 words) - 14:31, 12 June 2022
Fermat's Little Theorem
...t [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>. ...denotes [[Euler's totient function]]. In particular, <math>\varphi(p) = p-1</math> for prime numbers <math>p</math>. In turn, this is a special case of

16 KB (2,658 words) - 16:02, 8 May 2024
Parabola
xaxis(-9,9,Ticks(f, 1.0)); yaxis(-9,9,Ticks(f, 1.0));

3 KB (551 words) - 16:22, 13 September 2023
Chebyshev's Inequality
...\sum_{i=1}^{n}a_ib_i\right)\geq\left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)</math>. <math> b_n\geq b_{n-1}\geq ... \geq b_1 </math> then:

1 KB (214 words) - 20:32, 13 March 2022
Euler's Totient Theorem
...er [[relatively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>. ...hi(m)} \pmod{m} </math> <math> \implies </math> <math> a^{\phi (m)} \equiv 1 \pmod{m}</math> as desired. Note that dividing by <math> n_1 n_2 ... n_{\ph

3 KB (542 words) - 17:45, 21 March 2023
Geometric inequality
.../math> and [[perimeter]] <math>P</math>, then <math>\frac{4\pi A}{P^2} \le 1</math>. This means that given a perimeter <math>P</math> for a plane figure ...o <math>2\sin A \cos B<2\sin A</math>, which is equivalent to <math>\cos B<1</math>. Since this is always true for <math>0<B<180</math>, this inequality

7 KB (1,296 words) - 14:22, 22 October 2023
Brahmagupta's Formula
Substituting <math>\sin^2B=1-\cos^2B</math> results in <cmath>4[ABCD]^2=(1-\cos^2B)(ab+cd)^2=(ab+cd)^2-\cos^2B(ab+cd)^2</cmath>

3 KB (465 words) - 18:31, 3 July 2023
Ptolemy's theorem
== Proof 1 == ...<math> ABCDEFG </math>, prove that: <math> \frac{1}{AB}=\frac{1}{AC}+\frac{1}{AE} </math>.

7 KB (1,198 words) - 20:39, 9 March 2024
Elementary symmetric sum
...mbers is the sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of ...)^nS_n</math>, and the coefficient of the <math>x^k</math> term is <math>(-1)^{n-k}S_{n-k}</math>, where the symmetric sums are taken over the roots of

2 KB (275 words) - 12:51, 26 July 2023
Number theory/Advanced
<cmath>z'=\frac{az+b}{cz+d},\quad a,b,c,d\in\mathbb{Z}, ad-bc=1.</cmath>

5 KB (849 words) - 16:14, 18 May 2021
1959 IMO Problems/Problem 2
<math> \sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A,</math> given (a) <math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admi

3 KB (466 words) - 12:04, 12 April 2024
Harmonic mean
...> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}</math>. ..._1\cdot x_2 \cdots x_n}\ge \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}} </math>

1 KB (196 words) - 00:49, 6 January 2021
Euler's totient function
...(1-\frac{1}{p_1} \right) \left(1-\frac{1}{p_2} \right)\cdots \left(1-\frac{1}{p_m}\right).</cmath> ...fine the [[prime factorization]] of <math> n </math> as <math> n =\prod_{i=1}^{m}p_i^{e_i} =p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m} </math> where the <math>p

5 KB (898 words) - 19:12, 28 January 2024
Probability
..., <math>(\Omega, \mathfrak{a})</math>. <math>\mathit{P}:\mathfrak{a}\to [0,1]</math> is an assignment with certain properties (it is a special kind of [ <math>\mathit{P}(\{H, T\})=1</math>.

4 KB (588 words) - 12:47, 2 October 2022
LaTeX
<cmath>1+2+3+4+5+\sin \pi = \frac{5\cdot 6}{2}+0=15.</cmath>

1 KB (164 words) - 19:09, 14 February 2024
Divisibility
* The positive divisors of <math>35</math> are <math>1</math>, <math>5</math>, <math>7</math>, and <math>35</math>.

2 KB (277 words) - 16:21, 29 April 2023
Rearrangement Inequality
...in increasing order, then the maximum sum is just <math>-a_1b_k - a_2b_{k-1} + \ldots</math>. Thus, by negating all values the inequality follows.

5 KB (804 words) - 13:54, 26 January 2023
Real number
...umber line. It can have any value. Some examples of real numbers are:<math>1, 2, -23.25, 0, \frac{\pi}{\phi}</math>, and so on. Numbers that are not rea (1) <math>\alpha\neq\mathbb{Q}</math> and <math>\alpha</math> is [[bounded]] a

3 KB (496 words) - 23:22, 5 January 2022
Divisibility rules
Rule 1: Partition <math>N</math> into 3 digit numbers from the right (<math>d_3d_ [[Divisibility rules/Rule 1 for 7 proof | Proof]]

8 KB (1,315 words) - 18:18, 2 March 2024
Induction
...works for <math>n=1+1=2</math>, which in turn means it works for <math>n=2+1=3</math>, and so on. ...+1)}{2}</math> (the <math>n</math>th triangular number is defined as <math>1+2+\cdots +n</math>; imagine an [[equilateral polygon | equilateral]] [[tria

