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• The positive three-digit integer N has a ones digit of 0. What is the probability that N is divisible by 4? Express your answer as a common fraction.
  ...as odd digits, there is a <math>\boxed{1/2}</math> probability that <math>N</math> is divisible by 4.
  427 bytes (71 words) - 17:35, 12 June 2022

Page text matches

• Pythagorean Theorem
  ...n [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right tria In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>{} C </math>.
  5 KB (886 words) - 13:51, 15 May 2024
• 2016 AMC 10A Problems/Problem 9
  ...coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math>? <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\t
  2 KB (315 words) - 15:34, 18 June 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
• 2005 Alabama ARML TST Problems/Problem 7
  ...e series: <center><math>3+\frac{11}4+\frac 94 + \cdots + \frac{n^2+2n+3}{2^n}+\cdots</math>.</center> ...^{\infty} \left(\frac{2n}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)</math>
  1 KB (193 words) - 21:13, 18 May 2021
• Newton
  A '''Newton''' (abbreviated N) is the [[Système international|metric]] measure of [[force]], named for [ ...[[kilogram]] times one [[meter]] per [[second]] squared, or <math>\mathrm{N}=\mathrm{kg}\times \frac{\mathrm{m}}{\mathrm{s}^2}</math>. This is because
  665 bytes (96 words) - 23:17, 2 February 2021
• 1954 AHSME Problems/Problem 46
  In the diagram, if points <math>A, B</math> and <math>C</math> are points of tangency, then <math>x</math> eq pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
  3 KB (415 words) - 18:01, 24 May 2020
• 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
• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...>n</math> positive real weights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in
  3 KB (606 words) - 23:59, 1 July 2022
• Wilson's Theorem
  Suppose first that <math>p</math> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. ...ique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (560 words) - 22:51, 13 January 2024
• Arithmetic mean
  The '''arithmetic mean''' of a [[set]] of numbers (or variables) is the sum of all the numbers, divided by <center><math>AM=\frac{x_1+x_2+\cdots+x_n}{n}</math></center>
  699 bytes (110 words) - 12:44, 20 September 2015
• Polynomial
  ...algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. ===A More Precise Definition===
  6 KB (1,100 words) - 01:44, 17 January 2024
• 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 (1,129 words) - 17:57, 24 May 2024
• Factoring
  ...rs, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information abou Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
  3 KB (532 words) - 22:00, 13 January 2024
• Proof by contradiction
  ...fs look better --> This technique usually works well on problems where not a lot of information is known, and thus we can create some using proof by con ...d as a rational fraction of the form <math>\frac{b}{a}</math>, where <math>a</math> and <math>b</math> are two [[relatively prime]] integers. Now, sinc
  2 KB (374 words) - 14:01, 21 August 2022
• Mathematical problem solving
  ...ng]] involves using all the tools at one's disposal to attack a problem in a new way. == A Historical Example ==
  2 KB (314 words) - 06:45, 1 May 2014
• Pigeonhole Principle
  ...e''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one hole must contain two or more pigeons. This seemingly tri ...if <math>n</math> balls are to be placed in <math>k</math> boxes and <math>n>k</math>, then at least one box must contain more than one ball.
  11 KB (1,985 words) - 21:03, 5 August 2023
• Relatively prime
  ...ath> and <math>n</math> are relatively prime if and only if <math>\frac{m}{n}</math> is in lowest terms. ...th> and <math>n+1</math>, then it must divide their difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</math>.
  2 KB (245 words) - 15:51, 25 February 2020
• 1965 AHSME Problems/Problem 40
  ...^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: <math>\textbf{(A)}\ 4 \qquad
  3 KB (571 words) - 00:42, 22 October 2021
• 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
• Mill's Constant
  ...\lfloor\theta^{3^n}\rfloor</math> is always a prime number for all natural n. ...>\lfloor\theta^{3^n}\rfloor</math> is the prime number theorem where <math>n</math> can be any number and <math>\theta</math> is an element from an set
  794 bytes (105 words) - 01:59, 15 January 2022
• Combination
  ...of combinations of size <math>r</math> from an original set of size <math>n</math> ...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
• Cauchy-Schwarz Inequality
  ...th>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Ineq ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro
  13 KB (2,048 words) - 15:28, 22 February 2024
• Factorial
  ...[[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. ...</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
  10 KB (809 words) - 16:40, 17 March 2024
• Discriminant
  ...has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. ==Discriminant of polynomials of degree n==
  4 KB (734 words) - 19:19, 10 October 2023
• 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
• Chebyshev's Inequality
  ..._{i=1}^{n}a_ib_i\right)\geq\left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)</math>. <math> b_n\geq b_{n-1}\geq ... \geq b_1 </math> then:
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...is a positive integer [[relatively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>.
  3 KB (542 words) - 17:45, 21 March 2023
• Brute forcing
  ...e, through bashing calculations, and can actually sometimes be faster than a more creative approach, and is thus an important tool to have. ...te force would be to list all 91 possibilities (although this would not be a smart time to use brute force).
  1 KB (190 words) - 13:22, 5 May 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
• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  2 KB (275 words) - 12:51, 26 July 2023
• Number theory/Advanced
  ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
  5 KB (849 words) - 16:14, 18 May 2021
• Harmonic mean
  ...al number]]s <math> x_1, x_2... x_n </math> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}</math>. ...\frac{x_1+x_2+\ldots+x_n}{n}\ge \sqrt[n]{x_1\cdot x_2 \cdots x_n}\ge \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}} </math>
  1 KB (196 words) - 00:49, 6 January 2021
• Euler's totient function
  ...elatively prime]] to <math>n</math>. <math>\phi(n)</math> is read "phi of n." ...h>, one can compute <math>\phi(n)</math> using the formula <cmath>\phi(n)= n\left(1-\frac{1}{p_1} \right) \left(1-\frac{1}{p_2} \right)\cdots \left(1-\f
  5 KB (898 words) - 19:12, 28 January 2024
• Divisibility
  In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divisibility resides a ...that <math>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>.
  2 KB (277 words) - 16:21, 29 April 2023
• Rearrangement Inequality
  ...ss than or equal to exactly <math>{i}</math> of the other members of <math>A</math>, then <math>{b_k}</math> is also greater than or equal to exactly <m ...re more pennies than nickels, more nickels than dimes, and so on. This is a simple application of the rearrangement inequality. It is also an applicat
  5 KB (804 words) - 13:54, 26 January 2023
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...[[integer]]s (<math>\mathbb{Z}</math>), [[natural number]]s (<math>\mathbb{N}</math>) and [[irrational number]]s (sometimes, but not universally, denote
  3 KB (496 words) - 23:22, 5 January 2022
• Divisibility rules
  ...its of the number are divisible by <math>2^n</math>. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, A number is divisible by 3 or 9 if and only if the sum of its digits is divis
  8 KB (1,298 words) - 15:07, 23 May 2024
• Induction
  ...it works for <math>n=1+1=2</math>, which in turn means it works for <math>n=2+1=3</math>, and so on. ...nd show that if <math>{n=k}</math> gives the desired result, so does <math>n=k+2</math>. If you wish, you can similarly induct over the powers of 2.
  5 KB (768 words) - 20:45, 1 September 2022
• Greatest common divisor
  ...mon factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers. A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediatel
  2 KB (288 words) - 22:40, 26 January 2021
• Overcounting
  ...ea of strategic overcounting is fundamental to [[combinatorics]] and plays a role in incredibly important counting tools such as [[combinations]] and th An example of a classic problem is as follows:
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  ...s a [[counting]] technique that involves constructing an item belonging to a set. Along with the construction, one counts the total possibilities of eac '''Solution''': We can construct a four-digit by picking the first digit, then the second, and so on until the
  12 KB (1,896 words) - 23:55, 27 December 2023
• Jensen's Inequality
  Let <math>{F}</math> be a [[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb If <math>{F}</math> is a concave function, we have:
  3 KB (623 words) - 13:10, 20 February 2024
• 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
• Generating function
  ...se [[coefficient]]s, <math>c_0, c_1, c_2, \ldots</math>, give the terms of a [[sequence]] which is of interest. Therefore the power series (i.e. the gen ...se 0}+{n \choose 1}x + {n \choose 2}x^2+\cdots+</math><math>{n \choose n}x^n</math>.
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  ...x]] <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
• Recursion
  ...ely by the relations <math>F_0 = F_1 = 1</math> and <math>F_{n+1}=F_{n}+F_{n-1}</math>. (That is, each term is the sum of the previous two terms.) The ...> for <math>n > 0</math> also has the closed-form definition <math>a_n = 2^n</math>.
