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• In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle?
  In the diagram, the equilateral triangle has a base of <math>8</math> m. What is the perimeter of the triangle?
  111 bytes (20 words) - 04:34, 11 May 2021

Page text matches

• 2016 AMC 10A Problems/Problem 11
  <math>\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \ ...{2}</math>, and slide the left and right pieces next to each other to form a parallelogram with base <math>1</math> and height <math>\frac{8}{2}</math>.
  8 KB (1,016 words) - 00:17, 31 December 2023
• Mathematics summer program
  .... These programs mostly take place during the summer, but some last during a significant portion of the academic year. Selection is important in creatin ...programs listed below offer [[mathematics scholarships]] or financial aid. A few of them are free. When you discover more that do, please add that infor
  14 KB (1,869 words) - 20:06, 18 May 2024
• Mathematics, science, and technology scholarships
  ...t important such competition in the world. The 3 ISEF winners each receive a <dollar/>50,000 scholarship. [http://www.sciserv.org/isef website] ...in exchange for summer internships and a year of work after graduation in a DoD lab. [http://smart.asee.org/ website]
  7 KB (851 words) - 10:54, 29 January 2022
• Texas mathematics competitions
  *[[Texas A&M University math Contest]] [http://www.math.tamu.edu/teaching/undergrad/high *[http://www.math.tamu.edu/outreach/highschoolcontest/ Texas A&M University Math Contest] [http://www.math.tamu.edu/outreach/highschoolconte
  2 KB (255 words) - 07:55, 2 March 2023
• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato .../www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.
  24 KB (3,177 words) - 12:53, 20 February 2024
• Math Olympiads for Elementary and Middle Schools
  '''Math Olympiads for Elementary and Middle Schools''' ('''MOEMS''') is a large and popular [[mathematics competition]] for students in grades 4 thro ...e=Free Response|difficulty=1|breakdown=<u>Division E</u>: 1<br><u>Division M</u>: 1}}
  2 KB (215 words) - 02:54, 18 November 2020
• Math textbooks
  This '''Math textbooks''' page is for compiling a list of [[textbook]]s for mathematics -- not problem books, contest books, ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato
  7 KB (901 words) - 14:11, 6 January 2022
• Physics
  <strong>Physics</strong> is a branch of [[science]] that studies the properties of matter, energy, and ma Modern Physics is also a group of different subjects in physics:
  9 KB (1,355 words) - 07:29, 29 September 2021
• Newton
  A '''Newton''' (abbreviated N) is the [[Système international|metric]] measu ...er [[second]] squared, or <math>\mathrm{N}=\mathrm{kg}\times \frac{\mathrm{m}}{\mathrm{s}^2}</math>. This is because <math>F = ma</math>, which means fo
  665 bytes (96 words) - 23:17, 2 February 2021
• United States of America Mathematical Talent Search
  ...''United States of America Mathematical Talent Search''' ('''USAMTS''') is a [[mathematics competition]] in which students are challenged to write full Each problems are graded on a scale from 0 to 5. The full 5 points are awarded for correct, concise solut
  4 KB (613 words) - 13:08, 18 July 2023
• Knowledge bowl competitions
  * [[Texas A&M Regional Science Bowl]] [http://outreach.science.tamu.edu/sciencebowl.asp w
  2 KB (212 words) - 17:15, 2 April 2024
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• Joining an ARML team
  The SD team is entering just its third year of participation, so a permanent process of selecting team members has yet to be decided upon. The A student who wishes to attend practice should try to take the SDMO. If a student wishes to speak with one of the coaches for the team, they may do s
  21 KB (3,500 words) - 18:41, 23 April 2024
• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I M(t)=
  3 KB (606 words) - 23:59, 1 July 2022
• 1953 AHSME Problems/Problem 13
  A triangle and a trapezoid are equal in area. They also have the same altitude. If the base <math>\textbf{(A)}\ 36\text{ inches} \qquad
  878 bytes (143 words) - 20:56, 1 April 2017
• Simon's Favorite Factoring Trick
  ...of the equation and a product of variables with each of those variables in a linear term on the other side. An example would be: <cmath>xy+66x-88y=23333 ...Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
  7 KB (1,107 words) - 07:35, 26 March 2024
• Relatively prime
  ...</math> and <math>n</math> are relatively prime if and only if <math>\frac{m}{n}</math> is in lowest terms. Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, then
  2 KB (245 words) - 15:51, 25 February 2020
• Geometric mean
  .../math> and <math>c</math> is the number <math>a</math> such that <math>a + a = b + c</math>, while the geometric mean of the numbers <math>b</math> and ...length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
  2 KB (282 words) - 22:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...ized method/formula to find the number of [[element]]s in the [[union]] of a given group of [[set]]s, the size of each set, and the size of all possible ...counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem solving.
  9 KB (1,703 words) - 07:25, 24 March 2024
• Combination
  ...'binomial coefficient''', is a way of choosing <math>r</math> objects from a set of <math>n</math> where the order in which the objects are chosen is ir ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 11:43, 21 May 2021
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...tively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>.
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
  7 KB (1,296 words) - 14:22, 22 October 2023
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] <math>n</math> is defined to be the number of positive ...ere the <math>p_i </math> are distinct [[prime number]]s. Now, we can use a [[PIE]] argument to count the number of numbers less than or equal to <mat
  5 KB (898 words) - 19:12, 28 January 2024
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subse
  3 KB (496 words) - 23:22, 5 January 2022
• Euclidean algorithm
  ...orithm that finds the [[greatest common divisor]] (GCD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[int ...c idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</math>
  6 KB (924 words) - 21:50, 8 May 2022
• Binomial Theorem
  The '''Binomial Theorem''' states that for [[real]] or [[complex]] <math>a</math>, <math>b</math>, and [[non-negative]] [[integer]] <math>n</math>, <center><math>(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k</math></center>
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divis ...</math> must [[#Importance of Primes|have]] a prime factor, which leads to a direct contradiction.
  6 KB (985 words) - 12:38, 25 February 2024
• Function
  A '''function''' is a rule that maps one set of values to another set of values, assigning to eac ...is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
  10 KB (1,761 words) - 03:16, 12 May 2023
• Complementary counting
  ...ting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Combinatorial identity
  ...> objects must have some <math>0\le k\le r</math> objects from group <math>m</math> and the remaining from group <math>n</math>. .... Think of the left hand side as picking <math>k</math> men from the <math>m</math> total men and picking <math>r-k</math> women from the <math>n</math>
  12 KB (1,996 words) - 12:01, 18 May 2024
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
  5 KB (892 words) - 21:52, 1 May 2021
• Divisor
  ...hat <math>n=kd</math> or, in other words, if <math>\frac nd</math> is also a natural number (i.e <math>d</math> divides <math>n</math>). See [[Divisibil A common notation to indicate a number is a divisor of another is <math>n|k</math>. This means that <math>n</math> divi
  1 KB (274 words) - 19:50, 29 August 2023
• Base numbers
  ...each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal ...digits 0-9. Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above
  4 KB (547 words) - 17:23, 30 December 2020
• Multiple
  ...m</math> and <math>n</math> are integers. (In this case, <math>k</math> is a multiple of <math>n</math>, as well). .../math> exactly when <math>k</math> is [[divisibility | divisble by]] <math>m</math>.
  860 bytes (142 words) - 22:51, 26 January 2021
• Median of a triangle
  A '''median''' of a [[triangle]] is a [[cevian]] of the triangle that joins one [[vertex]] to the [[midpoint]] of In the following figure, <math>AM</math> is a median of triangle <math>ABC</math>.
  1 KB (185 words) - 20:24, 6 March 2024
• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
  8 KB (1,469 words) - 21:11, 16 September 2022
• Fibonacci sequence
  The '''Fibonacci sequence''' is a [[sequence]] of [[integer]]s in which the first and second terms are both e ...th> for <math>n \geq 3</math>. This is the simplest nontrivial example of a [[linear recursion]] with constant coefficients. There is also an explicit
  6 KB (957 words) - 23:49, 7 March 2024
• Chain Rule
  ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.) ...as <math>\Delta x</math> approaches <math>0</math>. This can be made into a rigorous proof. (But we do have to worry about the possibility that <math>
  12 KB (2,377 words) - 11:48, 22 July 2009
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe ...imes <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve.
  11 KB (2,082 words) - 15:23, 2 January 2022
• Logarithm
  ...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th
  4 KB (680 words) - 12:54, 16 October 2023
• Law of Cosines
  ...sines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle]]. It can be derived in several different ways, the most common ...h>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines
  6 KB (1,003 words) - 09:11, 7 June 2023
• Schur's Inequality
  Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>
  2 KB (398 words) - 16:57, 29 December 2021
• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we e ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
  3 KB (558 words) - 21:36, 11 December 2011
• Law of Sines
  The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines]] and the [[law of tan ...h>\triangle ABC</math>, where <math>a</math> is the side opposite to <math>A</math>, <math>b</math> opposite to <math>B</math>, <math>c</math> opposite
  4 KB (658 words) - 16:19, 28 April 2024
• Arithmetic sequence
  ...ithmetic sequence''', sometimes called an '''arithmetic progression''', is a [[sequence]] of numbers such that the difference between any two consecutiv ...and <math>c</math> are in arithmetic progression if and only if <math>b - a = c - b</math>.
  4 KB (736 words) - 02:00, 7 March 2024
• Homothety
  ...ath> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math> ...is <math>\frac{OD'}{OD}</math>. Additionally, <math>ABCD</math> and <math>A'B'C'D'</math> are homothetic with respect to <math>O</math>.
  3 KB (532 words) - 01:11, 11 January 2021
• 2005 USAMO Problems/Problem 4
  ...math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• 2000 AMC 12 Problems/Problem 1
  ...O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\
  2 KB (276 words) - 05:25, 9 December 2023
• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is
  9 KB (1,434 words) - 13:10, 20 February 2024
• Pascal's triangle
  ...ns the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. ...2 is <math>{4 \choose 2} = 6</math>. Pascal's Triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the character
  5 KB (838 words) - 17:20, 3 January 2023
• 1985 IMO Problems/Problem 5
  ...distinct points <math>K</math> and <math>N</math> respectively. Let <math>M</math> be the point of intersection of the circumcircles of triangles <math ...1 B M_2</math> is also cyclic with diameter <math>BO</math> and thus <math>M</math> must lie on the same circumcircle as <math>B</math>, <math>M_1</math
  3 KB (496 words) - 13:35, 18 January 2023
• Rational approximation
  The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational numb ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
  7 KB (1,290 words) - 12:18, 30 May 2019
• Euler's number
  '''Euler's number''' is a [[constant]] that appears in a variety of mathematical contexts. It is defined as the positive real number The <math>\ln</math> (natural logarithm) function is equivalent to a [[logarithm]] with base <math>e</math>. In addition, the function <math>\ex
  4 KB (764 words) - 21:09, 13 March 2022
• Goldbach Conjecture
  The '''Goldbach Conjecture''' is a yet unproven [[conjecture]] stating that every [[even integer]] greater tha In 1742, the Prussian mathematician Christian Goldbach wrote a letter to [[Leonhard Euler]] in which he proposed the following conjecture:
  7 KB (1,201 words) - 16:59, 19 February 2024
• Twin Prime Conjecture
  The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[infinite]]ly many pairs of [[twin ...twin primes. Unfortunately, it has been shown that this sum converges to a constant <math>B</math>, known as [[Brun's constant]]. This could mean eit
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• Riemann Hypothesis
  The '''Riemann Hypothesis''' is a famous [[conjecture]] in [[analytic number theory]] that states that all no ...n Hypothesis would hold. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>.
  2 KB (425 words) - 12:01, 20 October 2016
• Quadratic residues
  .../math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...lo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>
  5 KB (778 words) - 13:10, 29 November 2017
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (827 words) - 17:30, 21 February 2024
• Collatz Problem
  ...ath> as it was for <math>b</math>; any time the lead coefficient still has a factor of 2. ...<cmath>{6m+4\over 2}=3m+2</cmath> we can then observe that; only if <math>m</math> is even will another division by 2 be possible.
  1 KB (231 words) - 19:45, 24 February 2020
• Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (774 words) - 12:12, 29 October 2016
• Rhombus
  ...eral]] all of whose sides are [[congruent (geometry) | congruent]]. It is a special type of [[parallelogram]], and its properties (aside from those pro * If all of a rhombus' [[angle]]s are [[right angle]]s, then the rhombus is a [[square (geometry) | square]].
  3 KB (490 words) - 15:30, 22 February 2024
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
  9 KB (1,547 words) - 03:04, 13 January 2021
• Modular arithmetic/Introduction
  ...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  15 KB (2,396 words) - 20:24, 21 February 2024
• Perfect square
  ...a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect squares are <math>0, 1, 4, 9, 16, 25, 36< An integer <math>n</math> is a perfect square [[iff]] it is a [[quadratic residue]] [[modulo]] all but finitely [[prime]]s.
  954 bytes (155 words) - 01:14, 29 November 2023
• Modular arithmetic/Intermediate
  ...or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...Z}_n</math> for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[D
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 14
  ...n </math> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> deno triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0);
  6 KB (980 words) - 21:45, 31 March 2020
• 2006 AIME I Problems/Problem 11
  ...h]] <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 the rules: * The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math>
  3 KB (436 words) - 05:40, 4 November 2022
• 2022 AIME I Problems/Problem 1
  .../math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. for some constants <math>a,b,c</math> and <math>d.</math>
  4 KB (670 words) - 13:03, 13 November 2023
• 2006 AMC 12B Problems
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  {{AMC12 Problems|year=2006|ab=A}} <math> \mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D)
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  {{AMC12 Problems|year=2002|ab=A}} <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mat
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  ...O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2002 AMC 12B Problems
  ...number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit <math>\mathrm{(A)}\ 0
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 22
  ...0^n</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by <math>10</math>. What is the smallest possible v \mathrm{(A)}\ 489
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12A Problems/Problem 16
  ...nters <math>A</math> and <math>B</math> have radius 3 and 8, respectively. A [[common internal tangent line | common internal tangent]] intersects the c pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
  2 KB (286 words) - 10:16, 19 December 2021
• 2006 AMC 12A Problems/Problem 19
  ...] to the circles can be written in the form <math>y=mx+b</math> with <math>m>0</math>. What is <math>b</math>? <math> \mathrm{(A) \ } \frac{908}{119}\qquad \mathrm{(B) \ } \frac{909}{119}\qquad \mathrm{(C
  2 KB (253 words) - 22:52, 29 December 2021
• 2006 AMC 12A Problems/Problem 23
  ...h>S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence ...ath>x>0</math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>?