5 KB (768 words) - 20:45, 1 September 2022
Incenter/excenter lemma
...wever, <cmath>\angle BIL = \angle BAI + \angle ABI = \frac{1}{2} A + \frac{1}{2} B.</cmath> Hence, <math>\triangle BIL</math> is isosceles, so <math>LB

2 KB (291 words) - 16:31, 18 May 2021
Orthic triangle
[[Image:Acute_orthic_triangle.png|thumb|right|300px|Case 1: <math>\triangle ABC</math> is acute.]] ...enter of <math>\triangle DEF</math>, <math>\angle EDC = 90^{\circ} - \frac{1}{2} \angle D</math>. Thus, <math>\angle ADC = 90^{\circ}</math>, and becaus

8 KB (1,408 words) - 11:54, 8 December 2021
Triangle
{{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}} {{asy image|<asy>draw((0,0)--(1,0)--(0.5,0.5)--cycle);</asy>|right|An isosceles triangle.}}

4 KB (628 words) - 17:17, 17 May 2018
Greatest common divisor
...use the recursive formula <math>GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)</math>.

2 KB (288 words) - 22:40, 26 January 2021
Overcounting
Here is MATHCOUNTS 2008 National Target #1: Try to solve this. ...According to the formula we get 2 handshakes, but wait, we will have only 1 handshake between two persons. That means we have overcounted somewhere.

4 KB (635 words) - 12:19, 2 January 2022
Constructive counting
=== Example 1 === ...first digit is, we know that it removes one option, so there are <math>8 - 1 = 7</math> options for the second digit.

12 KB (1,896 words) - 23:55, 27 December 2023
Jensen's Inequality
...math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n=1</math>. Then Let <math>\bar{x}=\sum_{i=1}^n a_ix_i</math>.

3 KB (623 words) - 13:10, 20 February 2024
Casework
=== Example 1 === '''Case 1''': The word is one letter long. Clearly, there are <math>5</math> of these

5 KB (709 words) - 10:28, 19 February 2024
Euclidean algorithm
<math>r_{n-1} \pmod {r_n} \equiv 0</math> for <math>r_{k+1} < r_k < r_{k-1}</math> 

6 KB (924 words) - 21:50, 8 May 2022
Conic section
...here the distance from a line (the directrix) is some number <math>0 < e < 1</math> times the distance to some fixed point (the focus). ...nce away from a point (focus) and a line (called the directrix) (<math>e = 1</math>).

5 KB (891 words) - 01:14, 9 January 2023
Generating function
...ath>A(k)={n \choose k}</math>, then we have <math>{n \choose 0}+{n \choose 1}x + {n \choose 2}x^2+\cdots+</math><math>{n \choose n}x^n</math>. .....+{n \choose n}=2^n</math>(let <math>{x}=1</math>), also <math>{n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots</math>.

4 KB (659 words) - 12:54, 7 March 2022
Binomial Theorem
...ath>, with coefficients <math>1 = \binom{5}{0}</math>, <math>5 = \binom{5}{1}</math>, <math>10 = \binom{5}{2}</math>, etc. ...(a+b)^{n} = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}</math>, <cmath>(a+b)^{n+1} = (\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k})(a+b)</cmath>

5 KB (935 words) - 13:11, 20 February 2024
Prime number
...integer]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. Note that <math>1</math> is usually defined as being neither prime nor [[composite number|com

6 KB (985 words) - 12:38, 25 February 2024
Recursion
...early values of the sequence in terms of previous values: <math>F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on. ...defined recursively by <math>a_0 = 1</math> and <math>a_n = 2\cdot a_{n - 1}</math> for <math>n > 0</math> also has the closed-form definition <math>a_

2 KB (316 words) - 16:03, 1 January 2024
Function
...exactly one value in the second. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it ta ...'.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>.

10 KB (1,761 words) - 03:16, 12 May 2023
Complementary counting
...bility it does not happen is one. Thus, we have the identity <cmath>P(A) = 1 - P(A^c).</cmath> Like its counting analog, complementary probability often === Example 1 ===

8 KB (1,192 words) - 17:20, 16 June 2023
Combinatorial identity
<math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math> ...s added to the number of ways to choose <math>k</math> things from <math>n-1</math> things.

12 KB (1,996 words) - 12:01, 18 May 2024
Integer
...ath> are integers, then their sum <math>a+b</math>, their difference <math>a-b</math>, and their product <math>ab</math> are all integers (that is, the i ...</math> bits), which limits their maximum value (typically to <math>2^{31}-1</math> for signed <math>32</math>-bit integers). Integers in Python can be

2 KB (296 words) - 15:04, 5 August 2022
Prime factorization
* [[2000 AIME I Problems/Problem 1]]

3 KB (496 words) - 22:14, 5 January 2024
Composite number
...''' is a [[positive integer]] with at least one [[divisor]] different from 1 and itself. Some composite numbers are <math>4=2^2</math> and <math>12=2\t Every positive integer either is prime, composite, or 1.