  2 KB (316 words) - 16:03, 1 January 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
  <math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math> ...}{k}</math> is the binomial coefficient <math>\binom{n}{k} = {}_nC_k = C_k^n</math>.
  12 KB (1,996 words) - 12:01, 18 May 2024
• Prime factorization
  ...ental Theorem of Arithmetic]] tells us that every [[positive integer]] has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
  3 KB (496 words) - 22:14, 5 January 2024
• Composite number
  A comprehensive video explaining composite numbers: https://youtu.be/SMOGYNYD A '''composite number''' is a [[positive integer]] with at least one [[divisor]] different from 1 and its
  6 KB (350 words) - 12:58, 26 September 2023
• Natural number
  ...tegers (sometimes called [[whole number]]s). In particular, <math>\mathbb{N}</math> usually includes zero in the contexts of [[set theory]] and [[abstr
  1 KB (162 words) - 21:44, 13 March 2022
• 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
  ...\frac nd</math> is also a natural number (i.e <math>d</math> divides <math>n</math>). See [[Divisibility]] for more information. ...number is a divisor of another is <math>n|k</math>. This means that <math>n</math> divides <math>k</math>.
  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
• Partition of an interval
  A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals. Partition Let <math>[a,b]</math> be an interval of [[real number]]s.
  1 KB (178 words) - 20:34, 6 March 2022
• Henstock-Kurzweil integral
  ...ble generalizations of the [[Integral|Riemann integral]], but it also uses a strikingly simple and elegant idea. It was developed independently by [[Ral Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
  2 KB (401 words) - 09:46, 31 January 2018
• 2001 AMC 12 Problems/Problem 2
  Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
  1,007 bytes (165 words) - 00:28, 30 December 2023
• Multiple
  ...</math> are integers. (In this case, <math>k</math> is a multiple of <math>n</math>, as well). An equivalent phrasing is that <math>k</math> is a multiple of <math>m</math> exactly when <math>k</math> is [[divisibility |
  860 bytes (142 words) - 22:51, 26 January 2021
• Common factorizations
  \text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-2}+y^{n-1}) ...e be solved by [[Newton sums]] or problems that give a polynomial, and ask a question about the roots. Combined with [[Vieta's formulas]], these are ex
  2 KB (327 words) - 02:06, 28 April 2024
• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. The inequality states that in for four points <math>A, B, C, D </math> in the plane,
  3 KB (602 words) - 09:01, 7 June 2023
• Cevian
  ...(or the extension of that side). In the below diagram, <math>AD</math> is a cevian. label("$A$",(10,50),N);
  1 KB (194 words) - 01:35, 19 June 2018
• 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
  ...ost surprising places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for p ...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
• Phi
  <math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of .../math>^th term, the result approaches <math>\phi</math> as <math>n</math> increases.
  2 KB (302 words) - 14:04, 1 January 2024
• 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
• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: A common [[Brute forcing|bruteforce]] technique with inequalities is to clear
  8 KB (1,346 words) - 12:53, 8 October 2023
• Cardinality
  '''Cardinality''' is a property of [[set]]s. For [[finite]] sets, the cardinality of is the numbe ...4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (A)</math> are used.
  2 KB (263 words) - 00:54, 17 November 2019
• 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
• Polygon
  A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segment]]s. There ar A polygon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[co
  2 KB (372 words) - 19:04, 30 May 2015
• 2011 AMC 10B Problems/Problem 22
  ...ength <math>1</math> and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py <math> \textbf{(A)}\ 5\sqrt{2} - 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \fra
  4 KB (691 words) - 18:38, 19 September 2021
• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• 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) - 00:02, 20 May 2024
• Roots of unity
  ...ne the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ...e note that since we have an '''n'''th degree polynomial, there will be '''n''' complex roots.
  3 KB (558 words) - 21:36, 11 December 2011
• Angle bisector
  ...internal) angle bisector of <math>\angle BAC</math> is the line from <math>A</math> such that the angle between this line and <math>\overline{AB}</math> pair A,B,C,D,E,F;
  3 KB (575 words) - 15:27, 19 March 2023
• 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
• Sequence
  A '''sequence''' is an ordered list of terms. Sequences may be either [[fini ...or instance, the function <math>f(x) = x^2</math> defined on <math>\mathbb{N}</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldo
  2 KB (413 words) - 21:18, 13 November 2022
• Geometric sequence
  ...'geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between any two consecutive ter ...e with common ratio <math>2</math> and <math>100, -50, 25, -25/2</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3
  4 KB (644 words) - 12:55, 7 March 2022
• 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>.
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• Fermat's Last Theorem
  ...\geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...ater than the second to be the sum of two like powers. ''I have discovered a truly marvelous demonstration of this proposition that this margin is too n
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• 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>.
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• 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.)
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• Geometry
  '''Geometry''' is the field of [[mathematics]] dealing with figures in a given [[space]]. It is one of the two oldest branches of mathematics, along :''“Through any line and a point not on the line, there is exactly one line passing through that point
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• 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
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• Area
  ...size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
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• Floor function
  ...]</math>. The action of this function is the same as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything af ...+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor</math> for all real <math>(a,b)</math>.
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• Pascal's triangle
  ...ns the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. ...he sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
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• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
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• 1985 IMO Problems/Problem 5
  ...ath> and <math>BC</math> again at distinct points <math>K</math> and <math>N</math> respectively. Let <math>M</math> be the point of intersection of the .../math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KN</math> to
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• 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
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• Trigonometry
  ...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...definition of trigonometry is preferred in order to extend trigonometry to a complex domain.
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• 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
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• Complex analysis
  ...ctions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis. ...ly connected]] [[domain]] ''D'', and let <math>\Gamma\subseteq D</math> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interi
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• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says ...</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.
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• 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:
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• 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 ...uld hold. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>.
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• Quadratic residues
  ...] <math>m</math> if there is some integer <math>n</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>
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• 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 =
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• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some ...gorithm]] that every complex polynomial of degree <math>n</math> has <math>n</math> complex roots, counting multiplicities. In other words, every polyn
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• Collatz Problem
  ...sitive integer]] <math>n</math>, the sequence <math>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people h ==Properties of <math>f(n)</math> ==
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• Triangle Inequality
  The '''Triangle Inequality''' says that in a [[nondegenerate]] [[triangle]] <math>ABC</math>: ..._n = \left(\sum_{j=1}^n a_j\right) - a_i</math> for <math>i = 1, 2 \ldots, n</math>. Expressing the inequality in this form leads to <math>2a_i < P</ma
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• Majorization
  ...m_{i=1}^{k}a_i \ge \sum_{i=1}^{k}b_i </math>, with equality when <math>k = n </math>. If <math>\{a_i\} </math> and <math>\{b_i\} </math> are not necess ...for all <math> 1\le k \le n </math>, <math>\sum_{i=k}^n a_i \le \sum_{i=k}^n b_i</math>, with equality when <math>k=1 </math>. An interesting consequen
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• Surjection
  ..., the range is a subset of the codomain.) In adjectival form, we say that a function is ''surjective'' or ''onto''. ...defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is not one more than any other natural number.
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• Algebraic geometry
  ...> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine alge ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
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• Prime Number Theorem
  constants <math>A</math> and <math>B</math>, <cmath> \pi(x) \sim \frac{x}{A \log x - B} . </cmath>
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• 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
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• Infinite
  A [[set]] <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <mat In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element
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• 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 ==
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• Commutative property
  An [[operation]] (especially a [[binary operation]]) is said to have the '''commutative property''' or to ...eal number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over the
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• Permutation
  ...are <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> distinct objects. ...ite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
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• Bertrand's Postulate
  ...em. A more widely known version states that there is a prime between <math>n</math> and <math>2n</math>. ...closer look at the [[combinations|binomial coefficient]] <math>\binom{2n}{n}</math>. Assuming that the reader is familiar with that proof, the Bertrand
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• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
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• 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>.
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• Perfect square
  ...'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</mat ...th>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>
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• Uncountable
  ...d to be '''uncountable''' if there is no [[injection]] <math>f:S\to\mathbb{N}</math>. Assuming the [[Axiom of choice]], every set that is ''not'' uncoun ...f <math>A</math> (in other words, take an injection <math>f: A \to \mathbb{N}</math>, and denote <math>\omega_i = f(i)</math>).