  3 KB (466 words) - 22:40, 29 September 2023
• 2005 AMC 12B Problems/Problem 3
  <math>\textbf{(A) }\ \frac15 \qquad\textbf{(B) }\ \frac13 \qquad\textbf{(C) }\ \frac25 \qqua ...arrow \frac35 m = (\mbox{CDs})</math>. Thus, the money left over is <math>m-\frac35m = \frac25m</math>, so the answer is <math>\boxed{\textbf{(C) }\fra
  978 bytes (156 words) - 14:14, 14 December 2021
• 2005 AMC 12B Problems/Problem 12
  ...n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math> <math>\textbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qq
  2 KB (317 words) - 12:27, 16 December 2021
• 2005 AMC 12B Problems/Problem 19
  ...}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>? \mathrm{(A)}\ 88 \qquad
  2 KB (283 words) - 20:02, 24 December 2020
• 2005 AMC 12B Problems/Problem 21
  A positive integer <math>n</math> has <math>60</math> divisors and <math>7n</ <math>\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad
  888 bytes (140 words) - 20:04, 24 December 2020
• 2005 AMC 12B Problems/Problem 24
  ...</math> are relatively prime positive integers. What is the value of <math>m + n</math>? <math> \mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15}}}\qquad\mathrm{(C)}\ {{{16}}}\qquad\m
  4 KB (761 words) - 09:10, 1 August 2023
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...ath> with all <math>c_j \in \mathbb C</math> and <math>c_n \neq 0</math>, a general degree-<math>n</math> polynomial. The degree of <math>P(x)</math> i
  8 KB (1,427 words) - 21:37, 13 March 2022
• 2006 AMC 10A Problems
  {{AMC10 Problems|year=2006|ab=A}} <math>\mathrm{(A)}\ 31\qquad\mathrm{(B)}\ 32\qquad\mathrm{(C)}\ 33\qquad\mathrm{(D)}\ 34\qqu
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 15
  ...ns counterclockwise at <math>300 m/min</math> and uses the outer lane with a radius of <math>60</math> meters, starting on the same radial line as Odell <math>\textbf{(A) } 29\qquad\textbf{(B) } 42\qquad\textbf{(C) } 45\qquad\textbf{(D) } 47\qqu
  3 KB (532 words) - 17:49, 13 August 2023
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2006 USAMO Problems
  ...e that there exist integers <math>m</math> and <math>n</math> with <math>0<m<n<p</math> and if and only if <math>s</math> is not a divisor of <math>p-1</math>.
  3 KB (520 words) - 09:24, 14 May 2021
• 2022 USAJMO Problems/Problem 1
  For which positive integers <math>m</math> does there exist an infinite arithmetic sequence of integers <math>a <math>\bullet</math> <math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>;
  4 KB (792 words) - 00:29, 13 April 2024
• 2005 AIME II Problems/Problem 2
  ...h> and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </math> ...We need only calculate the probability the first and second person all get a roll of each type, since then the rolls for the third person are determined
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 3
  ...h> and <math> n </math> are [[relatively prime]] [[integer]]s. Find <math> m+n. </math> ...>10 = \frac a{1 + r}</math>. Then <math>a = 2005 - 2005r</math> and <math>a = 10 + 10r</math> so <math>2005 - 2005r = 10 + 10r</math>, <math>1995 = 201
  3 KB (581 words) - 07:54, 4 November 2022
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math>
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 8
  ...th> n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 12
  ...AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </math> Given that <math> B ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> ...ertices]] of the octahedron be <math>A, B, C, D, E, F</math> so that <math>A</math> and <math>F</math> are opposite each other and <math>AF = s\sqrt2</m
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME II Problems/Problem 11
  ...- \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math>m. </math> For <math>0 < k < m</math>, we have
  3 KB (499 words) - 18:52, 21 November 2022
• Line
  ...lope]] of the line and ''b'' is the [[y-intercept]]. Any two points define a line, and given specific <math>(x_1,y_1)</math> <math>(x_2,y_2)</math> on
  674 bytes (106 words) - 18:40, 9 May 2024
• 2005 AIME I Problems
  ...ngent to two circles adjacent to it. All circles are internally tangent to a circle <math> C </math> with radius 30. Let <math> K </math> be the area of ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 4
  ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 7
  ...ateral]] <math> ABCD,\ BC=8,\ CD=12,\ AD=10, </math> and <math> m\angle A= m\angle B = 60^\circ. </math> Given that <math> AB = p + \sqrt{q}, </math> wh label("$A$",(0,0),SW);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...ath> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...atement says that <math>2^{111\cdot(x_1 + x_2 + x_3)} = 4</math> so taking a [[logarithm]] of that gives <math>111(x_1 + x_2 + x_3) = 2</math> and <math
  1 KB (161 words) - 19:50, 2 January 2022
• 2005 AIME I Problems/Problem 10
  ...</math> are <math> (p,q). </math> The [[line]] containing the [[median of a triangle | median]] to side <math> BC </math> has [[slope]] <math> -5. </ma pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22);
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  ...ximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> We note that aligning the base of the semicircle with a side of the square is certainly non-optimal. If the semicircle is tangent
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math> ...ll-known that <math>\tau(n)</math> is odd if and only if <math>n</math> is a [[perfect square]]. (Otherwise, we can group [[divisor]]s into pairs whose
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 15
  ...r | divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math> pair A=(0,0),B=(26,0),C=IP(circle(A,10),circle(B,20)),D=(B+C)/2,I=incenter(A,B,C);
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  ..._2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relativ pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  pair A = origin; pair C = rotate(15,A)*(A+dir(-50));
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  ...<math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>. ...>9</math> operations; however, not all <math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradic
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 13
  ...> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> ...f the form <math>\frac m{19}</math> or <math>\frac n {17}</math> for <math>m, n > 0</math>.
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 12
  ...> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> The notation <math> [z] </math> denotes the [[floor function|gre Graphing this yields a series of [[rectangle]]s which become smaller as you move toward the [[orig
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 11
  ...ath> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...Using the [[Pythagorean Theorem]], we get <math>\ell = 5</math> and <math>A = 24\pi</math>.
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 10
  ...> and <math> n </math> are relatively prime positive integers. Find <math> m + n. </math> ...one unit away from <math>\overline{AC}</math>. Let this triangle be <math>A'B'C'</math>.
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME I Problems/Problem 9
  ...nd <math> n</math> are [[relatively prime]] positive integers. Find <math> m+n. </math> .../math> or <math>\overline{FG}</math>. <math>V_2</math> is a trapezoid with a right angle then, from which it follows that <math>V_1</math> contains one
  4 KB (618 words) - 20:01, 4 July 2013
• 2004 AIME I Problems/Problem 2
  ...and the greatest element of <math>B</math> is <math>99</math>. Find <math>m.</math> ...ath>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 5
  ...nd <math> n </math> are relatively prime positive integers. What is <math> m+n </math>? Let <math>q</math> be the number of questions Beta takes on day 1 and <math>a</math> be the number he gets right. Let <math>b</math> be the number he get
  3 KB (436 words) - 18:31, 9 January 2024
• 2004 AIME I Problems/Problem 8
  Define a regular <math> n </math>-pointed star to be the union of <math> n </math> l .../math> line segments intersects at least one of the other line segments at a point other than an endpoint,
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...A </math> and the greatest element of <math> B </math> is 99. Find <math> m. </math>
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 13
  ...> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> ...parallel CE, BC \parallel AD, </math> it follows that <math>ABCF</math> is a [[parallelogram]], and so <math>\triangle ABC \cong \triangle CFA</math>. A
  3 KB (486 words) - 22:15, 7 April 2023
• 2004 AIME II Problems/Problem 12
  ...th> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...{40} </math> whose binary expansions have exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] that A positive integer <math>n</math> has exactly two 1s in its binary representa
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 7
  ...> and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 3
  A solid rectangular block is formed by gluing together <math> N </math> [[con ...minimize <math>l\cdot m\cdot n</math> is to make <math>l</math> and <math>m</math> and <math>n</math> as close together as possible, which occurs when
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems/Problem 2
  ...ath> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...treated the choices as ordered; that is, Terry chose first one candy, then a second, and so on. We could also solve the problem using unordered choices
  2 KB (330 words) - 13:42, 1 January 2015
• 2004 AIME II Problems
  ...a, b, c, d, e, </math> and <math> f </math> are positive integers, <math> a </math> and <math> e </math> are relatively prime, and neither <math> c </m ...> and <math> n </math> are relatively prime positive integers, find <math> m+n. </math>
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B<
  7 KB (1,104 words) - 12:53, 6 July 2022
• 2019 AIME I Problems/Problem 2
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...ng how many ways there are to choose <math>2</math> distinct elements from a <math>20</math> element set such that no <math>2</math> elements are adjace
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math pair A=(0,0),B=(12,0),C=(4,5);
  6 KB (933 words) - 01:15, 19 June 2022
• 1987 AIME Problems
  ...)</math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of s ...er we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the p
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...>10^{99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>. ...C</math>, <math>\tan \angle CAB = 22/7</math>, and the altitude from <math>A</math> divides <math>BC</math> into segments of length <math>3</math> and <
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1991 AIME Problems
  Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerat Suppose <math>r^{}_{}</math> is a real number for which
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  ...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went :(a) the winner caught <math>15</math> fish;
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...\,</math> consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence ...ath>, where <math>m\,</math> and <math>n\,</math> are integers. Find <math>m + n\,</math>.
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> ...ath> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math>
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  ...<math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> ...allelogram]]. Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overli
  7 KB (1,084 words) - 02:01, 28 November 2023
• 1999 AIME Problems
  ...and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> ...all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  ...Let <math>A = (u,v)</math>, let <math>B</math> be the reflection of <math>A</math> across the line <math>y = x</math>, let <math>C</math> be the reflec ...icients of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...> and <math> n </math> are relatively prime positive integers. Find <math> m + n. </math> ...et <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? ...he largest number of students who could study both languages. Find <math>M-m</math>.
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R= ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...<math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 4
  ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< A=r*dir(45),B=(A.x,A.y-r);
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 6
  Firstly, we try to find a relationship between the numbers we're provided with and <math>49</math>. W ...ells us that <math>a^{42} \equiv 1 \pmod{49}</math> where <math>\text{gcd}(a,49) = 1</math>. Thus <math>6^{83} + 8^{83} \equiv 6^{2(42)-1}+8^{2(42)-1} <
  3 KB (361 words) - 20:20, 14 January 2023
• 1983 AIME Problems/Problem 14
  ...<math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. ...ath>C</math>, and with radius <math>6</math>) that intersects circle <math>A</math> at <math>Q</math>. The rest is just finding lengths, as follows.
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1983 AIME Problems/Problem 15
  ...s a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 5
  ...^{21}</math>. If we multiply the two equations together, we get that <math>a^4b^4 = 2^{36}</math>, so taking the fourth root of that, <math>ab = 2^9 = \ ...>\frac{\ln ab}{\ln 2} = 9</math>. The left-hand side can be interpreted as a base-2 logarithm, giving us <math>ab = 2^9 = \boxed{512}</math>.
  6 KB (863 words) - 16:10, 16 May 2024
• 1984 AIME Problems/Problem 6
  ...t <math>(14, 92)</math>, <math>(17, 76)</math>, and <math>(19, 84)</math>. A [[line]] passing through <math>(17,76)</math> is such that the total area o ...gruency]]; hence <math>M</math> lies on the line. The coordinates of <math>M</math> are <math>\left(\frac{19+14}{2},\frac{84+92}{2}\right) = \left(\frac
  6 KB (1,022 words) - 19:29, 22 January 2024
• 1984 AIME Problems/Problem 8
  ...1</math> and argument of the form <math>40m^\circ</math> for integer <math>m</math> (the ninth degree [[roots of unity]]). Now we simply need to find th {{AIME box|year=1984|num-b=7|num-a=9}}
  3 KB (430 words) - 19:05, 7 February 2023
• 1984 AIME Problems/Problem 9
  ...math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the [[volume]] of the tetrahedron in <mat path3 rightanglemark(triple A, triple B, triple C, real s=8)
  6 KB (947 words) - 20:44, 26 November 2021
• 1984 AIME Problems/Problem 11
  ...[probability]] that no two birch trees are next to one another. Find <math>m+n</math>. ...d remove <math>4</math> trees that aren't birch. What you are left with is a unique arrangement of <math>5</math> birch trees and <math>3</math> other t
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 14
  Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be ex ...prime quintuplet is <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answ
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1985 AIME Problems/Problem 10
  where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[ ...t the numbers of the form <math>\frac{m}{\textrm{lcm}(2, 4, 6, 8)} = \frac{m}{24}</math>. This gives us 24 calculations to make; we summarize the resul
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 8
  ...value]] of the difference, is as small as possible. For this minimum <math>M</math>, what is <math>100M</math>? ...6,A_7)</math> with the smallest <math>M</math>, and the rest would just be a piece of cake.