6 KB (350 words) - 12:58, 26 September 2023
Natural number
A '''natural number''' is any positive [[integer]]: <math>\text{1, 2, 3, 4, 5, 6, 7,\dots}</math>. The set of '''natural numbers''', denoted

1 KB (162 words) - 21:44, 13 March 2022
Circle
'''Case 1:''' The circle's area is greater than the triangle's area. ...han the radius so <math>a<r</math>. Therefore <math> P=\frac{1}{2}ap<\frac{1}{2}r\cdot 2\pi r=T</math>. However it cannot be both <math>P>T</math> and <

9 KB (1,581 words) - 18:59, 9 May 2024
Ellipse
MC(90,"\mbox{semiminor axis}",7,D((0,0)--(0,3),green+linewidth(1)),E); MC("\mbox{semimajor axis}",7,D((0,0)--(5,0),red+linewidth(1)),S);

5 KB (892 words) - 21:52, 1 May 2021
Divisor
...ath>d(n)=(\alpha_{1} + 1)\cdot(\alpha_{2} + 1)\cdot\dots\cdot(\alpha_{m} + 1)</math>. It is often useful to know that this expression grows slower than * <math>{\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor

1 KB (274 words) - 19:50, 29 August 2023
Base numbers
...number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...six <math>10^0</math>'s, the second digit tells us there are four <math>10^1</math>'s, the third digit tells us there are seven <math>10^2</math>'s, and

4 KB (547 words) - 17:23, 30 December 2020
Base numbers/Common bases
...ong computer programmers. It has just two digits: <math>0</math> and <math>1</math>. [[Hexadecimal]] is base 16. The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. One of its common uses is f

2 KB (351 words) - 10:39, 1 October 2015
Partition of an interval
...thcal{P}</math> is defined as <math>\|\mathcal{P}\|=\sup\{x_i-x_{i-1}\}_{i=1}^n</math> ...he set of ordered pairs <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math>.

1 KB (178 words) - 20:34, 6 March 2022
Henstock-Kurzweil integral
Consider the function <math>f:[0,1]\rightarrow\mathbb{R}</math> <math>f\left( \frac{1}{n}\right) =n\forall n\in\mathbb{N}</math>

2 KB (401 words) - 09:46, 31 January 2018
2001 AMC 12 Problems/Problem 1
{{duplicate|[[2001 AMC 12 Problems|2001 AMC 12 #1]] and [[2001 AMC 10 Problems|2001 AMC 10 #3]]}}

788 bytes (120 words) - 10:32, 8 November 2021
2001 AMC 12 Problems/Problem 2
{{AMC12 box|year=2001|num-b=1|num-a=3}}

1,007 bytes (165 words) - 00:28, 30 December 2023
Square (geometry)
A = (-1, 1); B = (1, 1);

1 KB (169 words) - 01:12, 13 June 2022
Mathematical Kangaroo
...ngaroo is an international mathematical competition for students in grades 1 through 12. The competition consists of a single round that is taken on th Any student that is currently enrolled in grades 1 through 12 (or homeschooled equivalent) is eligible to participate. They m

6 KB (936 words) - 15:38, 22 February 2024
American Scholastic Mathematics Association
Contest #1 - October 10, 2019

1 KB (153 words) - 13:11, 14 May 2019
Least common multiple
<math>12 = 2^2\cdot 3^1</math> <math>15 = 3^1\cdot 5^1</math>

2 KB (383 words) - 10:49, 4 September 2022
Common factorizations
...llet}&&x^{2n+1}+y^{2n+1}&=(x+y)(x^{2n}-x^{2n-1}y+x^{2n-2}y^2-\ldots-xy^{2n-1}+y^{2n})\\ \text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-2}+y^{n-1})

2 KB (327 words) - 02:06, 28 April 2024
Math Bee
...evel I: 0.5 - 1 Level II: 0.5 - 1.5 Level III: 1 - 1.5}} * Level I: For grades 1 and 2.

1 KB (197 words) - 10:59, 14 April 2024
Math Day at the Beach
== Individual Round - Part 1 == * 1 point is awarded for each question left blank

4 KB (644 words) - 12:56, 29 March 2017
Ptolemy's Inequality
BD = \frac{BA \cdot DC }{AP} \; (1) ...both sides of the inequality by <math>BD</math> and using equations <math>(1) </math> and <math>(2) </math> gives us

3 KB (602 words) - 09:01, 7 June 2023
Median of a triangle
A = (1, 2); draw(B--M, StickIntervalMarker(1));

1 KB (185 words) - 20:24, 6 March 2024
Pi
...e most surprising places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations fo ...rmula for pi is <math>4\left( \sum_{i = 0}^\infty (-1)^i \left(\frac{1}{2n+1}\right)\right) </math>. This can be computed to the desired degree of accur

8 KB (1,469 words) - 21:11, 16 September 2022
Phi
...ently in mathematical writing, often to represent the constant <math>\frac{1+\sqrt{5}}{2}</math>. (The Greek letter [[Tau]] (<math>\tau</math>) was also ...as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>.

2 KB (302 words) - 14:04, 1 January 2024
Fibonacci sequence
...term is the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...[[recursion|recursively]] as <math>F_1 = F_2 = 1</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n \geq 3</math>. This is the simplest nontrivial

6 KB (957 words) - 23:49, 7 March 2024
Muirhead's Inequality
...sequence <math>(5,1)</math> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</ <cmath>x^5y^1+y^5x^1\geq x^4y^2+y^4x^2</cmath>

8 KB (1,346 words) - 12:53, 8 October 2023
Cardinality
...y of <math>\{3, 4\}</math> is 2, the cardinality of <math>\{1, \{2, 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0.