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• Gamma function
  The '''Gamma function''' is a generalization of the notion of a [[factorial]] to [[complex number|complex numbers]]. ...plex numbers. We can then use the identity to extend the Gamma function to a [[meromorphic]] function on the full [[complex plane]], with simple poles a
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• Modular arithmetic/Intermediate
  ...th>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
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• Subset
  ...r than <math>B</math> itself. In the latter case, <math>A</math> is called a ''proper subset''. The following is a true statement:
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• 2006 AIME I Problems
  {{AIME Problems|year=2006|n=I}} In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat
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• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math ...the numbers in this list up in the following way: Whenever a positive and a negative number are adjacent in this progression, pair them up and remove t
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• 2006 AIME I Problems/Problem 14
  ...> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the 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);
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• 2006 AIME I Problems/Problem 13
  ...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. .../math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math>
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• 2006 AIME I Problems/Problem 12
  ...e have <math>a^3 + b^3 = (a+b)^3 \rightarrow ab(a+b) = 0</math>. But <math>a+b = 2\cos 4x\cos x</math>, so we require <math>\cos x = 0</math>, <math>\co ...A = \{150, 126, 162, 198, 112.5, 157.5\}</math> and thus <math>\sum_{x \in A} x = \boxed{906}</math>.
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• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o
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• 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>
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• 2006 AIME I Problems/Problem 10
  ...are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> ...h>, which can be easily solved to be <math>6x = 2y + 5</math>. Thus, <math>a^2 + b^2 + c^2 = \boxed{065}</math>.
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• 2006 AIME I Problems/Problem 9
  ...8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> <cmath>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 a+\log_8 (ar)+\ldots+\log_8 (ar^{11}) \\
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• 2006 AIME I Problems/Problem 8
  ...area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6),
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• 2006 AIME I Problems/Problem 7
  ...atio of shaded region <math> D </math> to the area of shaded region <math> A. </math> pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);
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• 2006 AIME I Problems/Problem 6
  ...cimal| decimals]] of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct [[digit]]s. Find the sum of the elements of <mat Another method, albeit a little risky, that can be used is to note that the numbers between 1 and 99
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• 2006 AIME I Problems/Problem 5
  ...n be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math> <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2
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• 2006 AIME I Problems/Problem 4
  ...oduct <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both t
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• 2006 AIME I Problems/Problem 3
  ...math> so we take <math>N_0 = 25</math> and <math>n = 2.</math> Then <cmath>N = 7 \cdot 10^2 + 25 = \boxed{725},</cmath> and indeed, <math>725 = 29 \cdot ...ow that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1,</math> <math>2</math> or <math>3</math> digits. Checkin
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• 2006 AIME I Problems/Problem 2
  ...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> Alternatively, for ease of calculation, let set <math>\mathcal{B}</math> be a 10-element subset of <math>\{1,2,3,\ldots,100\}</math>, and let <math>T</ma
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• 2006 AIME I Problems/Problem 1
  In [[quadrilateral]] <math> ABCD</math>, <math>\angle B </math> is a [[right angle]], [[diagonal]] <math>\overline{AC}</math> is [[perpendicular pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18));
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• 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>
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• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
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• 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)
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• 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)
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• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
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• 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)
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• 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) \
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• 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
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• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{
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• 2001 AMC 12 Problems
  <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D) Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits
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• 2002 AMC 12B Problems
  ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc <math>\mathrm{(A)}\ 0
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• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
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• 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
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• 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
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• 2006 AMC 12B Problems/Problem 1
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>(-1)^n=1</math> if n is even and <math>-1</math> if n is odd. So we have
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• 2006 AMC 12B Problems/Problem 14
  ...h>, <math>J</math> and <math>N</math> are all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? \mathrm{(A)}\ 1.05
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• 2006 AMC 12B Problems/Problem 18
  ...ne unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? \mathrm{(A)}\ 120
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• 2006 AMC 12B Problems/Problem 20
  \mathrm{(A)}\ \frac 18 For a fixed <math>k</math> the length of the interval <math>\left[ 10^k, \frac{10
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• 2006 AMC 12B Problems/Problem 22
  ...divisible by <math>10</math>. What is the smallest possible value of <math>n</math>? \mathrm{(A)}\ 489
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• 2006 AMC 12B Problems/Problem 23
  ...re <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? pair A = (B.y,0);
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• 2006 AMC 12B Problems/Problem 25
  ...tegers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</math>. If <math>a_1=999</math>, <math>a_2<999</math> and <math>a_{20 \mathrm{(A)}\ 165
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• 2006 AMC 12A Problems/Problem 15
  <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ ..., x must be in the form of <math>\frac{\pi}{2} + \pi n</math>, where <math>n</math> denotes any [[integer]].
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• 2006 AMC 12A Problems/Problem 17
  ...ies on the circle, on the same side of <math>\overline{BE}</math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqr label("A", (-s,0), W);
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• 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 <math>\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)</math>
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  <math> \mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qqu <cmath>\sum_{a+b+c=2006}{\frac{2006!}{a!b!c!}x^ay^bz^c}</cmath>
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  <math> \mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{ ...on can be solved fairly directly by casework and pattern-finding. We give a somewhat more general attack, based on the solution to the following proble
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  ...h>, and <math>f^{[n + 1]}(x) = f^{[n]}(f(x))</math> for each integer <math>n \geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> (\text {A}) \ 0 \qquad (\text {B}) \ 2005 \qquad (\text {C})\ 4010 \qquad (\text {D})
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12B Problems/Problem 12
  ...f <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 21
  ...e greatest integer <math>k</math> such that <math>7^k</math> divides <math>n</math>? <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 22
  A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is define <cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath>
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 24
  ...th> 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
• 2005 AMC 12B Problems/Problem 25
  ...the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its <math>\mathrm{(A)}\ \frac {5}{256}
  10 KB (1,840 words) - 21:35, 7 September 2023
• Power of a Point Theorem/Introductory Problem 4
  ...</math> of a given circle are perpendicular to each other and intersect at a right angle at <math>E</math> . Given that <math> BE = 16, DE = 4, </math> pair A = (3,0);
  1 KB (177 words) - 02:14, 26 November 2020
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...wice, triple roots three times, and so on, there are in fact exactly <math>n</math> complex roots of <math>P(x)</math>.
  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 12
  Rolly wishes to secure his dog with an 8-foot rope to a [[square (geometry) | square]] shed that is 16 feet on each side. His prel MP('8', (20,-8), N);
  3 KB (424 words) - 10:14, 17 December 2021
• 2006 AMC 10A Problems/Problem 16
  A circle of radius <math>1</math> is [[tangent]] to a circle of radius <math>2</math>. The sides of <math>\triangle ABC</math> ar D('B', (0,0),SW); D('C',(4*t,0), SE); D('A', (2*t, 8), N);
  5 KB (732 words) - 23:19, 19 September 2023
• 2006 AMC 10A Problems/Problem 17
  pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 19
  <math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 59\qquad\textbf{(D) } 89\qquad The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an
  2 KB (259 words) - 03:10, 22 June 2023
• Set
  ...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• 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
  ...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
• 1991 AJHSME Problems
  <math>\text{(A)}\ 222,222,222,222 \qquad <math>\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qqua
  17 KB (2,246 words) - 13:37, 19 February 2020
• 2022 USAJMO Problems/Problem 1
  ...<math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>; ...\nmid d</math> and <math>m|a+(n-1)d-gr^{n-1}</math> for all integers <math>n>1</math>.
  4 KB (792 words) - 00:29, 13 April 2024
• 2006 AIME II Problems/Problem 2
  .../math> is a positive integer. Find the number of possible values for <math>n</math>. ...equality]] and applying the well-known logarithmic property <math>\log_{c} a + \log_{c} b = \log_{c} ab</math>, we have that
  1 KB (164 words) - 14:58, 14 April 2020
• 2005 AIME II Problems/Problem 1
  ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...<math>{n \choose 6} = \frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1}</math>.
  1 KB (239 words) - 11:54, 31 July 2023
• 2005 AIME II Problems/Problem 2
  ...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
  ...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/Problem 4
  ...(a positive integer) is a divisor of one of the numbers. Therefore, <math>n</math> can be expressed as <math>{p_1}^{e_1}</math> or as <math>{p_2}^{e_2} ...to <math>5^{10}</math>) for <math>p_1=5</math>, and 1 possibility if <math>n=1</math>. From this case, there are <math>11+22+10+1=44</math> possibilitie
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  ...math> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> ...it has roots at <math>17</math> and <math>24</math>. Hence <math>P(x)-x+7=A(x-17)(x-24)</math>. In particular, this means that
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 5
  ...b) </math> of [[integer]]s such that <math> \log_a b + 6\log_b a=5, 2 \leq a \leq 2005, </math> and <math> 2 \leq b \leq 2005. </math> ...r <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>.