  2 KB (377 words) - 02:17, 16 February 2021
• 1986 AIME Problems/Problem 15
  ...<math>AB</math> is <math>60</math>, and that the [[median]]s through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</math> and <math> Let <math>\theta_1</math> be the angle that the median through <math>A</math> makes with the positive <math>x</math>-axis, and let <math>\theta_2<
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1986 AIME Problems/Problem 10
  ...rticipants to think of a three digit number <math>(abc)</math> where <math>a</math>, <math>b</math>, and <math>c</math> represent digits in base <math>1 ...math> be the number <math>100a+10b+c</math>. Observe that <math>3194+m=222(a+b+c)</math> so
  3 KB (565 words) - 16:51, 1 October 2023
• 1987 AIME Problems/Problem 15
  ...iangle]]s in the figure are [[similar]] to triangle <math>ABC</math>, it's a good idea to use [[area ratios]]. In the diagram above, <math>\frac {T_1}{T ...ath>, and the ratio between the sides is <math>\sqrt {441} = 21</math>. As a result, <math>AB = 21\sqrt {440} = \sqrt {AC^2 + BC^2}</math>. We now need
  5 KB (838 words) - 18:05, 19 February 2022
• 1987 AIME Problems/Problem 12
  ...ath>, where <math>n</math> is a [[positive integer]] and <math>r</math> is a [[positive]] [[real number]] less than <math>1/1000</math>. Find <math>n</m In order to keep <math>m</math> as small as possible, we need to make <math>n</math> as small as pos
  4 KB (673 words) - 19:48, 28 December 2023
• 1987 AIME Problems/Problem 11
  Let us write down one such sum, with <math>m</math> terms and first term <math>n + 1</math>: <math>3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)</math>.
  3 KB (418 words) - 18:30, 20 January 2024
• 1987 AIME Problems/Problem 7
  ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...of <math>m, j</math> equal to 3. This gives 7 possible triples <math>(j, m, p)</math>: <math>(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3
  3 KB (547 words) - 22:54, 4 April 2016
• 1987 AIME Problems/Problem 1
  ...[[non-negative]] [[integer]]s is called "simple" if the [[addition]] <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of s ...rrying over is allowed, the range of possible values of any digit of <math>m</math> is from <math>0</math> to the respective [[digit]] in <math>1492</ma
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 11
  ...mplex number]]s. A line <math>L</math> in the [[complex plane]] is called a mean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math ...>, <math>w_4 = 1 + 27i</math>, and <math>w_5 = - 14 + 43i</math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of th
  2 KB (422 words) - 00:22, 6 September 2020
• 1988 AIME Problems/Problem 5
  ...10^{99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>. ...is <math>\frac{m}{n} = \frac{144}{10000} = \frac{9}{625}</math>, and <math>m + n = \boxed{634}</math>.
  822 bytes (108 words) - 22:21, 6 November 2016
• 1989 AIME Problems/Problem 15
  ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math ...ath>YPL</math> have their areas in ratio <math>XP:PL</math> (as they share a common height from <math>Y</math>), and the same is true of triangles <math
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 14
  Given a positive [[integer]] <math>n</math>, it can be shown that every [[complex n <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 12
  Let <math>ABCD</math> be a [[tetrahedron]] with <math>AB=41</math>, <math>AC=7</math>, <math>AD=18</ma pair A,B,C,D,M,P,Q;
  2 KB (376 words) - 13:49, 1 August 2022
• 1989 AIME Problems/Problem 11
  ...ach between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference ...e let appear <math>n > 1</math> times. We let the arithmetic mean be <math>M</math>, and the sum of the numbers <math>\neq x</math> be <math>S</math>. T
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 6
  ...the same time Allie leaves <math>A</math>, Billie leaves <math>B</math> at a speed of <math>7</math> meters per second and follows the [[straight]] path pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  6 KB (980 words) - 15:08, 14 May 2024
• 1989 AIME Problems/Problem 4
  ...t <math>b+c+d</math> is a [[perfect square]] and <math>a+b+c+d+e</math> is a [[perfect cube]], what is the smallest possible value of <math>c</math>? ...re is a cubed term in <math>5^2 \cdot y^3</math>, <math>3^3</math> must be a factor of <math>c</math>. <math>3^35^2 = \boxed{675}</math>, which works as
  3 KB (552 words) - 12:41, 3 March 2024
• 1990 AIME Problems/Problem 14
  ...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 13
  ...tive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9 ...ith 9, let it be <math>9^k = m</math>. The integer <math>9^{k - 1} = \frac{m}{9}</math> must then be an n-digit number that begins with <math>1</math>.
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 8
  ...n two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules: 1) The marksman first chooses a column from which a target is to be broken.
  3 KB (491 words) - 04:24, 4 November 2022
• 2006 AMC 10B Problems/Problem 14
  ...e the roots of the equation <math> x^2-mx+2=0 </math>. Suppose that <math> a+\frac1b </math> and <math> b+\frac1a </math> are the roots of the equation <math>\textbf{(A) } \frac{5}{2}\qquad \textbf{(B) } \frac{7}{2}\qquad \textbf{(C) } 4\qquad
  2 KB (264 words) - 21:10, 19 September 2023
• 1991 AIME Problems/Problem 15
  ...2,\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> ...= 17\cdot 17 = 1\cdot 289.</math> The only possible value, then, for <math>m</math> is <math>145</math>, in which case <math>n^2 = 144</math>, and <math
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 12
  ...est terms, denote the [[perimeter]] of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. ...--B--C--D--cycle);draw(P--R..Q--S); draw(P--Q--R--S--cycle); label("\(A\)",A,NW);label("\(B\)",B,NE);label("\(C\)",C,SE);label("\(D\)",D,SW); label("\(P
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...ath>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. for (int a=1; a<24; a+=2)
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 9
  ...rac mn,</math> where <math>\frac mn</math> is in lowest terms. Find <math>m+n^{}_{}.</math> ...] root will work, so the value of <math>y = \frac{29}{15}</math> and <math>m + n = \boxed{044}</math>.
  10 KB (1,590 words) - 14:04, 20 January 2023
• 1991 AIME Problems/Problem 3
  ...this last equation and using the well-known fact <math>\log(a_{}^{}b)=\log a + \log b</math> (valid if <math>a_{}^{},b_{}^{}>0</math>), we have ...tain <math>k=m</math>, where <math> A_0 < A_1 < A_2 < \dots < A_{m-2} < A_{m-1} < A_m </math> and
  5 KB (865 words) - 12:13, 21 May 2020
• 1992 AIME Problems/Problem 15
  ...integer <math>m^{}_{}</math> such that the decimal representation of <math>m!</math> ends with exactly <math>n</math> zeroes. How many positive integers ...ght\rfloor + \left\lfloor \frac{m}{625} \right\rfloor + \left\lfloor \frac{m}{3125} \right\rfloor + \cdots</math>.
  2 KB (358 words) - 01:54, 2 October 2020
• 1992 AIME Problems/Problem 13
  ...[[distance formula]], we see that <math>\frac{ \sqrt{a^2 + b^2} }{ \sqrt{ (a-9)^2 + b^2 } } = \frac{40}{41}</math>. Simplifying gives <math>-a^2 -\frac{3200}{9}a +1600 = b^2</math>.
  4 KB (703 words) - 02:40, 29 December 2023
• 1992 AIME Problems/Problem 11
  ...ind the smallest positive integer <math>m^{}_{}</math> for which <math>R^{(m)}(l)=l^{}_{}</math>. Let <math>l</math> be a line that makes an angle of <math>\theta</math> with the positive <math>x</
  2 KB (404 words) - 19:24, 4 July 2013
• 1992 AIME Problems/Problem 9
  ...nd <math>n^{}_{}</math> are relatively prime positive integers, find <math>m+n^{}_{}</math>. ...0x=70\cdot 92,</math> hence <math>AP=x=\frac{161}{3}</math>. This gives us a final answer of <math>161+3=\boxed{164}</math>
  5 KB (874 words) - 10:27, 22 August 2021
• 1992 AIME Problems/Problem 1
  ...math>, less than <math>30</math>. Note that they come in pairs <math>(m,30-m)</math> which result in sums of <math>1</math>; thus the sum of the smalles Note that if <math>x</math> is a solution, then <math>(300-x)</math> is a solution. We know that <math>\phi(300) = 80.</math> Therefore the answer is
  1 KB (190 words) - 20:02, 23 February 2022
• 2006 AMC 10B Problems/Problem 25
  ...s. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy! <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 15
  ...>m\,</math> and <math>n\,</math> are relatively prime integers. Find <math>m + n\,</math>. pair A,B,C,H;
  3 KB (449 words) - 21:39, 21 September 2023
• 1993 AIME Problems/Problem 13
  ...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? Let <math>A</math> and <math>B</math> be Kenny's initial and final points respectively
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1993 AIME Problems/Problem 12
  ...P_n L}</math>. Given that <math>P_7 = (14,92)\,</math>, what is <math>k + m\,</math>? ...= \cfrac{\cfrac{\cfrac{P_1 + L_1}2 + L_2}2 + \cdots}{2\ldots} = \frac {(k,m)}{64} + \frac {L_1}{64} + \frac {L_2}{32} + \frac {L_3}{16} + \frac {L_4}8
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  ...elatively prime positive integers. What are the last three digits of <math>m+n\,</math>? The probability that the <math>n</math>th flip in each game occurs and is a head is <math>\frac{1}{2^n}</math>. The first person wins if the coin lands
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 8
  .../math> represents the same selection as the pair <math>\{b, c, d, e, f\},\{a, c\}.</math> ...ach possible selection is double counted, except the case where both <math>m</math> and <math>n</math> contain all <math>6</math> elements of <math>S.</
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 6
  ...ying, <math>9a = 10b + 9 = 11c + 19</math>. The relationship between <math>a,\ b</math> suggests that <math>b</math> is divisible by <math>9</math>. Als ...information given, it can be determined that, for positive integers <math>a, \ b, \ c</math>:
  3 KB (524 words) - 18:06, 9 December 2023
• 1993 AIME Problems/Problem 4
  ...,b,c,d)\,</math> with <math>0 < a < b < c < d < 500\,</math> satisfy <math>a + d = b + c\,</math> and <math>bc - ad = 93\,</math>? ...>(k-c)c - a(k-a) = (a-c)(a+c-k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - c) = (1,93),(3,31),(31,3),(93,1)</math>.
  8 KB (1,343 words) - 16:27, 19 December 2023
• 1994 AIME Problems/Problem 11
  ...imes10''\times19'',</math> are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes <math>4 ...h>15</math> are all multiples of <math>3</math>, the change will always be a multiple of <math>3</math>, so we just need to find the number of changes w
  4 KB (645 words) - 15:12, 15 July 2019
• 1994 AIME Problems/Problem 10
  ...> and <math>n\,</math> are relatively prime positive integers. Find <math>m+n.\,</math> ...= \frac{BC}{AB} = \frac{29^2 x}{29x^2} = \frac{29}{421}</math>, and <math>m+n = \boxed{450}</math>.
  3 KB (534 words) - 16:23, 26 August 2018
• 1994 AIME Problems/Problem 8
  The points <math>(0,0)\,</math>, <math>(a,11)\,</math>, and <math>(b,37)\,</math> are the vertices of an equilateral ...<math>b+37i</math> is then a rotation of <math>60</math> degrees of <math>a+11i</math> about the origin, so:
  5 KB (788 words) - 13:53, 8 July 2023
• 1994 AIME Problems/Problem 2
  ...}</math>, where <math>m</math> and <math>n</math> are integers. Find <math>m + n</math>. ...at point <math>E</math>), and draw <math>\overline{AO}</math>. We now have a [[right triangle]], with [[hypotenuse]] of length <math>20</math>. Since <m
  2 KB (272 words) - 03:53, 23 January 2023
• 1995 AIME Problems/Problem 15
  ...> and <math>n_{}</math> are relatively prime positive integers, find <math>m+n</math>. ...ince the first letter could be <tt>T</tt> or the sequence could start with a block of <tt>H</tt>'s, the total probability is that <math>3/2</math> of it
  6 KB (979 words) - 13:20, 11 April 2022
• 1995 AIME Problems/Problem 14
  ..._{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> ...), D=E+48*expi(7*pi/6), A=E+30*expi(5*pi/6), C=E+30*expi(pi/6), F=foot(O,B,A);
  3 KB (484 words) - 13:11, 14 January 2023
• 1995 AIME Problems/Problem 12
  ...th> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> // projection of point A onto line BC
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 9
  ...,</math> where <math>a</math> and <math>b</math> are integers. Find <math>a+b.</math> ...$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE);
  7 KB (1,181 words) - 13:47, 3 February 2023
• 1995 AIME Problems/Problem 7
  ...<math>m_{}</math> and <math>n_{}</math> [[relatively prime]], find <math>k+m+n.</math> {{AIME box|year=1995|num-b=6|num-a=8}}
  3 KB (427 words) - 09:23, 13 December 2023
• 1995 AIME Problems/Problem 5
  For certain real values of <math>a, b, c,</math> and <math>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d ...jugates of <math>m,n</math>. Let <math>m'</math> be the conjugate of <math>m</math>, and <math>n'</math> be the conjugate of <math>n</math>. Then,
  3 KB (451 words) - 15:02, 6 September 2021
• 1995 AIME Problems/Problem 3
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math> ...math>2!</math> ways to determine the order of the remaining two steps, for a total of <math>12</math> sequences that we have to exclude. This gives <mat
  3 KB (602 words) - 23:15, 16 June 2019
• 1995 AIME Problems/Problem 1
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> The sum of the areas of the [[square]]s if they were not interconnected is a [[geometric sequence]]:
  2 KB (302 words) - 19:29, 4 July 2013
• 1996 AIME Problems/Problem 14
  A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is ...oordinates of the diagonally opposite corner of the rectangle, where <math>a, b, c \in \mathbb{Z_{+}}</math>.
  5 KB (923 words) - 21:21, 22 September 2023
• 1996 AIME Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. pair B=(0,0), C=(15^.5, 0), A=IP(CR(B,30^.5),CR(C,6^.5)), E=(B+C)/2, D=foot(B,A,E);
  3 KB (521 words) - 01:18, 25 February 2016
• 1996 AIME Problems/Problem 12
  ...ath>\frac {330 \cdot 8! \cdot 5}{10!} = \frac {55}{3}</math>, and <math>m+n = \boxed{058}</math>. ...her way to think about this is that the total sum is 330. Because you have a total of <math>10 \cdot 9</math> possible "pairs" or "differences", the ave
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 6
  ...math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>. ...}{2} = 10</math> games in total, and every game can either end in a win or a loss. Therefore, there are <math>2^{10} = 1024</math> possible outcomes.