2 KB (263 words) - 00:54, 17 November 2019
Injection
...thbb R</math>, <math>f(x)=x^2</math> is not an injection (<math>f(-1)=f(1)=1</math>), the function <math>g:[0,\infty)\to\mathbb R</math>, <math>g(x)=x^2

1 KB (228 words) - 01:01, 17 November 2019
Fundamental Theorem of Calculus
...ve.) Interpret the distance that the object travels between times <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve. ...is the amount that <math>F</math> changes on the interval <math>[x_i, x_{i+1}]</math>, then <math>\Delta F_i \approx F'(x_i)\Delta x</math>.

11 KB (2,082 words) - 15:23, 2 January 2022
Polygon
...-- it will have "degenerated" from an <math>n</math>-gon to an <math>(n - 1)</math>-gon. (In the case of triangles, this will result in either a line

2 KB (372 words) - 19:04, 30 May 2015
Isoperimetric Inequalities
...h>A</math> and perimeter <math>P</math> then <math>\frac{4\pi A}{P^2} \leq 1</math>. This means that given a perimeter <math>P</math> for a plane figure

789 bytes (115 words) - 17:08, 29 December 2021
Writing scholarships
...s://seasonsbeachcottage.com/mental-health-education-scholarship-2023-round-1 Mental health Education Annual Scholarship 2023] Applicants must be current ...g Scholarship] Zampi, Inc. A small business marketing platform is offering 1,000 dollar scholarship. To enter, simple create a small business marketing

7 KB (1,039 words) - 18:45, 18 January 2024
National Scholarships
* [[Best Buy @15 Scholarship]] of <dollar/>1,000 for students in grades 9-12. [https://www.at15.com/contests_scholarship

4 KB (538 words) - 00:48, 28 January 2024
2011 AMC 10B Problems/Problem 22
A pyramid has a square base with sides of length <math>1</math> and has lateral faces that are equilateral triangles. A cube is plac == Solution 1 ==

4 KB (691 words) - 18:38, 19 September 2021
2011 AMC 10B Problems/Problem 4
== Solution 1==

1 KB (249 words) - 13:05, 24 January 2024
Bretschneider's formula
<math>[ABCD]=\frac{1}{4} \cdot \sqrt{4p^2q^2-(b^2+d^2-a^2-c^2)^2}</math>. ...vec{c} - \vec{d}</math>. The area of any such quadrilateral is <math>\frac{1}{2} |\vec{p} \times \vec{q}|</math>.

3 KB (566 words) - 03:51, 12 February 2021
Vector
pen p=linewidth(1); MA("\theta",(5,-1),(2,3),(4,6),0.3,9,yellow);

7 KB (1,265 words) - 13:22, 14 July 2021
Logarithm
...bers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}</cmath> would be a pain to have to calculate any ti <cmath>\log_4(5)\approx 1.160964047443681173935159715\ldots</cmath>

4 KB (680 words) - 12:54, 16 October 2023
Law of Cosines
==Proof 1== label("A",A,(1,0));

6 KB (1,003 words) - 09:11, 7 June 2023
Schur's Inequality
<cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath> The <math>r=1</math> case yields the well-known inequality:

2 KB (398 words) - 16:57, 29 December 2021
Trigonometric identities
* <math>\sin^2x + \cos^2x = 1</math> * <math>1 + \cot^2x = \csc^2x</math>

8 KB (1,397 words) - 21:55, 20 January 2024
Irrational number
...>\sqrt{2}</math> as the length of a diagonal of a square with side lengths 1 in the sixth century <math>B.C</math>. The Pythagoreans lived by the doctri <math>3\frac{10}{71}</math> <math>to</math> <math>3\frac{1}{7}</math>.

3 KB (368 words) - 19:26, 6 June 2015
Complex number
...numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (Se

5 KB (860 words) - 15:36, 10 December 2023
Rational number
...ecause every integer <math>a</math> can be represented as <math>a=\frac{a}{1}</math>

1 KB (207 words) - 15:51, 25 August 2022
Roots of unity
...the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ...ik}</math>. The magnitude of the RHS is 1, making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This lea

3 KB (558 words) - 21:36, 11 December 2011
Angle bisector
markangle(n=1,radius=20,D,A,F,green); markangle(n=1,radius=22,F,A,B,green);

3 KB (575 words) - 15:27, 19 March 2023
Without loss of generality
* [[2005_Austrian_Mathematical_Olympiad_Final_Round-Part 1/Problem 5]]

2 KB (280 words) - 15:30, 22 February 2024
Law of Sines
=== Method 1 === real r = 1;

4 KB (658 words) - 16:19, 28 April 2024
Sequence
...<math>\mathbb{N}</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>. A classic example of convergence would be to show that <math>1/n\to 0</math> as <math>n\to \infty</math>.