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 6
  ...original stack, the stack is named magical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original pos ...rds from each of piles A, B in front of card 131. This suggests that <math>n = 131 + 65 = 196</math>; the total number of cards is <math>196 \cdot 2 = \
  2 KB (384 words) - 00:31, 26 July 2018
• 2005 AIME II Problems
  {{AIME Problems|year=2005|n=II}} ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math>
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  ...\sqrt[2^n]{5} + 1)(\sqrt[2^n]{5} - 1) = (\sqrt[2^n]{5})^2 - 1^2 = \sqrt[2^{n-1}]{5} - 1 </math>. {{AIME box|year=2005|n=II|num-b=6|num-a=8}}
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 8
  ...> 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 9
  ...ath> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>? ...math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply De Moivre's Theorem.
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...> AB </math> with <math> AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </m ...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
  ...</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
  ...math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of reals such that <math>a_0 = 37, a_1 = 72, a_m = 0, </math> and Thus the product <math>a_{k}a_{k+1}</math> is a [[monovariant]]: it decreases by 3 each time <math>k</math> increases by 1.
  3 KB (499 words) - 18:52, 21 November 2022
• Euler's identity
  ...ers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties: ...states that for any [[real number]] <math>\theta</math> and integer <math>n</math>,
  3 KB (452 words) - 23:17, 4 January 2021
• 2005 AIME I Problems
  {{AIME Problems|year=2005|n=I}} ...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
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 1
  ...] 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 are ...f the triangle plus <math>r</math>. Thus, the radius of <math>C</math> has a length of <math>3r = 30</math>, and so <math>r = 10</math>. <math>K = 30^2\
  1 KB (213 words) - 13:17, 22 July 2017
• 2005 AIME I Problems/Problem 2
  ..., <math>(1002,2)</math> and <math>(2004,1)</math>, and each of these gives a possible value of <math>k</math>. Thus the requested number of values is <m {{AIME box|year=2005|n=I|num-b=1|num-a=3}}
  2 KB (303 words) - 01:31, 5 December 2022
• 2005 AIME I Problems/Problem 3
  ...s,, so <math>n</math> must be in the form <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math> ...h>47</math>) so there are <math> {15 \choose 2} =105</math> ways to choose a pair of primes from the list and thus <math>105</math> numbers of the first
  2 KB (249 words) - 09:37, 23 January 2024
• 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 ...<math>n \leq 14</math>. In fact, when <math>n = 14</math> we have <math>n(n + 7) = 14\cdot 21 = 294 = 17^2 + 5</math>, so this number works and no larg
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...e, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the numbe ...since after the first H no more tails can appear. The first H can occur in a maximum of eight times different positions, and then there is also the poss
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 6
  ...n the roots of the above equation are <math>x = 1 + i^n r</math> for <math>n = 0, 1, 2, 3</math>. The two non-real members of this set are <math>1 + ir If you think of each part of the product as a quadratic, then <math>((x-1)^2+\sqrt{2006})</math> is bound to hold the two
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  ...rilateral]] <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> label("$A$",(0,0),SW);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...h> 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 9
  ...> a,b, </math> and <math> c </math> are [[positive integer]]s, find <math> a+b+c+p+q+r. </math> ...}{6} = \frac{2}{3}</math> of all orientations, so from these cubes we gain a factor of <math>\left(\frac{2}{3}\right)^6</math>.
  4 KB (600 words) - 21:44, 20 November 2023
• 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
  ...mum 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> ...if and only if there are an odd number of perfect squares less than <math>n</math>. So <math>S(1), S(2)</math> and <math>S(3)</math> are odd, while <m
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 13
  A particle moves in the [[Cartesian plane]] according to the following rules: ...> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME I Problems/Problem 14
  ...unique [[square]] <math> S </math> such that each of the four points is on a different side of <math> S. </math> Let <math> K </math> be the area of <ma Consider a point <math>E</math> such that <math>AE</math> is [[perpendicular]] to <mat
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  ...| 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
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma 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
  ...itive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> an ...icting our assumption that <math>20</math> was the smallest value of <math>n</math>. Using [[complementary counting]], we see that there are only <math>
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> triple Oxy = (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  ...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> ...he 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|great 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
  ...h> 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
  ...d <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
  ...<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 1
  ...from left to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
  2 KB (374 words) - 14:53, 27 December 2019
• 2004 AIME I Problems/Problem 2
  ...The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find ...e median) must be <math>2</math>. 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 3
  ...gular and <math>12</math> of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the ...f the [[polyhedron]] determines either an [[edge]], a face [[diagonal]] or a space diagonal. We have <math>{26 \choose 2} = \frac{26\cdot25}2 = 325</ma
  1 KB (156 words) - 17:56, 1 January 2016
• 2004 AIME I Problems/Problem 4
  ...re. The [[midpoint]]s of the line segments in set <math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <ma ...midpoints lying on the sides determined by vertex <math>(0,0)</math> form a quarter-[[circle]] with [[radius]] 1.
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 5
  ...<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 6
  ...h>x_2x_40x_3</math>, <math>x_3x_40x_2</math>. Because we know that zero is a digit, there are <math>3\cdot{9\choose 3}=252</math> snakelike numbers whic ...ays to permute to make the number snakelike, b-a-c, or c-a-b. And, we pick a,b,c from 1 to 9, since 0 has already been chosen as one of the digits. So,
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  <cmath>\begin{align*}\left(\sum_{i=1}^{n} S_i\right)^2 &= \left(\sum_{i=1}^{n} S_i^2\right) + 2\left(\sum_{1 \le i \neq j}^{15} S_iS_j\right)\\ (-8)^2 &= and the right-hand sum comes from the formula for the sum of the first <math>n</math> perfect squares. Therefore, <math>|C| = \left|\frac{64-1240}{2}\righ
  6 KB (941 words) - 11:37, 27 May 2024
• 2004 AIME I Problems/Problem 8
  Define a regular <math> n </math>-pointed star to be the union of <math> n </math> line segments <math> P_1P_2, P_2P_3,\ldots, P_nP_1 </math> such tha .../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
  {{AIME Problems|year=2004|n=I}} ...from left to right. What is the sum of the possible remainders when <math> n </math> is divided by 37?
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...eated <math>8</math> more times. After the last fold, the strip has become a stack of <math>1024</math> unit squares. How many of these squares lie belo ...number of squares below the <math>n</math> square after the final fold in a strip of length <math>2^{k}</math>.
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...d from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulti ...</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 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
  ...> 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 11
  ...the cone is <math>125</math>, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex The easiest way is to unwrap the cone into a circular sector. Center the sector at the origin with one radius on the pos
  2 KB (268 words) - 22:20, 23 March 2023
• 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 ...and <math>y-x \equiv 3 \pmod 6 = 6n +3</math> for some whole number <math>n</math>.
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 9
  ...est term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...(n-1)f(n)</math>, <math>a_{2n+1} = f(n)^2</math>, where <math>f(n) = nx - (n-1)</math>.{{ref|1}}
  3 KB (537 words) - 18:03, 25 May 2024
• 2004 AIME II Problems/Problem 8
  We can [[divisor function | count the number of divisors]] of a number by multiplying together one more than each of the [[exponentiation | ...math>2^a\cdot 3^b\cdot 167^c</math>. Thus we need to find how many <math>(a,b,c)</math> satisfy
  2 KB (353 words) - 18:08, 25 November 2023
• 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 6
  ...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...> is divisible by <math>72</math> (however, since the denominator contains a <math>27</math>, the factors of <math>3</math> cancel, and it only really n
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  In order to complete a large job, <math>1000</math> workers were hired, just enough to complete th A train is traveling at <math>1000</math> miles per hour and has one hour to
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  First, let's count numbers with only a single digit. We have nine of these for each length, and four lengths, so ...2^n - 2</math> ways to arrange them in an <math>n</math>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> suc
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems/Problem 3
  ...the 1-cm cubes cannot be seen. Find the smallest possible value of <math> N. </math> ...xtra layer makes the entire block <math>4\times8\times12</math>, and <math>N= \boxed{384}</math>.