  3 KB (461 words) - 00:33, 16 May 2024
• 1997 AIME Problems/Problem 14
  ...h>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n</math>. Now, let <math>v</math> be the root corresponding to <math>m\theta=2m\pi/1997</math>, and let <math>w</math> be the root corresponding t
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 6
  ...1</math>, <math>A_n</math>, and <math>B</math> are consecutive vertices of a regular polygon? ...A_1A_n = \frac{(n-2)180}{n}</math>, <math>\angle A_nA_1B = \frac{(m-2)180}{m}</math>, and <math>\angle A_2A_1B = 60^{\circ}</math>. Since those three an
  3 KB (497 words) - 00:39, 22 December 2018
• 1997 AIME Problems/Problem 4
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> So <math>m+n = \boxed{17}</math>.
  2 KB (354 words) - 22:33, 2 February 2021
• 1997 AIME Problems/Problem 2
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard,
  3 KB (416 words) - 21:09, 27 October 2022
• 1998 AIME Problems/Problem 14
  ...r box, where <math>m, n,</math> and <math>p</math> are integers, and <math>m\le n\le p.</math> What is the largest possible value of <math>p</math>? <cmath>2mnp = (m+2)(n+2)(p+2)</cmath>
  2 KB (390 words) - 21:05, 29 May 2023
• 1998 AIME Problems/Problem 9
  ...math>c</math> is not divisible by the square of any [[prime]]. Find <math>a + b + c.</math> ...9 a.m.). The two mathematicians meet each other when <math>|M_1-M_2| \leq m</math>. Also because the mathematicians arrive between 9 and 10, <math>0 \l
  4 KB (624 words) - 18:34, 18 February 2018
• 1998 AIME Problems/Problem 4
  ...<math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> In order for a player to have an odd sum, he must have an odd number of odd tiles: that is
  5 KB (917 words) - 02:37, 12 December 2022
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ..."\(D\)",foot(A,B,C),NE);label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><a
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 14
  ...<math>n_{}</math> are relatively [[prime]] positive integers. Find <math>m+n.</math> pair A=(0,0),B=(13,0),C=IP(circle(A,15),circle(B,14));
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 13
  ...eam plays every other team exactly once. No ties occur, and each team has a <math>50 \%</math> chance of winning any game it plays. The [[probability] ...t beat the teams with 1 and 0 wins, and so on; thus, this uniquely defines a combination.
  2 KB (329 words) - 01:38, 6 October 2015
• 1999 AIME Problems/Problem 11
  ...prime positive integers that satisfy <math>\frac mn<90,</math> find <math>m+n.</math> ...tric identity|identity]] <math>\sin a \sin b = \frac 12(\cos (a-b) - \cos (a+b))</math>, we can rewrite <math>s</math> as
  4 KB (614 words) - 04:38, 8 December 2023
• 1999 AIME Problems/Problem 10
  ..._{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n.</math> ...<math>{10\choose3}</math> sets of 3 points which form triangles. However, a fourth distinct segment must also be picked. Since the triangle accounts fo
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 9
  ...and <math>n_{}</math> are relatively prime positive integers, find <math>m+n.</math> ...= (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 4
  ..._{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n.</math> pair A=intersectionpoint(Y--Z, y--z),
  3 KB (398 words) - 13:27, 12 December 2020
• 1999 AIME Problems/Problem 2
  ..._{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n</math>. ...45 + \frac{135}{19}}{10} = \frac{99}{19}</math>, and the solution is <math>m + n = \boxed{118}</math>.
  3 KB (423 words) - 11:06, 27 April 2023
• 1999 AIME Problems/Problem 8
  ...> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> This problem just requires a good diagram and strong 3D visualization.
  3 KB (445 words) - 19:40, 4 July 2013
• 2000 AIME I Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...,0)</math>. All these circles are [[homothety|homothetic]] with respect to a center at <math>(5,0)</math>.
  3 KB (571 words) - 00:38, 13 March 2014
• 2000 AIME I Problems/Problem 11
  .../math> be the sum of all numbers of the form <math>a/b,</math> where <math>a</math> and <math>b</math> are [[relatively prime]] positive [[divisor]]s of ...here <math>-3 \le x,y \le 3</math>. Thus every number in the form of <math>a/b</math> will be expressed one time in the product
  4 KB (667 words) - 13:58, 31 July 2020
• 2000 AIME I Problems/Problem 10
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>m + n</math>. {{AIME box|year=2000|n=I|num-b=9|num-a=11}}
  2 KB (319 words) - 22:26, 29 December 2022
• 2000 AIME I Problems/Problem 8
  ...math>p</math> is not divisible by the cube of any prime number. Find <math>m + n + p</math>. ...3}{4}\right)^{3}\right)^{1/3}=12-3\left(37^{1/3}\right)</math>. Thus <math>m+n+p=\boxed{052}</math>.
  4 KB (677 words) - 16:33, 30 December 2023
• 2000 AIME I Problems/Problem 7
  ...and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + n</math>. Thus, <math>z+\frac1y=\frac{5}{24}+\frac{1}{24}=\frac{1}{4}</math>, so <math>m+n=\boxed{005}</math>.
  5 KB (781 words) - 15:02, 20 April 2024
• 2000 AIME I Problems/Problem 5
  ...d <math>n</math> are [[relatively prime]] positive integers. What is <math>m + n</math>? ...e quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle of Inclusion-Exclusion]] still requires us to fin
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2001 AIME I Problems/Problem 15
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> ...hedron <math>DBEG</math>, and four are <math>\textit{long}</math>, joining a vertex of one tetrahedron to the diagonally opposite point from the other.
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 14
  ...get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail d ...represent a house that does not receive mail and <math>1</math> represent a house that does receive mail. This problem is now asking for the number of
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2001 AIME I Problems/Problem 13
  ...where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> Note that a cyclic quadrilateral in the form of an isosceles trapezoid can be formed fr
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circu
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 10
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> ...\frac {5\cdot 8\cdot 13 - 60}{60\cdot 59} = \frac {23}{177}\Longrightarrow m+n = \boxed{200}</math>.
  8 KB (1,187 words) - 02:40, 28 November 2020
• 2001 AIME I Problems/Problem 9
  ...and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n</math>. ...A=(0,0),B=(13,0),C=IP(CR(A,17),CR(B,15)), D=A+p*(B-A), E=B+q*(C-B), F=C+r*(A-C);
  4 KB (673 words) - 20:15, 21 February 2024
• 2001 AIME I Problems/Problem 7
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C);
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2001 AIME I Problems/Problem 6
  ...th>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m + n</math>. ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME I Problems/Problem 5
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. Denote the vertices of the triangle <math>A,B,</math> and <math>C,</math> where <math>B</math> is in [[quadrant]] 4 and
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2002 AIME I Problems/Problem 13
  ...and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. ...,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C));
  6 KB (974 words) - 13:01, 29 September 2023
• 2002 AIME I Problems/Problem 11
  ...and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. When a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for t
  3 KB (591 words) - 15:11, 21 August 2019
• 2002 AIME I Problems/Problem 4
  ...tegers <math>m</math> and <math>n</math> with <math>m<n</math>, find <math>m+n</math>. ...dfrac{1}{m+1}-\dfrac{1}{m+2}+\cdots +\dfrac{1}{n-1}-\dfrac{1}{n}=\dfrac{1}{m}-\dfrac{1}{n}</math>
  2 KB (320 words) - 07:55, 4 November 2022
• 2002 AIME I Problems/Problem 1
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> ...<math>\frac{10 \times 10}{10^3} = \frac 1{10}</math>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindrome
  3 KB (369 words) - 23:36, 6 January 2024
• 2003 AIME I Problems/Problem 15
  ...> and <math> n </math> are relatively prime positive integers. Find <math> m + n. </math> ...name of a point represent the mass located there. Since we are looking for a ratio, we assume that <math>AB=120</math>, <math>BC=169</math>, and <math>C
  8 KB (1,382 words) - 14:23, 29 December 2022
• 2003 AIME I Problems/Problem 14
  ...and <math> n </math> are [[relatively prime]] positive integers and <math> m < n, </math> contains the digits <math>2, 5</math>, and <math>1</math> cons ...<math>0</math> and <math>1</math>, we can rewrite this as <math>\frac{10^k m - nX}{n} = \frac{p}{q}</math>, where <math>q \le n</math>. Then the fractio
  3 KB (477 words) - 14:23, 4 January 2024
• 2003 AIME I Problems/Problem 11
  ...h> are [[positive integer]]s with <math> m + n < 1000, </math> find <math> m + n. </math> ...ssions and those three lengths not forming a [[triangle]] is equivalent to a violation of the [[triangle inequality]]
  2 KB (284 words) - 13:42, 10 October 2020
• 2003 AIME I Problems/Problem 10
  ...h> AC = BC </math> and <math> \angle ACB = 106^\circ. </math> Point <math> M </math> is in the interior of the triangle so that <math> \angle MAC = 7^\c pair A=(0,0), B=(2,0), C=(1,Tan(37)), M=IP(A--(2Cos(30),2Sin(30)),B--B+(-2,2Tan(23)));
  7 KB (1,058 words) - 01:41, 6 December 2022
• 2003 AIME I Problems/Problem 6
  ...re <math> m, n, </math> and <math> p </math> are [[integer]]s. Find <math> m + n + p. </math> ...e face diagonal and one space diagonal of the cube) and those which lie in a plane [[oblique]] to the edges of the cube, whose sides are three face diag
  3 KB (477 words) - 18:35, 27 December 2021
• 2003 AIME I Problems/Problem 5
  ...math> n </math> and <math> p </math> are [[relatively prime]], find <math> m + n + p. </math> ...allelepipeds that each share a face with the large parallelepiped and have a height of <math>1</math>, the <math>1/8</math> [[sphere]]s (one centered at
  2 KB (288 words) - 19:58, 4 July 2013
• 2003 AIME I Problems/Problem 2
  ...ath> n </math> are [[relatively prime]] [[positive integer]]s. Find <math> m + n. </math> For any <math>a</math>, <math>a^{2}-(a-1)^{2}=a^{2}-(a^{2}-2a+1)=2a-1</math>. We can cancel the [[divisor | factor]] of [[pi]] fr
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 14
  ...ive integers and n is not divisible by the square of any prime. Find <math>m + n.</math> ..., with height <math>8</math> and base <math>\frac{8}{\sqrt{3}}</math>) and a parallelogram (<math>ABDE</math>, with height <math>8</math> and base <math
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2003 AIME II Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>m + n.</math> ...o return to the original vertex, the net number of clockwise steps must be a multiple of 3, i.e., <math>\#CW - \#CCW \equiv 0 \pmod{3}</math>. Since <m
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2003 AIME II Problems/Problem 11
  ...and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> ...{5\sqrt{11}} {2}</math>. Since <math>ABC</math> is a right triangle, <math>M</math> is the circumcenter and thus, <math>CM=\frac{25} {2}</math>. We let
  5 KB (772 words) - 19:47, 1 August 2023
• 2003 AIME II Problems/Problem 6
  ...on of the two regions enclosed by the triangles <math>ABC</math> and <math>A'B'C'?</math> Since a <math>13-14-15</math> triangle is a <math>5-12-13</math> triangle and a <math>9-12-15</math> triangle "glued" together on the <math>12</math> side,
  5 KB (787 words) - 17:38, 30 July 2022
• 2003 AIME II Problems/Problem 4
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. <math>m+n = 1 + 27 = \boxed{28}</math>.
  3 KB (563 words) - 17:36, 30 July 2022
• 2002 AIME II Problems/Problem 15
  ...me, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...<math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• 2002 AIME II Problems/Problem 14
  ...and <math>n</math> are [[relatively prime]] positive integers, find <math>m+n</math>. ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr
  4 KB (658 words) - 19:15, 19 December 2021
• 2002 AIME II Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. pair A,B,C,D,E,X,P,Q,R;
  6 KB (935 words) - 13:23, 3 September 2021
• 2002 AIME II Problems/Problem 11
  .../math>, <math>n</math>, and <math>p</math> are positive integers and <math>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</ ...> Now we have that <math>ar^2 = \frac18</math> and we also have that <math>a+r = 1</math>. We can solve for <math>r</math> and the only appropriate valu
  4 KB (696 words) - 16:27, 22 March 2022
• 2002 AIME II Problems/Problem 10
  ...ath>, <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>. ...pi + \frac{\pi x}{180} \Rightarrow x = \frac{360\pi}{180-\pi} \Rightarrow (m,n) = (360,180)</math>.
  2 KB (336 words) - 19:30, 24 June 2020
• 2002 AIME II Problems/Problem 5
  ...-negative integers, for which <math>a^6</math> is not a divisor of <math>6^a</math>. ...n,m) for which <math>(2^n3^m)^{6}</math> is not a divisor of <math>6^{2^n3^m}</math>
  3 KB (515 words) - 14:46, 14 February 2021
• 2002 AIME II Problems/Problem 4
  ...blocks that are hexagons <math>1</math> unit on a side are used to outline a garden by placing the blocks edge to edge with <math>n</math> on each side. ...where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>.
  2 KB (268 words) - 07:28, 13 September 2020
• 2001 AIME II Problems/Problem 15
  ...and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. ...ath>E</math>, where the drilling starts, is at <math>(8,8,8)</math>. Using a little visualization (involving some [[similar triangles]], because we have
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 14
  ...>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < ...ich is equal to <math>\boxed {840}</math> degrees. We only want the sum of a certain number of theta, not all of it.