2 KB (413 words) - 21:18, 13 November 2022
Geometric sequence
...ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between cons ...ogression if and only if <math>a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}</math>. A similar definition holds for infinite geometric sequences. It ap

4 KB (644 words) - 12:55, 7 March 2022
Arithmetic sequence
...h>1, 2, 3, 4</math> is an arithmetic sequence with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with commo ...ogression if and only if <math>a_2 - a_1 = a_3 - a_2 = \cdots = a_n - a_{n-1}</math>. A similar definition holds for infinite arithmetic sequences. It a

4 KB (736 words) - 02:00, 7 March 2024
Fermat's Last Theorem
...n unique factorization in [[ring]]s of the form <math>\mathbb{Z}[\sqrt[n]{-1}]</math>. Unfortunately, this is not often the case. In fact, it has now

3 KB (453 words) - 11:13, 9 June 2023
Homothety
* A homothety with factor <math>-1</math> is a <math>180^\circ</math> rotation about the center.

3 KB (532 words) - 01:11, 11 January 2021
2005 USAMO Problems/Problem 4
...a piece of length <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? == Solution 1 ==

7 KB (1,276 words) - 20:51, 6 January 2024
2000 AMC 12 Problems/Problem 1
...0 AMC 12 Problems|2000 AMC 12 #1]] and [[2000 AMC 10 Problems|2000 AMC 10 #1]]}} == Solution 1 (Verifying the Statement)==

2 KB (276 words) - 05:25, 9 December 2023
Diophantine equation
...there will always be an infinite number of solutions when <math>\gcd(a,b)=1</math>. If <math>\gcd(a,b)\nmid c</math>, then there are no solutions to t .../math> is an [[odd]] number, then <math>m, \frac {m^2 -1}{2}, \frac {m^2 + 1}{2}</math> is a Pythagorean triple.

9 KB (1,434 words) - 13:10, 20 February 2024
Area
...ices to the [[circumcenter]]. This creates a triangle that is <math>\frac{1}{n},</math> of the total area (consider the regular [[octagon]] below as an ..._b^{-1})(H-h_c^{-1})}</math>, where <math>H=\frac{(h_a^{-1}+h_b^{-1}+h_c^{-1})}{2}</math> and the triangle has altitudes <math>h_a</math>, <math>h_b</ma

6 KB (1,181 words) - 22:37, 22 January 2023
Fraction
...the denominator such as <math>\frac{3}{3}</math> is always equal to <math>1</math>. ...d denominator by the same quantity (since we're essentially multiplying by 1).

3 KB (432 words) - 19:34, 11 June 2020
Functional equation
...'.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>. ...le of such a function is <math>f(x) = 1/x</math> because <math>f(f(x)) = f(1/x) = x</math>. Cyclic functions can significantly help in solving function

2 KB (361 words) - 14:40, 24 August 2021
Exponential form
A=(0,1); C=(1,0);

1 KB (238 words) - 22:51, 20 February 2022
Floor function
...loor+\left\lfloor a+\frac{1}{n}\right\rfloor+\ldots+\left\lfloor a+\frac{n-1}{n}\right\rfloor</cmath> ...te the largest integer not exceeding <math>x</math>. For example, <math>[2.1]=2</math>, <math>[4]=4</math> and <math>[5.7]=5</math>. How many positive i

3 KB (508 words) - 21:05, 26 February 2024
Pascal's triangle
...sum of the two numbers appearing above it. For example, <math>{5 \choose 1}+{5 \choose 2} = 5 + 10 = 15 = {6 \choose 2}</math>. This property allows ...math>{6 \choose 0}+{5 \choose 1}+{4 \choose 2}+{3 \choose 3} = 1 + 5 + 6 + 1 = 13 = F(7)</math>. A "shallow diagonal" is plotted in the diagram.

5 KB (838 words) - 17:20, 3 January 2023
Newton's Sums
<center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> <cmath>a_nP_1 + a_{n-1} = 0</cmath>

4 KB (690 words) - 13:11, 20 February 2024
Exponentiation
...words, we want to divide by 2 four times. Therefore, <math> 2^{-4}=\frac 1{2^4}.</math> * <math>b^0 = 1</math> if <math>b \neq 0</math>

5 KB (803 words) - 16:25, 10 August 2020
Rational approximation
...en denominator <math>q\ge 1</math> with an error not exceeding <math>\frac 1{2q}</math>. ...interval <math>\left[qx-\frac12,qx+\frac12\right]</math> has length <math>1</math> and, therefore, contains at least one integer. Choosing <math>p</mat

7 KB (1,290 words) - 12:18, 30 May 2019
Trigonometry
...d as the reciprocal of the sine of <math>A</math>. <cmath>\csc (A) = \frac{1}{\sin (x)} = \frac{\textrm{hypotenuse}}{\textrm{opposite}} = \frac{c}{a}.</ ...as the reciprocal of the cosine of <math>A</math>. <cmath>\sec (A) = \frac{1}{\cos (x)} = \frac{\textrm{hypotenuse}}{\textrm{adjacent}} = \frac{c}{b}.</

8 KB (1,217 words) - 20:15, 7 September 2023
Euler's number
...>e</math> such that <math>\ln e=1</math>, where <math>\ln x=\int_1^x \frac{1}{t} \, dt</math>. It has been shown to be both [[irrational]] and [[transce ...tion such that <math>\frac{d}{dx} \exp(x)=\exp(x)</math> and <math>\exp(0)=1</math> is equal to <math>e^x</math>.

4 KB (764 words) - 21:09, 13 March 2022
Exponential function
:<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}} :If <math>b > 1</math>, then the function will show growth.