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems/Problem 2
  ...h> 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/Problem 1
  ...b, c, d, e, </math> and <math> f </math> are [[positive integer]]s, <math> a </math> and <math> e </math> are [[relatively prime]], and neither <math> c ...f the radius (since the chord [[bisect]]s it), and the radius. Thus, it is a <math>30^\circ</math> - <math>60^\circ</math> - <math>90^\circ</math> [[tri
  2 KB (329 words) - 23:20, 4 July 2013
• 2004 AIME II Problems
  {{AIME Problems|year=2004|n=II}} ...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
  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) - 03:13, 27 May 2024
• 2019 AIME I Problems/Problem 2
  ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...2</math> elements are adjacent. Using the well-known formula <math>\dbinom{n-k+1}{k}</math>, there are <math>\dbinom{20-2+1}{2} = \dbinom{19}{2} = 171</
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ...h>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  ...]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>? The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t
  5 KB (847 words) - 15:48, 21 August 2023
• 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 sim ...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
  ...ts of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. Suppose that <math>|x_i| < 1</math> for <math>i = 1, 2, \dots, n</math>. Suppose further that
  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
• 1990 AIME Problems
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 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
  ...etc. If the candidate went <math>\frac{n^{2}}{2}</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how many miles was he from his star ...many contestants caught <math>n\,</math> fish for various values of <math>n\,</math>.
  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 ..., 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
  ...> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> ...h> 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
  ...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
  ...ath>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
  ...nd <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
  {{AIME Problems|year=2000|n=I}} ...he least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  {{AIME Problems|year=2001|n=I}} A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  {{AIME Problems|year=2002|n=I}} ...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  {{AIME Problems|year=2003|n=I}} <center><math> \frac{((3!)!)!}{3!} = k \cdot n!, </math></center>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  {{AIME Problems|year=2000|n=II}} ...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  {{AIME Problems|year=2001|n=II}} .../math> forms a perfect square. What are the leftmost three digits of <math>N</math>?
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  {{AIME Problems|year=2002|n=II}} Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  {{AIME Problems|year=2003|n=II}} ...is the sum of the other two. Find the sum of all possible values of <math>N</math>.
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 3
  If we were to expand by squaring, we would get a [[quartic Equation|quartic]] [[polynomial]], which isn't always the easiest ...he square root of a real number can't be negative, the only possible <math>n</math> is <math>5</math>.
  3 KB (532 words) - 05:18, 21 July 2022
• 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
  Let <math>a_n=6^{n}+8^{n}</math>. Determine the remainder upon dividing <math>a_ {83}</math> by <mat Firstly, we try to find a relationship between the numbers we're provided with and <math>49</math>. W
  3 KB (361 words) - 20:20, 14 January 2023
• 1983 AIME Problems/Problem 8
  ...is the largest <math>2</math>-digit [[prime]] factor of the integer <math>n = {200\choose 100}</math>? ...ppears twice in the denominator. Thus, we need <math>p</math> to appear as a factor at least three times in the numerator, so <math>3p<200</math>. The l
  2 KB (249 words) - 23:25, 11 May 2024
• 1983 AIME Problems/Problem 11
  The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to th triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  6 KB (971 words) - 15:35, 27 May 2024
• 1983 AIME Problems/Problem 13
  ...simply <math>5</math>. Find the sum of all such alternating sums for <math>n=7</math>.<!-- don't remove the following tag, for PoTW on the Wiki front pa Let <math>S</math> be a non-[[empty set | empty]] [[subset]] of <math>\{1,2,3,4,5,6\}</math>.
  5 KB (894 words) - 22:02, 5 April 2024
• 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 (2,151 words) - 17:48, 27 May 2024
• 1983 AIME Problems/Problem 15
  ...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);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 1
  A somewhat quicker method is to do the following: for each <math>n \geq 1</math>, we have <math>a_{2n - 1} = a_{2n} - 1</math>. We can substi A better approach to this problem is to notice that from <math>a_{1}+a_{2}+\c
  4 KB (576 words) - 21:03, 23 December 2023
• 1984 AIME Problems/Problem 2
  ...h>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. Any multiple of 15 is a multiple of 5 and a multiple of 3.
  1 KB (187 words) - 20:05, 29 May 2021
• 1984 AIME Problems/Problem 3
  A [[point]] <math>P</math> is chosen in the interior of <math>\triangle ABC</ pair A=(0,0),B=(12,0),C=(4,5);
  4 KB (726 words) - 13:39, 13 August 2023
• 1984 AIME Problems/Problem 4
  Let <math>S</math> be a list of positive integers--not necessarily distinct--in which the number <m Suppose that <math>S</math> has <math>n</math> numbers other than the <math>68,</math> and the sum of these numbers
  2 KB (319 words) - 03:38, 16 January 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 7
  ...nction]] f is defined on the [[set]] of [[integer]]s and satisfies <math>f(n)=\begin{cases} n-3&\mbox{if}\ n\ge 1000\\
  4 KB (617 words) - 22:09, 15 May 2024
• 1984 AIME Problems/Problem 8
  ...another factor to make the equation easier to solve. If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math ...la says that <math>(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta</math> and <math>\frac{360}{3} = 120</math>. ~programmeruser
  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
  ...robability]] 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
  ...>21</math>, ... , and in general <math>9 + 6n</math> for nonnegative <math>n</math> are odd composites. We now have 3 cases: ...th> can be expressed as <math>9 + (9+6n)</math> for some nonnegative <math>n</math>. Note that <math>9</math> and <math>9+6n</math> are both odd composi
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1985 AIME Problems/Problem 14
  ...h of the two players earned <math>\frac{1}{2}</math> point if the game was a tie. After the completion of the tournament, it was found that exactly half ...tal, for an average of 9. Thus we must have <math>n > 10</math>, so <math>n = 15</math> and the answer is <math>15 + 10 = \boxed{25}</math>.
  5 KB (772 words) - 22:14, 18 June 2020
• 1985 AIME Problems/Problem 13
  ...<math>a_{n+1}</math>. Find the maximum value of <math>d_n</math> as <math>n</math> ranges through the [[positive integer]]s. ...h>2n+1</math> if it is going to divide the entire [[expression]] <math>100+n^2+2n+1</math>.
  4 KB (671 words) - 20:04, 6 March 2024
• 1985 AIME Problems/Problem 12
  ...when it has crawled exactly <math>7</math> meters. Find the value of <math>n</math>. ...> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 10
  where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[ ...o <math>\frac n2</math> to <math>\frac {n+1}2</math> for any integer <math>n</math> (same reasoning as above). So now we only need to test every 10 numb
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 5
  ...is chosen so that <math>a_n = a_{n - 1} - a_{n - 2}</math> for each <math>n \ge 3</math>. What is the sum of the first 2001 terms of this sequence if t ...h>n</math> times, <math>a_{j + 6n} = a_j</math> for all [[integer]]s <math>n</math> and <math>j</math>.
  2 KB (410 words) - 13:37, 1 May 2022
• 1985 AIME Problems/Problem 4
  ...division points closest to the opposite vertices. Find the value of <math>n</math> if the the [[area]] of the small square is exactly <math>\frac1{1985 ...2 - 2n + 1 = 1985</math>. Solving this [[quadratic equation]] gives <math>n = \boxed{32}</math>.
  3 KB (484 words) - 21:40, 2 March 2020
• 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 13
  ...tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For ex ...tches to <tt>H</tt> four times; hence it follows that our string will have a structure of <tt>THTHTHTH</tt>.
  4 KB (772 words) - 21:09, 7 May 2024
• Analytic continuation
  ...sum_{n=1}^{\infty}{n} = \frac{-1}{12}</math> and <math>\sum_{n=1}^{\infty}{n^2} = 0</math>. Interestingly, even though these properties seem to be only
  1 KB (180 words) - 20:12, 19 August 2015
• 1986 AIME Problems/Problem 11
  ...ix} ,</math> where the term <math>\dbinom{n}{k}</math> is negated if <math>n+k</math> is odd. ...\\ \sum_{n=2}^{17} \tbinom{n}{2} \\ \vdots \\ -\sum_{n=17}^{17} \tbinom{n}{17} \end{bmatrix} = \begin{bmatrix} \tbinom{18}{1} \\ - \tbinom{18}{2} \\
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 10
  ...>. Play the role of the magician and determine <math>(abc)</math> if <math>N= 3194</math>. ...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
• 1986 AIME Problems/Problem 9
  pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425)); pair Da=IP(Circle(A,289),A--B),E=IP(Circle(C,324),B--C),Ea=IP(Circle(B,270),B--C);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 8
  ...</math> [[logarithm]]s of all the [[proper divisor]]s (all [[divisor]]s of a number excluding itself) of <math>1000000</math>. What is the integer neare ...d(n))/2}</math>, where <math>d(n)</math> is the number of divisor of <math>n</math>.