  2 KB (380 words) - 15:03, 22 July 2018
• 2001 AIME II Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. pair A=(0,0), D=(x,0), B=IP(CR(A,8),CR(D,10)), E=(-3x/5,0), C=IP(CR(E,16),CR(D,64/5));
  4 KB (743 words) - 03:32, 23 January 2023
• 2001 AIME II Problems/Problem 12
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...hedra. However, the volume of the tetrahedra being constructed decrease by a factor of <math>\frac 18</math>. Thus we have the recursion <math>\Delta P_
  2 KB (380 words) - 00:28, 5 June 2020
• 2001 AIME II Problems/Problem 11
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...ac{1 - \frac{141}{729}}{2} = \frac{98}{243}</math>, so the answer is <math>m+n = \boxed{341}</math>.
  3 KB (415 words) - 23:25, 20 February 2023
• 2001 AIME II Problems/Problem 9
  ...and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + n</math>. ...riction. Therefore, the probability of obtaining a grid that does not have a <math>2 \times 2</math> red square is <math>\frac{417}{512}</math>, and <ma
  8 KB (1,207 words) - 20:04, 5 September 2023
• 2001 AIME II Problems/Problem 6
  ...elatively prime positive integers and <math>m < n</math>. Find <math>10n + m</math>. .../math>. By the [[Pythagorean Theorem]], the radius of <math>\odot O = OC = a\sqrt{2}</math>.
  4 KB (772 words) - 19:31, 6 December 2023
• 2001 AIME II Problems/Problem 4
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...,\frac{3b}{10}\right)</math>. By the midpoint formula, we have <math>\frac{a+b}2=8</math> and <math>\frac{15a}{16}+\frac{3b}{20}=6</math>. Solving for <
  2 KB (240 words) - 20:34, 4 July 2013
• 2001 AIME II Problems/Problem 2
  ...he largest number of students who could study both languages. Find <math>M-m</math>. For <math>m = S \cap F</math> to be smallest, <math>S</math> and <math>F</math> must be
  2 KB (252 words) - 00:54, 10 January 2024
• 2000 AIME II Problems/Problem 14
  ...m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math> is an integer, <math>0\le f_i Thus for all <math>m\in\mathbb{N}</math>,
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2000 AIME II Problems/Problem 13
  ...and <math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>. ...ic (or in this case, nearly symmetric) polynomials is to divide through by a power of <math>x</math> with half of the polynomial's degree (in this case,
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 12
  ...and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>. ...eron's Formula]] the area of <math>\triangle ABC</math> is (alternatively, a <math>13-14-15</math> triangle may be split into <math>9-12-15</math> and <
  3 KB (532 words) - 13:14, 22 August 2020
• 2000 AIME II Problems/Problem 11
  ...ath> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...on of <math>D</math> across that perpendicular. Then <math>ABCD'</math> is a [[parallelogram]], and <math>\overrightarrow{AB} = \overrightarrow{D'C}</ma
  4 KB (750 words) - 22:55, 5 February 2024
• 2000 AIME II Problems/Problem 3
  ...and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + n.</math> ...s, the answer is <math>\frac{54+1}{703} = \frac{55}{703}</math>, and <math>m+n = \boxed{758}</math>.
  1 KB (191 words) - 04:27, 4 November 2022
• 2000 AIME II Problems/Problem 1
  ...th> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. Therefore, <math> m+n=1+6=\boxed{007}</math>
  2 KB (292 words) - 13:33, 4 April 2024
• 2005 AMC 10B Problems/Problem 14
  ...lateral <math> \triangle ABC</math> has side length <math>2</math>, <math> M</math> is the midpoint of <math> \overline{AC}</math>, and <math> C</math> pair A = 2*dir(60);
  5 KB (882 words) - 22:12, 30 April 2024
• 2005 AMC 10B Problems/Problem 24
  ...math> satisfy <math>x^2 - y^2 = m^2</math> for some positive integer <math>m</math>. What is <math>x + y + m</math>?
  5 KB (845 words) - 19:23, 17 September 2023
• 2006 USAMO Problems/Problem 1
  ...that there exist integers <math>m</math> and <math>n</math> with <math>0 < m < n < p</math> and if and only if <math>s </math> is not a divisor of <math>p-1 </math>.
  3 KB (506 words) - 17:54, 22 June 2023
• 2006 USAMO Problems/Problem 3
  ...ath>m</math>, let <math>p(m)</math> be the greatest prime divisor of <math>m</math>. By convention, we set <math>p(\pm 1)=1</math> and <math>p(0)=\infty Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  ...ow that if <math>k\ge 2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math ...igher power of 2). Thus we must have <math>M < i_0</math>, since otherwise a number divisible by <math>2^{i_0}</math> is visited before <math>2^{i_0} k_
  7 KB (1,280 words) - 17:23, 26 March 2016
• 2006 USAMO Problems/Problem 6
  ...th>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point. ...imilarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>.
  5 KB (986 words) - 22:46, 18 May 2015
• 2006 AMC 12A Problems/Problem 3
  <math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 24\qqu ...rac{m}{30}=\frac{3}{5}</math>. Solving for <math>m</math>, we obtain <math>m=\boxed{\textbf{(B) }18}.</math>
  1 KB (155 words) - 17:30, 16 December 2021
• 2006 AMC 12A Problems/Problem 7
  <math> \mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ ...>. Therefore, <math>2.32d=23.2</math>, and <math>d=10</math>. Then <math>m=.72(10)=7.2</math>. Mary will be <math>8</math> on her next birthday. The
  901 bytes (144 words) - 22:46, 5 February 2016
• Centroid
  ...fectly balanced. (The centroid does not have to be in the figure, however. A condition under which the centroid must be inside the figure is when the fi ...midpoint of the side opposite that [[vertex]]. Also, the three medians of a triangle divide the triangle into six regions of equal [[area]].
  3 KB (612 words) - 22:32, 25 February 2024
• Ring
  ...lgebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multipl * <math>(R,\cdot)</math> is a [[monoid]];
  6 KB (994 words) - 06:16, 8 April 2015
• Holomorphic function
  A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a rather than at points, for when we consider the behavior of a function
  9 KB (1,537 words) - 21:04, 26 July 2017
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• 2005 AMC 10B Problems
  ...t a price of five for &#36;<math>2</math>. They sell all the candy bars at a price of two for &#36;<math>1</math>. What was the profit, in dollars? <math>\mathrm{(A)} 100 \qquad \mathrm{(B)} 200 \qquad \mathrm{(C)} 300 \qquad \mathrm{(D)} 4
  12 KB (1,874 words) - 21:20, 23 December 2020
• 2000 AMC 12 Problems/Problem 8
  <math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{ ...qual to <math>4</math>. Thus, the original sequence can be generated from a quadratic function.
  8 KB (1,202 words) - 16:17, 10 May 2024
• Angle bisector theorem
  ...ng with [[Stewart's theorem]] it can be shown that <math>AD^2 = b\cdot c - m \cdot n</math> ...^^C^^D,blue);label("$b$",(A+C)/2,(1,0));label("$c$",(A+B)/2,(0,1));label("$m$",(B+D)/2,(0,-1));label("$n$",(D+C)/2,(0,-1)); </asy>
  3 KB (438 words) - 14:20, 4 March 2023
• Sigma notation
  '''Sigma notation''', also known as ''summation notation'', provides a method for writing long, complicated, sometimes [[infinite]] sums neatly an ...sum_{k=m}^n a(k)</math> for the sum <math>a(m)+a(m+1)+a(m+2)+\ldots+a(n-1)+a(n)</math>.
  2 KB (335 words) - 17:17, 8 February 2024
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 1
  ...math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note <math>[ABC]</math> denotes the area of <math>\triangle ABC< ...ave <math>17^2 = y^2 - x^2 = (y - x)(y + x)</math>, a [[factor]]ization as a product of two different [[positive integer]]s, so we must have <math>y - x
  2 KB (329 words) - 15:53, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 2
  ...is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>. ...gives us <math>m = 18564 - 7 - 42 - 42 - 105 = 18368</math> so <math>\star(m) = 1 + 8 + 3 + 6 + 8 = 026</math>.
  1 KB (188 words) - 15:53, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 7
  ...where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math> (note: <math>[ABC]</math> denotes the area of <math>\triangle ABC< ...lic, since <math>\angle AEB=\angle AFB</math>. We then have, from Power of a Point, that <math>CE\cdot CA=CF\cdot CB</math>. In other words, <math>1\cdo
  3 KB (518 words) - 16:54, 25 November 2015
• Mock AIME 1 2006-2007 Problems/Problem 8
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. pair A=expi(pi*5/12)+expi(0)+expi(pi/2), B=expi(pi*5/12), C=(0,0), D=expi(0), E=ex
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• Mock AIME 1 2006-2007 Problems/Problem 9
  ...istinct]] values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s, c ...distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then comp
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• Mock AIME 1 2006-2007 Problems/Problem 10
  ...where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>. Radius <math>a=\frac{3}{7}</math>, radius <math>b=\frac{6}{11}</math>, radius <math>c=\fra
  1 KB (236 words) - 23:58, 24 April 2013
• Mock AIME 1 2006-2007 Problems/Problem 12
  ...atisfies <math>14m+1=k</math>. Find the number of possible values of <math>m</math> between <math>0</math> and <math>10^{2007}</math>. ...}</math> and the order of 10 mod 13 is 6. Thus, we get one value of <math>m</math> each time <math>n = 6j + 1</math>. There are <math>335</math> such
  2 KB (249 words) - 18:14, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...f(m,49)</math> is an integer, find the sum of all possible values of <math>m</math>. ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• Mock AIME 1 2006-2007 Problems/Problem 15
  ...are relatively prime positive integers. Compute the last 3 digits of <math>m+n</math> ...ath>'s in one section (Which means that it considers <math>10</math> to be a digit). Thus, we need to subtract off the invalid "numbers" for the overcou
  3 KB (414 words) - 13:45, 19 February 2016
• Mock AIME 1 2006-2007/Problems
  ...math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note that <math>[ABC]</math> denotes the area of <math>\triangle ...is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>.
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 3
  ...[positive integer]]s <math>n</math> such that <math>n^2+12n-2007</math> is a [[perfect square]]. Find the [[remainder]] when <math>S</math> is divided b ...</math> and [[factoring]] the left-hand side, we get <math>(n + m + 6)(n - m + 6) = 2043</math>. <math>2043 = 3^2 \cdot 227</math>, which can be split
  1 KB (198 words) - 10:50, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 7
  ...is <math>m\sqrt{n},</math> where <math>n</math> is squarefree, find <math>m+n.</math> ...ther. Since <math>34\pi</math> is <math>1/3</math> of the circumference of a circle with radius <math>51</math>, we must have that <math>\angle AOB=\fra
  1 KB (231 words) - 18:10, 10 July 2014
• Mock AIME 2 2006-2007 Problems/Problem 12
  ...th>a,b,c</math> are [[relatively prime]] [[positive integer]]s, find <math>a+b+c.</math> <math>m\angle DAC=m\angle DBC \Rightarrow ABCD</math> is a cylic quadrilateral.
  2 KB (311 words) - 10:53, 4 April 2012
• Mock AIME 2 2006-2007 Problems
  ...r positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. ...ll positive integers <math>n</math> such that <math>n^2+12n-2007</math> is a perfect square. Find the remainder when <math>S</math> is divided by <math>
  5 KB (848 words) - 23:49, 25 February 2017
• Mock AIME 2 2006-2007 Problems/Problem 13
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math> Let the aces divide the 48 other cards into 5 "urns", with a, b, c, d, and e non-aces between each of these 5 dividers, respectively.
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• Expected value
  Given an event with a variety of different possible outcomes, the '''expected value''' is what on A video that goes over the type of Expected value, practical examples, and pr
  5 KB (789 words) - 20:56, 10 May 2024
• Rolle's Theorem
  Let <math>f:[a,b]\rightarrow\mathbb{R}</math> ...h>f</math> be continous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>
  1 KB (270 words) - 12:13, 30 May 2019
• 1959 IMO Problems
  <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 admitted for square root
  3 KB (480 words) - 11:57, 17 September 2012
• 1959 IMO Problems/Problem 3
  Let <math>a,b,c</math> be real numbers. Consider the quadratic equation in <math>\cos{x <math>a\cos ^{2}x + b\cos{x} + c = 0.</math>
  2 KB (300 words) - 03:20, 1 August 2019
• 1977 Canadian MO Problems
  ...uation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> ...on the circumference of the circle such that the angle <math>OPA</math> is a maximum.
  3 KB (560 words) - 19:23, 10 March 2015
• 1959 IMO Problems/Problem 4
  Construct a right triangle with a given hypotenuse <math>c</math> such that the median drawn to the hypotenus ...segment from any point on the circle to the midpoint of the hypotenuse is a radius.)
  6 KB (939 words) - 17:31, 15 July 2023
• 1959 IMO Problems/Problem 5
  ...respective centers <math>P </math> and <math>Q </math>, intersect at <math>M </math> and also at another point <math>N </math>. Let <math>N' </math> de (a) Prove that the points <math>N </math> and <math>N' </math> coincide.
  2 KB (408 words) - 01:40, 2 January 2023
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 7/Problem 1
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of LC, th
  2 KB (288 words) - 21:17, 11 October 2013
• 2006 Romanian NMO Problems/Grade 8/Problem 2
  Let <math>n</math> be a positive integer. Prove that there exists an integer <math>k</math>, <math> ...frac{(X-Y)^{2}-(X+Y)}{2} \\ &= \frac{m^{2}-\left(m+\binom{m}{2} - n+\binom{m}{2} - n\right)}{2} \\ &= n.\end{align*}</cmath>
  1 KB (261 words) - 00:08, 6 June 2011
• 2006 Romanian NMO Problems/Grade 9/Problem 2
  ...BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the [[midpoint]] of <math>BC</math>. The points <math>E,F,P</math ...ath>\triangle BDC</math>. Now assume that P' is the point such that DP' is a median of <math>\triangle EDF</math> (it is on <math>EF</math>). It is on D
  3 KB (488 words) - 14:05, 15 December 2022
• Midpoint
  In [[Euclidean geometry]], the '''midpoint''' of a [[line segment]] is the [[point]] on the segment equidistant from both endp ...M</math>, <math>AM=BM=\frac{1}{2}AB</math>. Alternatively, any point <math>M</math> on <math>\overline{AB}</math> such that <math>AM=BM</math> is the mi
  4 KB (596 words) - 17:09, 9 May 2024
• P versus NP
  .... <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math> ver Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2005 AMC 10A Problems/Problem 1
  <math> \textbf{(A) } 2\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 10\qqu Let <math>m</math> be Mike's bill and <math>j</math> be Joe's bill.