2 KB (312 words) - 15:57, 6 March 2022
Mathematical Olympiad Summer Program
...three instructional sessions: 8:30 AM - 10:00 AM, 10:15 AM - 11:45 AM, and 1:15 PM - 2:45 PM. Classes usually consist of a lecture followed by a problem ...ered in place of the afternoon lecture, and is graded with comments within 1-2 days.

6 KB (936 words) - 10:37, 27 November 2023
Complex analysis
...h> f^{(n)}(z_0)=\frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}. </cmath>

2 KB (271 words) - 22:06, 12 April 2022
Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
...cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power. ...s greater than or equal to 1. This creates the indeterminate form of <math>1^{\infty}</math>. Then, we can say that the limit as x goes to 0, the result

5 KB (912 words) - 20:06, 14 March 2023
Harmonic series
<math>\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots</math> <math>\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots</math> , though, approaches <math>\ln 2</math>.

2 KB (334 words) - 20:52, 13 March 2022
Goldbach Conjecture
He considered 1 to be a prime number, a [[mathematical convention|convention]] subsequently ...m</math> and <math>n-m</math> simultaneously being prime to be <math>\frac{1}{\ln m \cdot \ln (n-m)}</math>. This heuristic is non-rigorous for a numbe

7 KB (1,201 words) - 16:59, 19 February 2024
Twin Prime Conjecture
...1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\cdots</math></center> ...y if one of variables is 0 will the factorization be trivial (contain only 1 and itself).

2 KB (308 words) - 02:27, 1 May 2024
Fallacy
=== 2 = 1 === <div style="text-align:center"><math> 2 = 1 </math> (dividing by <math>a^2-ab</math>)</div>

2 KB (429 words) - 08:27, 5 June 2013
Riemann Hypothesis
...[[root |zero]]s of the [[Riemann zeta function]] have [[real part]] <math>1/2</math>. From the [[functional equation]] for the zeta function, it is eas ...atement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>.

2 KB (425 words) - 12:01, 20 October 2016
Quadratic residues
...\mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ residue\ modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ no ...left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}</math>. This is known as the [[Quadratic Reciprocity Theorem]].

5 KB (778 words) - 13:10, 29 November 2017
Power of a Point Theorem
''Proof.'' We have already proven the theorem for a <math>1</math>-sphere (i.e., a circle), so it only remains to prove the theorem for ==== Problem 1 ====

5 KB (827 words) - 17:30, 21 February 2024
Fundamental Theorem of Algebra
...loss of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. ...absolute values of the coefficients of <math>P</math>, so that <math>R \ge 1</math>. Then for <math>\lvert z \rvert \ge R</math>,

5 KB (832 words) - 14:22, 11 January 2024
Collatz Problem
...>, the sequence <math>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people have described it as the easies Observing that if <math>n=2m+1</math> then <math>3n+1=6m+4</math>, as well as: <cmath>{6m+4\over 2}=3m+2</cmath> we can then obse

1 KB (231 words) - 19:45, 24 February 2020
Algebraic number
...ynomial]] with [[rational]] [[coefficient]]s. Examples include <math>\frac{1}{3}</math>, <math>\sqrt{2}+\sqrt{3}</math>, <math>i</math>, and <math>\frac

1,006 bytes (151 words) - 21:56, 22 April 2022
Incenter
D=intersectionpoint(A+0.1*expi((angle(B-A)+angle(C-A))/2)--A+20*expi((angle(B-A)+angle(C-A))/2), circ dot(A^^B^^C^^D^^I);label("A",A,(0,1));label("B",B,(-1,0));label("C",C,(1,0));

2 KB (381 words) - 19:38, 24 November 2011
Triangle Inequality
...+ \ldots + a_n = \left(\sum_{j=1}^n a_j\right) - a_i</math> for <math>i = 1, 2 \ldots, n</math>. Expressing the inequality in this form leads to <math

2 KB (268 words) - 03:02, 3 January 2021
Majorization
...for all <math> 1 \le k \le n </math>, <math> \sum_{i=1}^{k}a_i \ge \sum_{i=1}^{k}b_i </math>, with equality when <math>k = n </math>. If <math>\{a_i\} ...h>\sum_{i=k}^n a_i \le \sum_{i=k}^n b_i</math>, with equality when <math>k=1 </math>. An interesting consequence of this is that the finite sequence <m

2 KB (288 words) - 22:48, 5 July 2023
Surjection
...function <math>f: \mathbb N \to\mathbb N</math> defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is

794 bytes (131 words) - 22:39, 13 May 2020
International Mathematical Olympiad
...|region=International|type=Proof|difficulty=5.5 - 10|breakdown=Problem 1/4: 6.5 Problem 2/5: 7.5-8 Problem 3/6: 9.5 Pr * Gold - the top 1/12 of individual scores.

3 KB (490 words) - 03:32, 23 July 2023
Prime Number Theorem
<cmath> \lim_{x\to \infty} \frac{\pi(x) \log x}{x} = 1 . </cmath> with <math>A(x)</math> tending to some constant number around 1.08366.

10 KB (1,729 words) - 19:52, 21 October 2023
...a_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative reals with sum of 1, then ...f{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]].