  3 KB (487 words) - 20:52, 16 September 2020
• 1986 AIME Problems/Problem 7
  ...powers of 3, in base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Therefore, we can ...4</math>th term. Also, note that the <math>k</math>th term after the <math>n</math>th power of 3 is equal to the power plus the <math>k</math>th term in
  5 KB (866 words) - 00:00, 22 December 2022
• 1986 AIME Problems/Problem 6
  The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t ...nd solve <math>\frac{n(n+1)}{2} = 1986</math>. The positive root for <math>n \approx \sqrt{3972} \approx 63</math>. Quickly testing, we find that <math>
  2 KB (336 words) - 14:13, 6 September 2020
• 1986 AIME Problems/Problem 5
  ...]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>? ...<math>900</math>. The greatest [[integer]] <math>n</math> for which <math>n+10</math> divides <math>900</math> is <math>\boxed{890}</math>; we can doub
  2 KB (338 words) - 19:56, 15 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 14
  ...+ 4b^4</math> can be factored as <math>\left(a^2 + 2b^2 - 2ab\right)\left(a^2 + 2b^2 + 2ab\right).</math> Each of the terms is in the form of <math>x^4 ...of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math>
  7 KB (965 words) - 10:42, 12 April 2024
• 1987 AIME Problems/Problem 13
  ...ng]] order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, a Suppose that <math>n = 40</math>, and that the terms of the initial sequence <math>r_1, r_2, \do
  3 KB (514 words) - 21:27, 31 December 2023
• 1987 AIME Problems/Problem 12
  ...s a [[positive]] [[real number]] less than <math>1/1000</math>. Find <math>n</math>. In order to keep <math>m</math> as small as possible, we need to make <math>n</math> as small as possible.
  4 KB (673 words) - 19:48, 28 December 2023
• 1987 AIME Problems/Problem 11
  ...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 8
  ...is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>? <cmath>\begin{align*}104(n+k) &< 195n< 105(n+k)\\
  2 KB (393 words) - 16:59, 16 December 2020
• 1987 AIME Problems/Problem 7
  ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...ath>c = 2^p5^q</math> for some [[nonnegative]] [[integer]]s <math>j, k, m, n, p, q</math>. Dealing first with the powers of 2: from the given condition
  3 KB (547 words) - 22:54, 4 April 2016
• 1987 AIME Problems/Problem 3
  ...ositive]] [[integer|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 ...product of the distinct proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances:
  3 KB (511 words) - 09:29, 9 January 2023
• 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 sim ...ath> to the respective [[digit]] in <math>1492</math> (the values of <math>n</math> are then fixed). Thus, the number of [[ordered pair]]s will be <math
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 13
  ...ath> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. Let <math>F_n</math> represent the <math>n</math>th number in the Fibonacci sequence. Therefore,
  10 KB (1,585 words) - 03:58, 1 May 2023
• 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 \sum_{k = 1}^n (z_k - w_k) = 0.
  2 KB (422 words) - 00:22, 6 September 2020
• 1988 AIME Problems/Problem 10
  ...ron lie in the interior of the polyhedron rather than along an [[edge]] or a [[face]]? The polyhedron described looks like this, a truncated cuboctahedron.
  6 KB (902 words) - 17:40, 19 May 2024
• 1988 AIME Problems/Problem 9
  A little bit of checking tells us that the units digit must be 2. Now our cub ...value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution.
  6 KB (893 words) - 08:15, 2 February 2023
• 1988 AIME Problems/Problem 5
  ...99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>. ...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
• 1988 AIME Problems/Problem 4
  Suppose that <math>|x_i| < 1</math> for <math>i = 1, 2, \dots, n</math>. Suppose further that What is the smallest possible value of <math>n</math>?
  2 KB (394 words) - 10:21, 27 January 2024
• 1988 AIME Problems/Problem 3
  ...ge of base formula]], which states <math>\log_a b = \frac{\log_k b}{\log_k a}</math> for arbitrary <math>k</math>. We wish to convert this expression into one which has a uniform base. Let's scale down all the powers of 8 to 2.
  3 KB (481 words) - 21:52, 18 November 2020
• 1988 AIME Problems/Problem 2
  ...ts of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. {{AIME box|year=1988|num-b=1|num-a=3}}
  696 bytes (103 words) - 19:16, 27 February 2018
• 1988 AIME Problems/Problem 1
  .../math> from <math>1</math> to <math>9</math>, the number of ways to choose a set of <math>x</math> buttons is <math>\sum^{9}_{k=1}{10 \choose k}</math>. Now we can use the identity <math>\sum^{n}_{k=0}{n \choose k}=2^{n}</math>.
  1 KB (181 words) - 18:23, 26 August 2019
• 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 label("$X$", X, N);
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 14
  ...ressed in the base <math>-n+i</math> using the integers <math>0,1,2,\ldots,n^2</math> as digits. That is, the equation <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 Let the mode be <math>x</math>, which we let appear <math>n > 1</math> times. We let the arithmetic mean be <math>M</math>, and the sum
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 10
  ...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find pair A = (0,0), B = (3, 0), C = (1, 4);
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 9
  ...h that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>. n^5&\equiv0\pmod{2}, \\
  6 KB (874 words) - 15:50, 20 January 2024
• 1989 AIME Problems/Problem 8
  ...tic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x f(1)&=\phantom{42}a+b+c&&=1, \\
  8 KB (1,146 words) - 04:15, 20 November 2023
• 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
• 1989 AIME Problems/Problem 3
  ...eger]] and <math>d</math> is a single [[digit]] in [[base 10]]. Find <math>n</math> if <center><math>\frac{n}{810}=0.d25d25d25\ldots</math></center>
  3 KB (499 words) - 22:17, 29 March 2024
• 1989 AIME Problems/Problem 2
  Ten [[point]]s are marked on a [[circle]]. How many distinct [[convex polygon]]s of three or more sides ca ...er of such subsets. There are <math>2^{10} = 1024</math> total subsets of a ten-member [[set]], but of these <math>{10 \choose 0} = 1</math> have 0 mem
  911 bytes (135 words) - 08:30, 27 October 2018
• 1989 AIME Problems/Problem 1
  ...have that <math>a=868</math> as per the equation <math>(a+1)^2 = a \cdot (a+2) +1</math>. ...prove that one more than the product of four consecutive integers must be a perfect square:
  4 KB (523 words) - 00:12, 8 October 2021
• 1990 AIME Problems/Problem 15
  Find <math>ax^5 + by^5</math> if the real numbers <math>a,b,x,</math> and <math>y</math> satisfy the equations ...h>(ax^n + by^n)(x + y) = (ax^{n + 1} + by^{n + 1}) + (xy)(ax^{n - 1} + by^{n - 1})</cmath>
  4 KB (644 words) - 16:24, 28 May 2023
• 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 ...n that there must be at least either one or two n-digit power of 9 for all n.
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 11
  ...r which <math>n^{}_{}!</math> can be expressed as the [[product]] of <math>n - 3_{}^{}</math> [[consecutive]] positive integers. ...\sqrt{a!}</math>, which decreases as <math>a</math> increases. Thus, <math>n = 23</math> is the greatest possible value to satisfy the given conditions.
  3 KB (519 words) - 09:28, 28 June 2022
• 1990 AIME Problems/Problem 10
  ...The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity. How many distinct elements are in <math>C_{ ...ent values. All solutions for <math>zw</math> will be in the form of <math>n^{8k_1 + 3k_2}</math>.
  3 KB (564 words) - 04:47, 4 August 2023
• 1990 AIME Problems/Problem 9
  A [[fair]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>\frac{ ...3} + {9\choose2} + {{10}\choose1} + {{11}\choose0} = 144</math>. There are a total of <math>2^{10}</math> possible flips of <math>10</math> coins, makin
  3 KB (425 words) - 12:36, 12 May 2024
• 1990 AIME Problems/Problem 7
  .../math> can be written in the form <math>ax+2y+c=0_{}^{}</math>. Find <math>a+c_{}^{}</math>. MP("P",P,N,f);MP("Q",Q,W,f);MP("R",R,E,f);
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1990 AIME Problems/Problem 6
  ...ample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate ...s the number of fish in May. Solving for <math>n</math>, we see that <math>n = \boxed{840}</math>
  2 KB (325 words) - 13:16, 26 June 2022
• 1990 AIME Problems/Problem 5
  ...gral divisors, including <math>1_{}^{}</math> and itself. Find <math>\frac{n}{75}</math>. ...to the least power. Therefore, <math>n = 2^43^45^2</math> and <math>\frac{n}{75} = \frac{2^43^45^2}{3 \cdot 5^2} = 16 \cdot 27 = \boxed{432}</math>.