  909 bytes (134 words) - 19:05, 25 December 2022
• 2005 AMC 10A Problems/Problem 7
  ...les apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When the <math> \textbf{(A) } 4\qquad \textbf{(B) } 5\qquad \textbf{(C) } 6\qquad \textbf{(D) } 7\qqua
  946 bytes (151 words) - 17:28, 10 August 2023
• 2005 AMC 10A Problems/Problem 10
  ...one solution for <math>x</math>. What is the sum of those values of <math>a</math>? <math> \textbf{(A) }-16\qquad\textbf{(B) }-8\qquad\textbf{(C) } 0\qquad\textbf{(D) }8\qquad\t
  3 KB (443 words) - 18:05, 29 March 2023
• Pi notation
  ...present the product of terms. It is similar to [[Sigma notation]] but uses a capital letter Pi. ...e product <math>a(m)\cdot a(m+1)\cdot a(m+2)\cdot a(m+3)\cdots a(n-1)\cdot a(n)</math>.
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• 2015 IMO Problems/Problem 6
  ...cmath>for all integers <math>m</math> and <math>n</math> such that <math>n>m\ge N</math>.<br /><br /> ...'''</u> ''Let <math>T</math> be a non-negative integer parameter. If given a sequence <math>a_1,a_2,\dots</math> that satisfies the conditions:''<br />
  4 KB (833 words) - 01:33, 31 December 2019
• Volume
  The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volu The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math>
  3 KB (523 words) - 20:24, 17 August 2023
• Committee forming
  First, we'll introduce committee forming with a simple example. How many committees of 3 people can be formed from a group of 12 people?
  3 KB (485 words) - 19:49, 16 July 2018
• Perpendicular bisector
  ...the '''perpendicular bisector''' of a [[line segment]] <math>AB</math> is a [[line]] <math>l</math> such that <math>AB</math> and <math>l</math> are [[ ...hat plane. The [[set]] of lines which are perpendicular bisectors of form a plane which is the plane perpendicularly bisecting <math>AB</math>.
  2 KB (367 words) - 15:20, 1 January 2014
• Orthocenter
  The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude|altitudes]]. I ''Note: The orthocenter's existence is a trivial consequence of the trigonometric version of [[Ceva's Theorem]]; how
  5 KB (829 words) - 13:11, 20 February 2024
• Free group
  ...rics]]. Let <math>I</math> be any [[nonempty]] index [[set]]. Informally, a free group on <math>I</math> is the collection of [[finite]] strings of cha ...group operation repeatedly in this process.) Showing that this string is a right inverse is equally straightforward. The proof that this holds for ev
  2 KB (454 words) - 17:54, 16 March 2012
• Angle
  An '''angle''' is the [[union]] of two [[ray]]s with a common [[endpoint]]. The common endpoint of the rays is called the [[verte ...most common form is <math>\angle ABC</math>, read "angle ABC", where <math>A,C</math> are points on the sides of the angle and <math>B</math> is the ver
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• 1970 IMO Problems
  Let <math>M</math> be a point on the side <math>AB</math> of <math>\triangle ABC</math>. Let <math ...1}</math> and <math>A_{n}</math> are numbers in the system with base <math>a</math> and <math>B_{n-1}</math> and <math>B_{n}</math> are numbers in the s
  3 KB (558 words) - 00:17, 10 December 2022
• 1970 IMO Problems/Problem 1
  Let <math>M</math> be a point on the side <math>AB</math> of <math>\triangle ABC</math>. Let <math <math> r \left[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) \right] = c</math>,
  2 KB (380 words) - 22:12, 19 May 2015
• Multiplicative function
  ...m \cdot n) = f(m) \cdot f(n)</math> for all ''[[relatively prime]]'' <math>m, n</math>. ...on the [[real numbers]] by <math>f(x) = x^2</math> is a simple example of a multiplicative function.
  3 KB (450 words) - 12:59, 21 July 2009
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>
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• 1981 IMO Problems/Problem 3
  ...ng <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. ...h>, since if we had <math>n < m</math>, then <math>n^2 -nm -m^2 = n(n-m) - m^2 </math> would be the sum of two negative integers and therefore less than
  1 KB (248 words) - 10:23, 13 May 2019
• 1987 IMO Problems/Problem 6
  ...rime. Since there must be a factor of <math>m</math> less than <math>\sqrt{m}</math>. Now if <math>N</math> has a factor which divides <math>2r-i</math> in the range <math>-2s</math> to <ma
  2 KB (430 words) - 13:03, 24 February 2024
• 1981 IMO Problems/Problem 4
  ...nsecutive [[positive integer]]s such that the largest number in the set is a [[divisor]] of the [[least common multiple]] of the remaining <math>n-1</ma ...3 must be divisible by a number that is greater than two and is a power of a prime.
  3 KB (516 words) - 09:43, 28 March 2012
• Roman numerals
  1000 - '''M''' ''(mille)'' ...for 4, IV is used. To "subtract," a smaller-value letter is placed before a larger-value letter. This done to make numbers smaller (e.g. IX instead of
  865 bytes (140 words) - 13:58, 24 March 2019
• Asymptote: Basics
  ...nded with the "[/asy]" tag. Each command in Asymptote must be separated by a semicolon (;), similar to programming languages like C and Java. This conve To write comments (Lines that are not interpreted by Asymptote), start a line with two forward slashes.
  5 KB (801 words) - 20:44, 6 May 2024
• Least upper bound
  ...h>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>, it is also the [[maximum]] 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
• United States of America Computing Olympiad
  ...Olympiad''' includes a series of seven internet programming contests and a training camp, used to eventually choose the team that will represent the U ...4 problems lasting 4-5 hours. Competitors may participate at any time over a weekend for four hours as long as those hours are continuous.
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• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
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• 1961 IMO Problems/Problem 1
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  5 KB (916 words) - 18:15, 26 March 2024
• Nine-point circle
  ...s ''Euler's circle'' or ''Feuerbach's circle'') of a given [[triangle]] is a circle which passes through 9 "significant" points: "The nine-point circle is tangent to the incircle, has a radius equal to half the circumradius, and its center is the midpoint of th
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• 2003 AMC 10A Problems
  {{AMC10 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
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• Fermat point
  ...>\triangle ABC</math> (with no angle more than <math>120^{\circ}</math> is a point <math>P</math> which has the minimum total distance to three [[vertic pair A=(2,4), B=(1,1), C=(6,1);
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• Balkan MO Problems and Solutions, with authors
  Triangle ABC has a right angle at C. The internal bisectors of angles BAC and ABC meet BC and CA at P and Q respectively. The points M and N are the feet of the perpendiculars from P and Q
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• Chinese Remainder Theorem
  The '''Chinese Remainder Theorem''' is a [[number theory | number theoretic]] result. ...esidue class mod <math>m</math> with a residue class mod <math>n</math> is a residue class mod <math>mn</math>.
  6 KB (1,022 words) - 14:57, 6 May 2023
• Matrix
  ...where <math>A</math> and <math>B</math> are the sets <math>A=\{1,2,\ldots,m\}</math> and <math>B=\{1,2,\ldots,n\}</math>. ...order is said to be <math>m \times n</math>, and it is written as <math>A_{m \times n}</math>.
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• 1985 IMO Problems
  A circle has center on the side <math>AB</math> of the cyclic quadrilateral < ...y prime natural numbers, <math>k < n</math>. Each number in the set <math>M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white. It is g
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• 1985 IMO Problems/Problem 2
  ...y prime natural numbers, <math>k < n</math>. Each number in the set <math>M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white. It is g (i) for each <math> i \in M </math>, both <math>i </math> and <math>n-i </math> have the same color;
  1 KB (267 words) - 00:11, 12 July 2020
• 1985 IMO Problems/Problem 3
  ...<math>P</math> of degree less than <math>2^m </math>, <math> w(P\cdot Q_{2^m}) = 2w (P)</math>. {{IMO box|year=1985|num-b=2|num-a=4}}
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• 2001 AMC 8 Problems
  ...class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ball. If it takes him <math>2</math> seconds to paint one dimple, how <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D)
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• 1985 AIME Problems
  When a right triangle is rotated about one leg, the volume of the cone produced is ...>b</math>, and <math>c</math> are positive integers which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>.
  7 KB (1,071 words) - 19:24, 23 February 2024
• Pythagorean triple
  ...a triple of [[positive integer]]s, <math>(a, b, c)</math> such that <math>a^2 + b^2 = c^2</math>. Pythagorean triples arise in [[geometry]] as the sid ...2 = c^2</math> defining the Pythagorean triples satisfies <math>a, b < c < a + b</math>. Thus, any triple of positive integers satisfying this equation
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• 2003 AMC 12A Problems/Problem 5
  ...MC10</math> and <math>AMC12</math> is <math>123422</math>. What is <math>A+M+C</math>? <math> \mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 11\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D)
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• 2005 Alabama ARML TST Problems/Problem 3
  ...>300\pi</math> square units. Find the number of units in the [[length]] of a [[edge | side]] of the [[triangle]]. ...lso the [[centroid]] of the triangle and <math>COM</math> is a [[median of a triangle | median]], so <math>O</math> trisects <math>CO</math> and <math>R
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• Perfect power
  ...''. For example, <math>64 = 8^2 = 4^3 = 2^6</math>, so <math>64</math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th power. ...erfect <math>1</math>st power" is a meaningless property: every integer is a <math>1</math>st power of itself.
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• 1987 IMO Problems/Problem 2
  ...</math>, the feet of these perpendiculars being <math>K </math> and <math>M </math> respectively. Prove that the quadrilateral <math>AKNM </math> and ...e cyclic quadrilateral <math>ABNC</math> gives <math>\angle BNC=180-\angle A</math>. Thus, we are to prove that
  4 KB (736 words) - 15:39, 21 September 2014
• 1987 IMO Problems/Problem 4
  ...ath>f(f(n)) = n + k</math> for all <math>n</math>, where <math>k</math> is a fixed positive integer, then <math>k</math> must be even. If <math>k = 2h</ ...1)) = m + k(s + 2)</math>. But <math>f(m + k) = n + k</math>, so <math>n = m + k(s + 1) > n</math>. Contradiction.
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• 1987 IMO Problems
  ...ath> onto itself. An element <math>i </math> in <math>S </math> is called a fixed point of the permutation <math>f </math> if <math>f(i) = i </math>.) ...</math>, the feet of these perpendiculars being <math>K </math> and <math>M </math> respectively. Prove that the quadrilateral <math>AKNM </math> and
  3 KB (459 words) - 14:24, 17 September 2023
• 2006 Seniors Pancyprian/2nd grade/Problem 4
  ...adrilateral <math>ABCD</math>, that has no parallel sides, is inscribed in a circle, its sides <math>DA</math>, <math>CB</math> meet at <math>E</math> a .../math> intersect the sides of the quadrilateral at the points <math>K , L, M ,N</math> prove that
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• 2002 IMO Shortlist Problems/N3
  .../math>, then <math>km </math> has at least twice as many divisors as <math>m </math>. ..., <math>ka </math> is clearly a divisor of <math>km </math>, but not <math>m </math>.
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• De Moivre's Theorem
  '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It a ...ath>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer.
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• Mock AIME 4 2006-2007 Problems/Problem 15
  ...f any [[prime]]. Find the greatest [[integer]] less than or equal to <math>m + \sqrt{n}</math>. We set up a trivial coordinate bash. Let A = 0,0, C = 48,0, B = 83/2, 13sqrt3/2. We find the coordinates of the circum
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• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} <math>\textbf{(A)}\ {-}\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{
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• 2006 Canadian MO Problems/Problem 2
  Let <math>ABC</math> be an [[acute angle]]d [[triangle]]. Inscribe a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is ...}{AB}</math> and let <math>H</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. Then by [[similarity]], <math>\frac{AE}{AC} = \
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• 2006 Canadian MO Problems/Problem 5
  The vertices of a right triangle <math>ABC</math> inscribed in a circle divide the circumference into three arcs. ...ht angle is at <math>A</math>, so that the opposite arc <math>BC</math> is a semicircle while arc <math>AB</math> and arc <math>AC</math> are
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• 2000 AMC 12 Problems/Problem 12
  ...the maximum value of [mathjax]A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C[/mathjax]?<!-- don't remove the following tag, for PoTW on the Wiki [katex] \mathrm{(A) \ 62 } \qquad \mathrm{(B) \ 72 } \qquad \mathrm{(C) \ 92 } \qquad \mathrm{
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• 2007 AIME II Problems/Problem 10
  .../math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0 or 6 elements, probability: <math>\frac{2}{2^6} =
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• Greatest lower bound
  ...h>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>, it is also the [[minimum]] of <math>S</math>. If <math>m \not\in S</math>, then <math>S</math> has no minimum.
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• 2005 AMC 10A Problems/Problem 23
  ...ts on the circle such that <math>DC \perp AB</math> and <math>DE</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
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• 2002 USAMO Problems/Problem 5
  ...h>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by ...ive]] (<math> a \leftrightarrow b \leftrightarrow c </math> implies <math> a \leftrightarrow c </math> ).
  7 KB (1,194 words) - 15:39, 28 March 2015
• Mock AIME 1 Pre 2005 Problems
  ...and <math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 2 Pre 2005/Problems
  ...y that <math>x + \frac{1}{x} = 3</math>. Let <math>S_m = x^m + \frac{1}{x^m}</math>. Determine the value of <math>S_7</math>. ...and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>.