4 KB (774 words) - 12:12, 29 October 2016
Infinite
...ot empty and cannot be put into bijection with any set of the form <math>\{1, 2, \ldots, n\}</math> for a [[positive integer]] <math>n</math>. <math>\sum_{i = 3}^{\infty}{(2i - 1)}</math>

1 KB (186 words) - 23:19, 16 August 2013
Isosceles trapezoid
{{asy image|<asy>draw((0,1)--(1,5)--(3,5)--(4,1)--cycle);</asy>|right|An isosceles trapezoid.}}

577 bytes (81 words) - 10:33, 18 April 2019
Mock AMC
! scope="row" | '''Mock AMC #1''' | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9572 1-5]

51 KB (6,175 words) - 20:58, 6 December 2023
Mock AIME
...Y2QwOTc3NWZiYjY0LnBkZg==&rn=TWlsZG9yZiBNb2NrIEFJTUUucGRm Mildorf Mock AIME 1] ** [[Mock AIME 1 2006-2007]]

8 KB (906 words) - 17:30, 26 April 2024
Commutative property
...n <math>AB</math> and <math>BA</math> are matrices of sizes <math>1 \times 1</math> and <math>2 \times 2</math>, respectively.

2 KB (301 words) - 17:46, 16 March 2012
Permutation
...ath>. We often drop the brackets and commas, so the permutation <math>\{2,1,3\}</math> would just be represented by <math>213</math>. ...ent objects we can choose from. For the second element, there are <math>n-1</math> objects we can choose, <math>n-2</math> for the third, and so on. I

3 KB (422 words) - 11:01, 25 December 2020
Algebraic topology
...ith <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1( ...>g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> One can check that this is indeed [[well-defined]].

3 KB (479 words) - 15:35, 1 December 2015
Bertrand's Postulate
<math>\frac{2^{2n}}{(2n+1)}\le{\binom{2n}{n}}\le <math>\left(\prod_{n<p\le{2n}}p\right)\ge \frac{4^{\frac n3}}{(2n+1)(2n)^{\sqrt {2n}}}</math>

2 KB (309 words) - 21:43, 11 January 2010
Riemann zeta function
<cmath>\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}= 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots</cmath>

9 KB (1,547 words) - 03:04, 13 January 2021
Mock USAMO
...ww.artofproblemsolving.com/Forum/viewtopic.php?p=423926#p423926 Mock USAMO 1 2006] ...http://www.artofproblemsolving.com/Forum/viewtopic.php?t=200572 Mock USAMO 1 2008]

2 KB (205 words) - 19:56, 4 March 2020
Modular arithmetic/Introduction
<math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math> This is the way in which we count in '''modulo 12'''. When we add <math>1</math> to <math>11</math>, we arrive back at <math>0</math>. The same is t

15 KB (2,396 words) - 20:24, 21 February 2024
Perfect square
...th> so that <math>m^2=n</math>. The first few perfect squares are <math>0, 1, 4, 9, 16, 25, 36</math>. ...th> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>

954 bytes (155 words) - 01:14, 29 November 2023
Convex function
...Equivalently, <math>f </math> is convex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math>, <center><math>\lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right) </math>.</center>

2 KB (417 words) - 00:10, 20 February 2016
Uncountable
Suppose that the set <math>A=\{x\in\mathbb{R}:0<x< 1\}</math> is countable. Let <math>\{\omega_1, \omega_2, \omega_3, ...\}</mat ...ath>. However, <math>\omega</math> is clearly a real number between 0 and 1, a [[contradiction]]. Thus our assumption that <math>A</math> is countable

2 KB (403 words) - 20:53, 13 October 2019
Gamma function
...<math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dt</cmath> ...f by one. Since <math>\Gamma(1)=1</math>, we therefore have <math>\Gamma(n+1)=n!</math> for nonnegative integers <math>n</math>. But with the integral,

840 bytes (137 words) - 22:26, 22 June 2009
Modular arithmetic/Intermediate
* Subtraction: <math>a-c\equiv b-d\pmod {n}</math>. <math>\{ \ldots, -5, -2, 1, 4, 7, \ldots \}</math>

14 KB (2,317 words) - 19:01, 29 October 2021
Right triangle
A = (0, 1); B = (1, 0);

3 KB (499 words) - 23:41, 11 June 2022
Subset
...thbb{C} \subset \mathbb{C}\cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}</math>

1 KB (217 words) - 09:32, 13 August 2011
2006 AMC 10A
* [[2006 AMC 10A Problems/Problem 1]]

2 KB (180 words) - 18:06, 6 October 2014
Limit
...the value of the function <math>f(x)</math> at <math>x = 0</math> is <math>1</math>, the limit <math>\lim_{x\rightarrow 0} f(x)</math> is, in fact, zero ...ons <math>d_A(a,b)</math> and <math>d_B(a,b)</math> are both simply <math>|a-b|</math>. We then obtain the following definition commonly found in calculu

7 KB (1,325 words) - 13:51, 1 June 2015
2006 AIME I Problems
== Problem 1 == [[2006 AIME I Problems/Problem 1|Solution]]

7 KB (1,173 words) - 03:31, 4 January 2023
2006 AIME I Problems/Problem 15
...</math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006} === Solution 1 ===

6 KB (910 words) - 19:31, 24 October 2023
2006 AIME I Problems/Problem 14
...s <math>4</math> feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let <math> h </math> be the height in feet of t == Solution 1 ==

6 KB (980 words) - 21:45, 31 March 2020
2006 AIME I Problems/Problem 13
...or each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 s == Solution 1 ==

10 KB (1,702 words) - 00:45, 16 November 2023
2006 AIME I Problems/Problem 11
...e with [[edge]]-[[length]] <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to ...th>, so there are <math>2\cdot 3^6 = 1458</math> towers using blocks <math>1, 2, \ldots, 8</math>, so the answer is <math>\boxed{458}</math>.