  1 KB (175 words) - 03:45, 21 January 2023
• 1990 AIME Problems/Problem 3
  ...lar polygon|regular]] <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea ...the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.
  3 KB (516 words) - 19:18, 16 April 2024
• 1990 AIME Problems/Problem 1
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. We need <math>n - T = 500</math>, where <math>n</math> is an integer greater than 500 and <math>T</math> is the set of numb
  2 KB (283 words) - 23:11, 25 June 2023
• 1991 AIME Problems/Problem 15
  <math>\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},</math> ...}</math> for which <math>S_n^{}</math> is also an integer. Find this <math>n^{}_{}</math>.
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 13
  ...s that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\frac{1}{2}</math> that both are red or both a ...e present case <math>t\leq 1991</math>, and so one easily finds that <math>n=44</math> is the largest possible integer satisfying the problem conditions
  7 KB (1,328 words) - 20:24, 5 February 2024
• 1991 AIME Problems/Problem 12
  ...t terms, denote the [[perimeter]] of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. ...\(Q\)",Q,E);label("\(R\)",R,SW);label("\(S\)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW); </asy></center>
  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) - 17:46, 24 May 2024
• 1991 AIME Problems/Problem 10
  ...ath>S_b^{}</math> in alphabetical order. When <math>p</math> is written as a [[fraction]] in [[irreducible fraction|lowest terms]], what is its [[numera ...es the partial sums of <math>P_b</math> (in other words, <math>S_b = \sum_{n=1}^{b} P_b</math>):
  5 KB (813 words) - 06:10, 25 February 2024
• 1991 AIME Problems/Problem 9
  ...c mn,</math> where <math>\frac mn</math> is in lowest terms. Find <math>m+n^{}_{}.</math> ...ot 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
• 2006 AMC 10B Problems/Problem 18
  ...<math>a_n=\frac{a_{n-1}}{a_{n-2}} </math> for each positive integer <math> n \ge 3 </math>. What is <math> a_{2006} </math>? <math> \mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \f
  1 KB (158 words) - 01:33, 29 May 2023
• 1991 AIME Problems/Problem 6
  Suppose <math>r^{}_{}</math> is a [[real number]] for which ...57}{100}\rfloor</math>, which is also the first term of this sequence with a value of <math>8</math>, so <math>8 \le r + \frac{57}{100} < 8.01</math>. S
  3 KB (447 words) - 17:02, 24 November 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 ...ft[\prod_{j=1}^{k}\frac{(N-j+1)x}{j}\right]=\sum_{j=1}^{k}\log\left[\frac{(N-j+1)x}{j}\right]\, .
  5 KB (865 words) - 12:13, 21 May 2020
• 1991 AIME Problems/Problem 2
  pair A=(0,0),B=(4,0),C=(4,3),D=(0,3); D(A--B--C--D--cycle);
  4 KB (595 words) - 12:51, 17 June 2021
• 1992 AIME Problems/Problem 15
  ...that the decimal representation of <math>m!</math> ends with exactly <math>n</math> zeroes. How many positive integers less than <math>1992</math> are n Note that if <math>m</math> is a multiple of <math>5</math>, <math>f(m) = f(m+1) = f(m+2) = f(m+3) = f(m+4)<
  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
  ...t <math>R^{(1)}(l)=R(l)^{}_{}</math> and <math>R^{(n)}(l)^{}_{}=R\left(R^{(n-1)}(l)\right)</math>. Given that <math>l^{}_{}</math> is the line <math>y=\ 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
  ...<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 8
  ...</math>. Suppose that all of the terms of the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</ ...are reminiscent of differentiation; from the condition <math>\Delta(\Delta{A}) = 1</math>, we are led to consider the differential equation
  5 KB (778 words) - 21:36, 3 December 2022
• 1992 AIME Problems/Problem 4
  <cmath>\frac{\dbinom{n}{k-1}}{3} = \frac{\dbinom{n}{k}}{4} = \frac{\dbinom{n}{k+1}}{5}.</cmath> Taking the first part, and using our expression for <math>n</math> choose <math>k</math>,
  3 KB (476 words) - 14:13, 20 April 2024
• 1992 AIME Problems/Problem 3
  ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>.500</math>. During the weekend, sh ...atches won, so that <math>\frac{n}{2n}=\frac{1}{2}</math>, and <math>\frac{n+3}{2n+4}>\frac{503}{1000}</math>.
  2 KB (251 words) - 08:05, 2 January 2024
• 1992 AIME Problems/Problem 2
  A [[positive integer]] is called ascending if, in its [[decimal representatio ..., since the only position for 0 is at the leftmost end of the number, i.e. a leading 0), there is exactly one ascending number with those digits.
  2 KB (336 words) - 05:18, 4 November 2022
• 2006 AMC 10B Problems/Problem 22
  ...math>, <math>J</math>, and <math>N</math> are positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? <math> \mathrm{(A) \ } \$ 1.05\qquad \mathrm{(B) \ } \$ 1.25\qquad \mathrm{(C) \ } \$ 1.45\qq
  2 KB (394 words) - 00:51, 25 November 2023
• 2006 AMC 10B Problems/Problem 23
  A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. T pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A);
  5 KB (861 words) - 00:53, 25 November 2023
• 1993 AIME Problems/Problem 15
  ...</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 14
  ...ath>\sqrt{N}\,</math>, for a positive integer <math>N\,</math>. Find <math>N\,</math>. ...h> and <math>D(x,-3)</math> for nonnegative <math>x,y</math>. Then this is a rectangle, so <math>OA=OB</math>, or <math>16+y^2=9+x^2</math>, so <math>x^
  3 KB (601 words) - 09:25, 19 November 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
  ...and <math>P_n\,</math> is the most recently obtained point, then <math>P_{n + 1}^{}</math> is the midpoint of <math>\overline{P_n L}</math>. Given tha ...e coordinates stay within the triangle. We have <cmath>P_{n-1}=(x_{n-1},y_{n-1}) = (2x_n\bmod{560},\ 2y_n\bmod{420})</cmath>
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  ...atively prime positive integers. What are the last three digits of <math>m+n\,</math>? ...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 heads on an odd numbered f
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 9
  ...on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as <math>199 ...n</math> will only occupy the same point on the circle if <math>\frac12(n)(n + 1)\equiv \frac12(1993)(1994) \pmod{2000}</math>.
  3 KB (488 words) - 02:06, 22 September 2023
• 1993 AIME Problems/Problem 8
  .../math> represents the same selection as the pair <math>\{b, c, d, e, f\},\{a, c\}.</math> ...ion is double counted, except the case where both <math>m</math> and <math>n</math> contain all <math>6</math> elements of <math>S.</math> So our final
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 7
  ...he brick parallel to the sides of the box. If <math>p</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? There is a total of <math>P(1000,6)</math> possible ordered <math>6</math>-tuples <mat
  5 KB (772 words) - 09:04, 7 January 2022
• 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 5
  ...r [[integer]]s <math>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. What is the [[coefficient]] of <math>x\,</math> in <math>P_{20} Using the formula for the sum of the first <math>n</math> numbers, <math>1 + 2 + \cdots + 20 = \frac{20(20+1)}{2} = 210</math>
  2 KB (355 words) - 13:25, 31 December 2018
• 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
• 1993 AIME Problems/Problem 3
  ...many contestants caught <math>n\,</math> fish for various values of <math>n\,</math>. <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\
  2 KB (364 words) - 00:05, 9 July 2022
• 1993 AIME Problems/Problem 2
  ...ast, etc. If the candidate went <math>n^{2}_{}/2</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how many miles was he from his star The N/S displacement is
  2 KB (241 words) - 11:56, 13 March 2015
• 1994 AIME Problems/Problem 15
  .../math> are folded onto <math>P.\,</math> Let us call <math>P_{}^{}</math> a fold point of <math>\triangle ABC\,</math> if these creases, which number t ...BM_2 \sim \triangle ABC</math>, we see that thse segments respectively cut a <math>120^{\circ}</math> arc in the circle with radius <math>18</math> and
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 14
  A beam of light strikes <math>\overline{BC}\,</math> at point <math>C\,</math ...= MP("B",(0,0),NW), C = MP("C",D((1,0))), A = MP("A",expi(alpha * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*expi(beta * pi/180);
  2 KB (303 words) - 00:03, 28 December 2017
• 1994 AIME Problems/Problem 13
  ...omega</math> where <math>\omega = e^{i(\pi n/5+\pi/10)}</math> where <math>n</math> is an integer. {{AIME box|year=1994|num-b=12|num-a=14}}
  3 KB (375 words) - 23:46, 6 August 2021
• 1994 AIME Problems/Problem 12
  A fenced, rectangular field measures <math>24</math> meters by <math>52</math ...row. Then <math>6|n</math>, and our goal is to maximize the value of <math>n</math>.