  6 KB (990 words) - 15:23, 11 November 2009
• Mock AIME 3 Pre 2005 Problems
  A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib
  5 KB (795 words) - 16:03, 17 October 2021
• Mock AIME 3 Pre 2005 Problems/Problem 9
  ...and <math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. ...rpendicular from D to AB N, and the foot of the perpendicular from C to AB M. By similarity, AN=2x/17. Also, AM=x/2. Since <math>\triangle</math>AND and
  2 KB (278 words) - 16:32, 27 December 2019
• Mock AIME 3 Pre 2005 Problems/Problem 11
  ...> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. ...ath>B</math> to <math>E</math> and <math>F</math> to <math>D</math>, so by a fundamental theorem of spiral similarity <math>\triangle BDE\sim\triangle F
  2 KB (325 words) - 15:32, 22 March 2015
• Mock AIME 3 Pre 2005 Problems/Problem 15
  ...> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. The fraction looks remarkably apt for a trigonometric substitution; namely, define <math>\theta</math> such that <m
  2 KB (312 words) - 10:38, 4 April 2012
• 2007 AIME I Problems/Problem 8
  We can see that <math>Q_1</math> and <math>Q_2</math> must have a [[root]] in common for them to both be [[factor]]s of the same cubic. Let this root be <math>a</math>.
  4 KB (728 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 15
  ...l, and <math>r</math> is an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math>r</math>. Denote the length of a side of the triangle <math>x</math>, and of <math>\overline{AE}</math> as <
  4 KB (673 words) - 22:14, 6 August 2022
• 2004 USAMO Problems/Problem 4
  ...from on to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players. ...ce writes a number in a useless square, then Bob writes a higher number in a non-useless square in the same row on his next turn.
  2 KB (430 words) - 13:40, 4 July 2013
• 2007 AIME II Problems/Problem 4
  ...n produce <math>150</math> widgets and <math>m</math> whoosits. Find <math>m</math>. ...nto the third equation to find that <math>1050 = 150 + 2m</math>, so <math>m = \boxed{450}</math>.
  1 KB (207 words) - 11:12, 20 April 2024
• 2007 AIME II Problems/Problem 5
  ...tively prime]], we don’t have to worry about crossing an intersection of a horizontal and vertical line at one time. We must also account for the firs ...e formed. In the second row, we see that the line <math>y = 8</math> gives a <math>x</math> value of <math>\frac{2007 - 8(223)}{9} = 24 \frac 79</math>,
  4 KB (559 words) - 00:38, 3 January 2023
• 2007 AIME II Problems/Problem 12
  Suppose that <math>x_0 = a</math>, and that the common [[ratio]] between the terms is <math>r</math>. ...<math>a</math> and <math>r</math> must be powers of 3. Denote <math>3^x = a</math> and <math>3^y = r</math>. We find that <math>8x + 28y = 308</math>.
  5 KB (829 words) - 12:22, 8 January 2024
• 2007 AIME II Problems/Problem 15
  ...and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math> pair A,B,C,X,Y,Z,P,Q,R;
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems/Problem 14
  Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> ...2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1</math>. Because <math>f(0) = 1</math>, the product of all the roots
  7 KB (1,335 words) - 17:44, 25 January 2022
• 2007 AIME II Problems
  ...rs among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains ...of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2005 USAMO Problems
  ...sible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. ...th>. Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle.
  4 KB (609 words) - 09:24, 14 May 2021
• 2002 USA TST Problems
  Let <math> \displaystyle ABC </math> be a triangle. Prove that \cos\frac{A-B}{2} + \cos\frac{B-C}{2} + \cos\frac{C-A}{2}.
  3 KB (544 words) - 06:58, 3 August 2017
• 1961 IMO Problems/Problem 5
  ...h> where ''M'' is the midpoint of ''BC''. Prove that the construction has a solution if and only if ...arc BD we have <math>\angle BPD = \omega</math>. Draw a circle with center A and radius AC, and the point of intersection of this circle and the major a
  1 KB (205 words) - 04:12, 7 June 2021
• Newton's Inequality
  ..., x'_{m-1} </math> with the same symmetric averages <math> d_0, \ldots, d_{m-1} </math>. ...ve, real roots, i.e., for some non-negative reals <math> x'_1, \ldots, x'_{m-1} </math>,
  5 KB (830 words) - 04:05, 28 January 2023
• 1989 USAMO Problems/Problem 2
  ...laying in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players. ...e keep deleting edges of vertices with degree greater than <math>1</math> (a maximum of <math>8</math> such edges), then we are left with at least <math
  3 KB (438 words) - 01:19, 27 December 2023
• 2003 USAMO Problems
  A convex polygon <math> \mathcal{P} </math> in the plane is dissected into sm A = a_0,a_1,a_2,\dots, a_n
  3 KB (487 words) - 09:21, 14 May 2021
• 2003 USAMO Problems/Problem 1
  ...+ a </math> is divisible by 5, which is true when <math> k \equiv -3^{n-1}a \pmod{5} </math>. Since there is an odd digit in each of the residue class ...even digits. We could then halve all of the digits in that number to get a second multiple of <math> 5^n</math> with at most n digits that only uses t
  4 KB (736 words) - 22:17, 3 March 2023
• 2003 USAMO Problems/Problem 3
  A = a_0,a_1,a_2,\dots, a_n t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)
  3 KB (636 words) - 13:39, 4 July 2013
• 2007 USAMO Problems
  Let <math>n</math> be a positive integer. Define a sequence by setting <math>a_1=n</math> and, for each <math>k>1</math>, lett ...the Euclidean plane consists of all points <math>(m,n)</math>, where <math>m</math> and <math>n</math> are integers. Is it possible to cover all grid po
  3 KB (539 words) - 13:42, 4 July 2013
• 2007 USAMO Problems/Problem 1
  ...m Vandervelde'') Let <math>n</math> be a [[positive]] [[integer]]. Define a [[sequence]] by setting <math>a_1 = n</math> and, for each <math>k>1</math> ...integers, so <math>b_{k+1} \le b_k</math>. As the <math>b_k</math>'s form a [[non-increasing]] sequence of positive integers, they must eventually beco
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2007 USAMO Problems/Problem 2
  ...clidean plane]] consists of all [[point]]s <math>(m,n)</math>, where <math>m</math> and <math>n</math> are [[integer]]s. Is it possible to cover all gr ...ngent]] circles with radius greater than or equal to 5, one can always fit a circle with radius greater than <math>\frac{1}{\sqrt{2}}</math> between tho
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 3
  (''András Gyárfás'') Let <math>S</math> be a [[set]] containing <math>n^2+n-1</math> [[element]]s, for some [[positive]] ...ad <math>k(n+1)-1</math> elements, then we must have k disjoint subsets in a class.
  6 KB (1,071 words) - 08:40, 21 October 2020
• 2007 USAMO Problems/Problem 5
  ...nteger <math>m</math> then <math>(x^7 + 1)/(x + 1)</math> is composite. As a consequence, <math>x^7 + 1</math> has at least two more prime factors than &= \{(x + 1)^3 - 7^m(x^2 + x + 1)\}\{(x + 1)^3 + 7^m(x^2 + x + 1)\}.
  2 KB (240 words) - 09:47, 7 August 2014
• Nesbitt's Inequality
  ...ely cited, has many instructive proofs. It states that for positive <math>a, b, c </math>, \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge \frac{3}{2}
  7 KB (1,224 words) - 16:21, 24 October 2022
• 2007 BMO Problems/Problem 3
  Find all positive integers <math>n </math> such that there exists a permutation <math>\sigma </math> on the set <math> \{ 1, \ldots, n \} </mat is a rational number.
  4 KB (682 words) - 10:53, 13 January 2016
• 2007 Cyprus MO/Lyceum/Problems
  <math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ -2\qquad\mathrm{(C)}\ 1\qquad\mathrm{(D)}\ -1\qquad <math>\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 4^x\qquad\mathrm{(C)}\ 2\cdot4^x\qquad\mathrm{(D)}\
  13 KB (1,990 words) - 08:29, 19 December 2009
• 2007 Cyprus MO/Lyceum/Problem 21
  ...\,\mathrm{m}</math>, <math>AC=4\,\mathrm{m}</math> and <math>BC=5\,\mathrm{m}</math>, then the length of the strap is ...4\pi)\,\mathrm{m}\qquad \mathrm{(D) \ } \left(12+\frac\pi5\right)\,\mathrm{m}\qquad \mathrm{(E) \ } \mathrm{None\,of\,these}</math>
  1 KB (175 words) - 22:21, 23 July 2020
• 2007 Cyprus MO/Lyceum/Problem 22
  ...h>AB = a</math> , <math>DC = 2a</math> . If <math>AD = 3a</math> and <math>M</math> is the midpoint of the side <math>BC</math>, then <math>AM</math> eq <math> \mathrm{(A) \ } \frac{3a}{2}\qquad \mathrm{(B) \ } \frac{3a}{\sqrt{2}}\qquad \mathrm{(
  979 bytes (166 words) - 02:33, 19 January 2024
• Pascal's Theorem
  pair A=dir(20), B=dir(40), D(L(D(IP(D(L(A,B,0.5,7)),D(L(E,D,0.5,1.5)))),
  4 KB (712 words) - 21:57, 25 April 2020
• 2004 IMO Shortlist Problems/N2
  ...le \psi(mn) = \psi(m)\psi(n) </math> for every two relatively prime <math> m,n \in \mathbf{N} </math>. <br> ...thbf{N} </math> the equation <math> \displaystyle \psi(x) = ax </math> has a solution. <br>
  6 KB (1,007 words) - 09:10, 29 August 2011
• Dirichlet convolution
  .../math> be relatively prime. We wish to prove that <math>(f*g)(mn) = (f*g)(m)\cdot (f*g)(n) </math>. ...</math>, <math>d_md_n = d </math>. Thus <math>p </math> is bijective. As a result of our claim, we have the identity
  3 KB (613 words) - 21:40, 21 June 2009
• 2005 IMO Shortlist Problems/C2
  ...th> persons buy sombreros (directly or indirectly), then that someone wins a free instructional video. Prove that if <math> \displaystyle n </math> per ...>. Since this is clear for <math>m=0</math>, let us WLOG assume that <math>m</math> is positive. Under this assumption, we will now prove the stronger b
  3 KB (538 words) - 09:12, 29 August 2011
• Difference of squares
  ...ic | arithmetic modulo]] <math>m</math> for any [[positive integer]] <math>m</math>, or more generally in any [[commutative ring]]. However, due to [[ma
  738 bytes (108 words) - 01:27, 25 January 2023
• LaTeX:Layout
  ...his allows you to use the same code in many source files by just including a single line in each source file. ...font size, which is 10pt by default but can be increased to 11pt or 12pt. A reference on other options for this command can be found [http://www.nada.k
  30 KB (5,171 words) - 10:16, 4 April 2021
• LaTeX:Commands
  | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2} Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than
  12 KB (1,898 words) - 15:31, 22 February 2024
• LaTeX:Help
  ==== I'm using a Macintosh. What do I do? ==== ...stall LaTeX and imagemagick on your server to use this. That site also has a TeX Converter that doesn't require LaTeX and Imagemagick.
  7 KB (1,109 words) - 20:01, 8 September 2022
• Chicken McNugget Theorem
  ...>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. ...The proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.
  17 KB (2,748 words) - 19:22, 24 February 2024
• Sophie Germain Identity
  <div style="text-align:center;"><math>a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)</math></div> ...he [[factoring]], we begin by [[completing the square]] and then factor as a [[difference of squares]]:
  2 KB (222 words) - 15:04, 30 December 2023
• 2007 AMC 12A Problems/Problem 22
  <math>\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \ :Subcase a: <math>T(n) = 1970 + u + 17 + u + 8 + u = 1995 + 3u = 2007 \Longrightarrow
  15 KB (2,558 words) - 19:33, 4 February 2024
• 2005 AMC 12A Problems/Problem 8
  Let <math>A,M</math>, and <math>C</math> be digits with <cmath>(100A+10M+C)(A+M+C) = 2005</cmath>
  635 bytes (96 words) - 21:19, 3 July 2013
• Green's Theorem
  '''Green's Theorem''' is a result in [[real analysis]]. It is a special case of [[Stokes' Theorem]].
  2 KB (381 words) - 12:12, 30 May 2019
• 2006 AIME II Problems/Problem 6
  ...d <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> pair B = (0, 0), C = (1, 0), D = (1, 1), A = (0, 1);
  7 KB (1,067 words) - 12:23, 8 April 2024
• 2006 AIME II Problems/Problem 5
  ...d <math> n </math> are [[relatively prime]] positive integers, find <math> m+n. </math> ...th>A</math> and a 6 on die <math>B</math> or a 6 on die <math>A</math> and a 1 on die <math>B</math>:
  5 KB (712 words) - 12:10, 5 November 2023
• 2006 AIME II Problems/Problem 15
  ...<math> n </math> is not divisible by the square of any prime, find <math> m+n.</math> Let <math>\triangle XYZ</math> be a triangle with sides of length <math>x, y</math> and <math>z</math>, and sup
  4 KB (725 words) - 17:18, 27 June 2021
• 2006 AIME II Problems/Problem 10
  ...> and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> ...We let this probability be <math>p</math>; then the probability that <math>A</math> and <math>B</math> end with the same score in these five games is <m
  6 KB (983 words) - 13:42, 8 December 2021
• 2006 AIME II Problems
  ...vex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \ang ...\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1988 IMO Problems/Problem 6
  ...>b</math> be positive integers such that <math>ab + 1</math> divides <math>a^{2} + b^{2}</math>. Show that \frac {a^{2} + b^{2}}{ab + 1}
  4 KB (720 words) - 12:26, 7 April 2024
• 1998 PMWC Problems
  Find a pair of consecutive Triangular Numbers and the difference between a pair of consecutive Square Numbers whose difference are both <math>11</math ...<math>OB = OA=3</math> cm. <math>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2003 PMWC Problems
  pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); draw((M--N)^^(origin--a)^^(origin--b));
  7 KB (918 words) - 16:15, 22 April 2014
• 1971 Canadian MO Problems
  <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</mat <math>ABCD</math> is a quadrilateral with <math>AD=BC</math>. If <math>\angle ADC</math> is greate
  3 KB (519 words) - 08:58, 13 September 2012
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• 1961 IMO Problems/Problem 6
  ...o not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? ...\prime B^\prime C^\prime</math>. It is obvious that the centroid of <math>A^\prime B^\prime C^\prime</math> can be any point on plane <math>\epsilon</m
  2 KB (301 words) - 23:29, 18 July 2016
• 1999 USAMO Problems/Problem 5
  ...ut producing SOS then the game is a draw. Prove that the second player has a winning strategy. ...if <math>l > m</math> he fills an S in <math>l + 3</math> and if <math>l < m</math> he fills an S in <math>l-3</math>.