3 KB (436 words) - 05:40, 4 November 2022
2006 AIME I Problems/Problem 10
...math> c </math> are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> for(i=0; i<3; i=i+1) {

4 KB (731 words) - 17:59, 4 January 2022
2006 AIME I Problems/Problem 9
== Solution 1 == ...s <math>1 + \frac{1001-11}{22}=1 + \frac{990}{22}=46</math>. (We must add 1 because both endpoints are being included.) So the answer is <math>\boxed{0

4 KB (651 words) - 18:27, 22 May 2021
2006 AIME I Problems/Problem 8
...1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(F--G--(2.1,0));

5 KB (730 words) - 15:05, 15 January 2024
2006 AIME I Problems/Problem 7
for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray);

4 KB (709 words) - 01:50, 10 January 2022
2006 AIME I Problems/Problem 6
== Solution 1 == ...ple, <math>.\overline{123}+.\overline{876}=.\overline{999} \Longrightarrow 1</math>.

2 KB (237 words) - 19:14, 20 November 2023
2006 AIME I Problems/Problem 5
== Solution 1 ==

3 KB (439 words) - 18:24, 10 March 2015
2006 AIME I Problems/Problem 4
...th>'s at the right end of the decimal representation of the product <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is di ...rom <math>100</math>. Thus, our final total is <math>970 + 76 + 51 + 26 + 1 = 1124</math>, and the answer is <math>\boxed{124}</math>.

2 KB (278 words) - 08:33, 4 November 2022
2006 AIME I Problems/Problem 3
...en its leftmost [[digit]] is deleted, the resulting integer is <math>\frac{1}{29}</math> of the original integer. === Solution 1 ===

4 KB (622 words) - 03:53, 10 December 2022
2006 AIME I Problems/Problem 2
...et]] <math> \mathcal{A} </math> be a 90-[[element]] [[subset]] of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elemen ...ible values of S, so the number of possible values of S is <math>4995-4095+1=901</math>.

1 KB (189 words) - 20:05, 4 July 2013
2022 AIME I Problems/Problem 1
==Solution 1 (Linear Polynomials)== P(16) &= &512 + 16a + b &= 54, \hspace{20mm}&&(1) \\

4 KB (670 words) - 13:03, 13 November 2023
2021 JMC 10
==Problem 1== What is the value of <cmath>\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?</cmath>

12 KB (1,784 words) - 16:49, 1 April 2021
2006 AMC 12B Problems
== Problem 1 == What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>?

13 KB (2,058 words) - 12:36, 4 July 2023
2006 AMC 12A
* [[2006 AMC 12A Problems/Problem 1]]

1 KB (168 words) - 21:51, 6 October 2014
2004 AMC 12A
* [[2004 AMC 12A Problems/Problem 1]]

2 KB (186 words) - 17:35, 16 December 2019
2005 AMC 12A
** [[2005 AMC 12A Problems/Problem 1|Problem 1]]

2 KB (202 words) - 21:30, 6 October 2014
2002 AMC 12A
* [[2002 AMC 12A Problems/Problem 1]]

1 KB (158 words) - 21:33, 6 October 2014
2003 AMC 12A
* [[2003 AMC 12A Problems/Problem 1]]

1 KB (162 words) - 21:52, 6 October 2014
2006 AMC 12A Problems
== Problem 1 == [[2006 AMC 12A Problems/Problem 1|Solution]]

15 KB (2,223 words) - 13:43, 28 December 2020
2005 AMC 12A Problems
== Problem 1 == (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10

13 KB (1,971 words) - 13:03, 19 February 2020
2004 AMC 12A Problems
== Problem 1 == Alicia earns <math> 20</math> dollars per hour, of which <math>1.45\%</math> is deducted to pay local taxes. How many cents per hour of Ali

13 KB (1,953 words) - 00:31, 26 January 2023
2003 AMC 12A Problems
== Problem 1 == <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 40

13 KB (1,955 words) - 21:06, 19 August 2023
2002 AMC 12A Problems
== Problem 1 == [[2002 AMC 12A Problems/Problem 1|Solution]]

12 KB (1,792 words) - 13:06, 19 February 2020
2000 AMC 12 Problems
== Problem 1 == [[2000 AMC 12 Problems/Problem 1|Solution]]

13 KB (1,948 words) - 12:26, 1 April 2022
2001 AMC 12 Problems
== Problem 1 == [[2001 AMC 12 Problems/Problem 1|Solution]]

13 KB (1,957 words) - 12:53, 24 January 2024
2002 AMC 12B Problems
== Problem 1 == [[2002 AMC 12B Problems/Problem 1|Solution]]

10 KB (1,547 words) - 04:20, 9 October 2022

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