  2 KB (433 words) - 13:35, 4 July 2013
• 1990 USAMO Problems
  A certain state issues license plates consisting of six digits (from 0 throug A sequence of functions <math>\, \{f_n(x) \} \,</math> is defined recursively
  3 KB (386 words) - 20:47, 3 July 2013
• 1999 AHSME Problems
  <math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm <math>\text{(A) All equilateral triangles are congruent to each other.}</math>
  13 KB (1,945 words) - 18:28, 19 June 2023
• Proof by contrapositive
  '''Proof by contrapositive''' is a method of [[proof writing|proof]] in which the [[contrapositive]] of the de ...= 2m+1</math>. Now then, we compute the sum <math>x+y = 2k + 2m + 1 = 2(k+m) + 1</math>, which is an odd integer by definition.
  1 KB (262 words) - 17:31, 21 October 2009
• Mandelbrot historical results
  The following is a collection of results for the [[Mandelbrot Competition]]. <tr><td>Jeffrey Chen</td><td>A&M Consolidated High School</td><td align=center>14</td><td align=center>44</t
  16 KB (2,223 words) - 14:34, 25 November 2009
• 1960 IMO Problems/Problem 1
  ...sible by <math>11</math>. Thus we get: <math>b = a + c</math> or <math>b = a + c - 11</math>. ...>q = 0,1,2,3,4</math>. Then <cmath>10a+2q=2a^2+4aq+8q^2</cmath><cmath>5a+q=a^2+2aq+4q^2</cmath>
  4 KB (751 words) - 05:01, 17 August 2022
• Asymptote: CSE5
  ...of two paths without common points, which may be useful if you try to draw a picture just from its verbal description and do not want to make any calcul CSE5 contains doubles of each function; a function with a full name and a function with an abbreviated name.
  4 KB (607 words) - 15:09, 30 June 2020
• Factor Theorem
  ...theorem''' is a theorem regarding the relationships between the factors of a polynomial and its roots. ...y, you can determine whether a number in the form <math>f(a)</math> (<math>a</math> is constant, <math>f</math> is polynomial) is <math>0</math> using p
  3 KB (508 words) - 12:31, 2 May 2024
• Summation
  A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notatio ...+c_{b}</math>. Here <math>i</math> refers to the index of summation, <math>a</math> is the lower bound, and <math>b</math> is the upper bound.
  3 KB (482 words) - 16:39, 8 October 2023
• Isaac Newton
  '''Isaac Newton''' (1643 – 1727) was a famous British [[physics|physicist]] and [[mathematician]]. His most famous ...irst, though he made use of rather odd notation. Ironically, Newton earned a knighthood from Queen Anne in 1705 not because of this great discovery, but
  2 KB (391 words) - 13:36, 22 November 2020
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Let <math>A</math> be a [[partially ordered set]].
  9 KB (1,669 words) - 19:02, 1 August 2018
• Relation
  ...e structure that can be placed on a pair or collection of [[set]]s to give a notion of "relatedness" to the [[element]]s of those sets. ...ldots, m - 1 \}</math> that the two numbers are congruent [[modulo]] <math>m</math>.
  2 KB (445 words) - 08:16, 21 August 2009
• Ideal
  In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorp ...xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as
  8 KB (1,389 words) - 23:44, 17 February 2020
• 2004 AMC 12A Problems/Problem 23
  A [[polynomial]] Which of the following quantities can be a non zero number?
  3 KB (517 words) - 14:13, 5 September 2021
• 2004 AMC 12A Problems/Problem 25
  ...ositive integers and <math>n</math> is as small as possible. What is <math>m</math>? <math>\text {(A)} 98 \qquad \text {(B)} 101 \qquad \text {(C)} 132\qquad \text {(D)} 798\qq
  3 KB (480 words) - 14:50, 17 August 2020
• Work
  ...er a certain distance. In the case of a constant force <math>F</math> over a distance <math>d</math>, the amount of work done is found using the equatio ...done (J), <math>F</math> is the force (N), <math>d</math> is the distance (m), and <math>\theta</math> is the angle between the direction of force and t
  1 KB (192 words) - 00:46, 16 July 2018
• 1995 AHSME Problems/Problem 4
  ...<math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math> <math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C)
  2 KB (232 words) - 22:30, 16 February 2018
• Mass
  ...e of the [[gravity]] on a body, mass is the amount of matter. Note that in a standard gravitational field, mass is directly [[proportion]]al to weight. ...s calculated by <math>M=mv</math>, where <math>M</math> is momentum, <math>m</math> is mass, and <math>v</math> is [[velocity]].
  1 KB (188 words) - 22:44, 10 October 2013
• 1976 USAMO Problems/Problem 5
  prove that <math>x-1</math> is a factor of <math>P(x)</math>. ...>m</math> is an integer less than <math>n</math> and <math>P_0, \dotsc, P_{m-1}</math> and <math>S</math> are polynomials satisfying
  3 KB (572 words) - 17:14, 16 August 2015
• Gravity
  ...is defined as the force of attraction between two bodies with [[mass]] as a result of the curving of [[space]]-[[time]] about them. It is the weakest o *The attractive force in [[Newton]]s between two bodies with masses <math>m</math> and <math>n</math> whose center of masses are distance <math>d</math
  2 KB (318 words) - 17:36, 10 March 2014
• Graph (graph theory)
  ...[[empty set | nonempty]] [[set]] of [[vertex|vertices]] that are joined by a number (possibly zero) of [[edge]]s. Graphs are frequently represented gra ...' 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 graphs with multiple
  8 KB (1,428 words) - 10:26, 27 August 2020
• 2000 AMC 12 Problems/Problem 13
  ...ut were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? <math>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qqu
  6 KB (996 words) - 18:14, 29 June 2023
• 2000 AMC 12 Problems/Problem 21
  ...right triangles. The area of one of the two small right triangles is <math>m</math> times the area of the square. The [[ratio]] of the area of the other ...xtbf {(A)}\ \frac{1}{2m+1} \qquad \textbf {(B)}\ m \qquad \textbf {(C)}\ 1-m \qquad \textbf {(D)}\ \frac{1}{4m} \qquad \textbf {(E)}\ \frac{1}{8m^2}</ma
  5 KB (804 words) - 01:22, 13 May 2024
• 2000 AMC 12 Problems/Problem 24
  ...ters at <math>B</math> and <math>A</math>, respectively, then there exists a circle tangent to both <math>\overarc {AC}</math> and <math>\overarc{BC}</m label("A", (0,0), W);
  2 KB (263 words) - 19:59, 18 April 2024
• 2007 AMC 10A Problems/Problem 23
  ...[[ordered pair]]s <math>(m,n)</math> of positive [[integer]]s, with <math>m \ge n</math>, have the property that their squares differ by <math>96</math <math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad
  2 KB (352 words) - 14:20, 3 July 2023
• 2007 AMC 10A Problems/Problem 17
  ...that <math>75m = n^{3}</math>. What is the minimum possible value of <math>m + n</math>? <math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qq
  1 KB (201 words) - 08:04, 11 February 2023
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 2007 AMC 10A Problems/Problem 10
  The Dunbar family consists of a mother, a father, and some children. The [[average]] age of the members of the family <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad
  2 KB (411 words) - 20:54, 19 August 2023
• 1995 AHSME Problems
  ...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1951 AHSME Problems
  The percent that <math>M</math> is greater than <math>N</math> is: ...\qquad \textbf{(D) \ } \frac {M - N}{M} \qquad \textbf{(E) \ } \frac {100(M + N)}{N} </math>
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1990 USAMO Problems/Problem 4
  Let a <math>k</math>-good sequence be a sequence of distinct integers <math>\{ a_i \}_{i=1}^k</math> such that for ...a</math>, and furthermore, the terms of each of these sequences constitute a permutation of <math>k</math> consecutive integers.
  3 KB (529 words) - 19:15, 18 July 2016
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  <math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1991 USAMO Problems/Problem 4
  ...h>\,n\,</math> are positive integers. Prove that <math>\,a^m + a^n \geq m^m + n^n</math>. ...h to analyze the ratio <math>\,(a^N - N^N)/(a-N),</math> for real <math>\, a \geq 0 \,</math> and integer <math>\, N \geq 1</math>.]
  2 KB (349 words) - 07:39, 19 July 2016
• 1991 USAMO Problems/Problem 3
  ...m statement is false for some integer <math>n \ge 1</math>. Then there is a least <math>n</math>, which we call <math>b</math>, for which the statement ...statement is false for <math>n=k</math>. Since <math>k<b</math>, this is a contradiction. Therefore the problem statement is true. <math>\blacksquar
  3 KB (559 words) - 19:18, 5 June 2023
• 1991 USAMO Problems/Problem 2
  where "<math>\Sigma</math>" denotes a sum involving all nonempty subsets <math>S</math> of <math>\{1,2,3, \ldots, Let <math>N(m)</math> denote the set <math>\{1, \dotsc, m\}</math>. Since <math>\sigma(\varnothing)</math>, the empty sum, is equal
  3 KB (490 words) - 07:38, 19 July 2016
• 1991 USAMO Problems
  ...1991]] [[USAMO]]. There were five questions administered in one three-and-a-half-hour session. ...th> angle <math>\,C\,</math> is obtuse, and the three side lengths <math>\,a,b,c\,</math> are integers. Determine, with proof, the minimum possible per
  3 KB (512 words) - 19:17, 18 July 2016
• 2002 AMC 12B Problems/Problem 20
  ...ht-angled triangle]] with <math>m\angle XOY = 90^{\circ}</math>. Let <math>M</math> and <math>N</math> be the [[midpoint]]s of legs <math>OX</math> and <math>\mathrm{(A)}\ 24
  3 KB (447 words) - 15:02, 17 August 2023
• 2002 AMC 12B Problems/Problem 23
  ...th>\overline{BC}</math> and the [[Median of a triangle|median]] from <math>A</math> to <math>\overline{BC}</math> have the same length. What is <math>BC <math>\mathrm{(A)}\ \frac{1+\sqrt{2}}{2}
  4 KB (598 words) - 09:45, 23 January 2024
• 2002 AMC 12B Problems/Problem 1
  ...number <math>M</math>, all of whose digits are distinct. The number <math>M</math> doesn't contain the digit <math>\mathrm{(A)}\ 0
  1 KB (185 words) - 19:54, 23 June 2023
• 2008 iTest Problems/Problem 97
  ...ted, and no two people are ever on the same spot. If <math>m</math> leaves a remainder of <math>1</math> when divided by <math>5</math>, how many possib ...the number of ways to do so with <math>n</math> students. Then, if we add a <math>n+1</math>st student at the right end, that student can either stay s
  3 KB (497 words) - 13:29, 20 October 2019
• 2001 AMC 12 Problems/Problem 4
  <math>\textbf{(A)}\ 5\qquad \textbf{(B)}\ 20\qquad \textbf{(C)}\ 25\qquad \textbf{(D)}\ 30\q ...> be the mean of the three numbers. Then the least of the numbers is <math>m-10</math>
  2 KB (311 words) - 21:53, 10 February 2024
• 2007 AMC 10A Problems/Problem 18
  ...at <math>\overline{AG}</math> and <math>\overline{CH}</math> meet at <math>M</math>. What is the area of quadrilateral <math>ABCM</math>? pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1
  6 KB (867 words) - 00:17, 20 May 2023
• 1974 USAMO Problems/Problem 1
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. ...P</math> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</cmath> (Why?)
  7 KB (1,291 words) - 20:30, 27 April 2020
• 1995 USAMO Problems
  A calculator is broken so that the only keys that still work are the <math>\, ...math>\, CC_2 \,</math> are concurrent, i.e. these three lines intersect at a point.
  3 KB (540 words) - 13:31, 4 July 2013
• 2004 AMC 12B Problems/Problem 23
  <math>\mathrm{(A)}\ 250,\!000 ...mber of possible products <math>t = rs = r(1002 - r)</math>. Because <math>m=t+1002^2</math> is an integer, we also note that <math>t</math> must be an
  3 KB (533 words) - 14:52, 29 October 2023
• 1990 USAMO Problems/Problem 5
  ...\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle. ...of the altitude from <math>A</math>: <math>A</math>, <math>H</math>, <math>A'</math> are concurrent, where <math>H</math> is the orthocentre.
  3 KB (604 words) - 20:52, 24 October 2018
• 1995 USAMO Problems/Problem 2
  A calculator is broken so that the only keys that still work are the <math>\, ...>0</math>, then the some finite sequence of buttons will yield <math>\sqrt{m/n}</math>.
  3 KB (516 words) - 00:18, 6 April 2020
• 2004 AMC 12B Problems/Problem 21
  ...r all points <math>(x,y)</math> on the ellipse. What is the value of <math>a+b</math>? <math>\mathrm{(A)}\ 3
  2 KB (256 words) - 16:23, 13 March 2023
• 1994 USAMO Problems
  ...ve, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> for <math>\, m = 1,2,3, \ldots \; \;</math>. Prove that, for each positive integer <math> ...the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in
  2 KB (391 words) - 07:58, 19 July 2016
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024

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