  3 KB (473 words) - 17:06, 1 January 2024
• 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 5
  ...<math>n\,</math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let ...a three-digit number (so <math>5 \equiv 005</math>), and since our <math>p(n)</math> ignores all of the zero-digits, replace all of the <math>0</math>s
  2 KB (275 words) - 19:27, 4 July 2013
• 1994 AIME Problems/Problem 4
  Find the positive integer <math>n\,</math> for which ...floor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994
  2 KB (264 words) - 13:33, 11 August 2018
• 1994 AIME Problems/Problem 3
  ...n (and easy to show) that the sum of two consecutive triangular numbers is a perfect square; that is, <cmath>T_{n-1} + T_n = n^2,</cmath>
  2 KB (252 words) - 11:12, 3 July 2023
• 1994 AIME Problems/Problem 2
  ...ath>, 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
• 1994 AIME Problems/Problem 1
  ...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 ...h term, so <math>n = 4 + (997-1) \cdot 3 = 2992</math>. The value of <math>n^2 - 1 = 2992^2 - 1 \pmod{1000}</math> is <math>\boxed{063}</math>.
  946 bytes (139 words) - 21:05, 1 September 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 13
  Let <math>f(n)</math> be the integer closest to <math>\sqrt[4]{n}.</math> Find <math>\sum_{k=1}^{1995}\frac 1{f(k)}.</math> ...}\right)^4 \right\rfloor</math> values of <math>n</math> for which <math>f(n) = k</math>. Expanding using the [[binomial theorem]],
  2 KB (287 words) - 01:25, 12 December 2019
• 1995 AIME Problems/Problem 12
  ...> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> // n = normal to plane
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 10
  ...that is not the sum of a positive integral multiple of <math>42</math> and a positive composite integer? ...that are multiples of <math>42</math> greater than them, until they reach a composite number.
  3 KB (436 words) - 19:26, 2 September 2023
• 1995 AIME Problems/Problem 9
  ...,</math> where <math>a</math> and <math>b</math> are integers. Find <math>a+b.</math> dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE
  7 KB (1,181 words) - 13:47, 3 February 2023
• 1995 AIME Problems/Problem 8
  Thus, for a given value of <math>y</math>, we need the number of multiples of <math>y(y ...y+1</math>. We can add <math>k</math> to each side in order to factor out a <math>y+1</math>. So, <math>ky+k+1 \equiv k \mod y+1</math> or <math>k(y+1)
  4 KB (646 words) - 17:37, 1 January 2024
• 1995 AIME Problems/Problem 7
  ...ath>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 6
  ...=2^{31}3^{19}.</math> How many positive [[integer]] [[divisor]]s of <math>n^2</math> are less than <math>n_{}</math> but do not divide <math>n_{}</math ...39=\boxed{589}</math> factors of <math>n^2</math> that do not divide <math>n</math>.
  2 KB (407 words) - 08:14, 4 November 2022
• 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 ...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 4
  ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le ...), F=B+3*expi(acos(1/3)), P=IP(F--F+3*(D-F),CR(A,9)), Q=IP(F--F+3*(F-D),CR(A,9));
  3 KB (605 words) - 11:30, 5 May 2024
• 1995 AIME Problems/Problem 3
  ...h> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math> If the object took <math>4</math> steps, then it must have gone two steps <tt>N</tt> and two steps <tt>E</tt>, in some permutation. There are <math>\frac{4
  3 KB (602 words) - 23:15, 16 June 2019
• 1995 AIME Problems/Problem 1
  ...> 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 15
  ...=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D); D(MP("A",A,N)--MP("B",B)--MP("C",C)--MP("D",D,N)--cycle); D(B--D); D(A--C); D(MP("O",O,SE));
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 13
  ...h> 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
  ...imes because there are <math>5</math> places <math>a_n</math> and <math>a_{n + 1}</math> can be. To find all possible values for <math>|a_n - a_{n - 1}|</math> we have to compute
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 11
  ...the product of the [[root]]s of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive [[imaginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\the This is just a slight variation of Solution 1.
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1996 AIME Problems/Problem 10
  So <math>19x = 141 +180n</math>, for some integer <math>n</math>. {{AIME box|year=1996|num-b=9|num-a=11}}
  4 KB (503 words) - 15:46, 3 August 2022
• 1996 AIME Problems/Problem 9
  A bored student walks down a hall that contains a row of closed lockers, numbered <math>1</math> to <math>1024</math>. He ope ...ith <math>2^n</math> lockers). It follows that <math>L_{n} = 2^{n} +2 -2L_{n-1}</math> as they are corresponding lockers. We can compute <math>L_1=2</ma
  3 KB (525 words) - 23:51, 6 September 2023
• 1996 AIME Problems/Problem 7
  ...r schemes are equivalent if one can be obtained from the other by applying a [[rotation]] in the plane board. How many inequivalent color schemes are po Note that a pair of yellow squares will only yield <math>2</math> distinct boards upon
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 6
  ...th>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
• 1996 AIME Problems/Problem 4
  ...th>x</math> centimeters directly above an upper [[vertex]], the cube casts a shadow on the horizontal surface. The area of the shadow, which does not in label("$1$",(unit/2,0,unit),N);
  2 KB (257 words) - 17:50, 4 January 2016
• 1996 AIME Problems/Problem 3
  ...[[integer]] <math>n</math> for which the expansion of <math>(xy-3x+7y-21)^n</math>, after like terms have been collected, has at least 1996 terms. ...>y</math> and so none of the terms will need to be collected. Hence <math>(n+1)^2 \ge 1996</math>, the smallest square after <math>1996</math> is <math>
  3 KB (515 words) - 04:29, 27 November 2023
• 1996 AIME Problems/Problem 2
  ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer? ...<math>n</math> must satisfy these [[inequality|inequalities]] (since <math>n < 1000</math>):
  1 KB (163 words) - 19:31, 4 July 2013
• 1997 AIME Problems/Problem 14
  ...n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n</math>. ...\pi/1997</math>, and let <math>w</math> be the root corresponding to <math>n\theta=2n\pi/ 1997</math>. Then
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 11
  ...h>x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}</math>. What is the greatest integer that does not exceed <math>100x ...m_{n=1}^{44} \sin n} = \frac{\sum_{n=46}^{89} \sin n}{\sum_{n=1}^{44} \sin n} = \frac {\sin 89 + \sin 88 + \dots + \sin 46}{\sin 1 + \sin 2 + \dots + \s
  10 KB (1,514 words) - 14:35, 29 March 2024
• 1997 AIME Problems/Problem 6
  ...1</math>, <math>A_n</math>, and <math>B</math> are consecutive vertices of a regular polygon? ...regular polygon formula, we have <math>\angle A_2A_1A_n = \frac{(n-2)180}{n}</math>, <math>\angle A_nA_1B = \frac{(m-2)180}{m}</math>, and <math>\angle
  3 KB (497 words) - 00:39, 22 December 2018
• 1997 AIME Problems/Problem 4
  ...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 3
  ...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 ...e trial and error on factors of 1000. If <math>9x - 1 = 100</math>, we get a non-integer. If <math>9x - 1 = 125</math>, we get <math>x=14</math> and <ma
  2 KB (375 words) - 19:34, 4 August 2021
• 1997 AIME Problems/Problem 2
  ...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 15
  ...rdered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after t We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected w
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 14
  ..., 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 13
  ...f the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given that <math>S_8 = - 176 - 64i</math> and <math> S_9 = p + q ...; hence we can just append a <math>9</math> to any of those subsets to get a new one.
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 11
  Three of the edges of a cube are <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CD} triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20);
  7 KB (1,084 words) - 11:48, 13 August 2023
• 1998 AIME Problems/Problem 8
  ...[negative]] term encounted. What positive integer <math>x</math> produces a sequence of maximum length? ...ont> || <math>1000 - x</math> || <math>2x - 1000</math><font color="white">a</font> || <math>2000 - 3x</math> || <math>5x - 3000</math> || <math>5000 -
  2 KB (354 words) - 19:37, 24 September 2023
• 1998 AIME Problems/Problem 7
  ...er]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math> Find <math>\frac n{100}.</math> ...ition, so now we can finish with plain stars as bars, which gives us <math>n= {50\choose3}</math>. Computing this and dividing by 100 gives us an answer
  5 KB (684 words) - 11:41, 13 August 2023
• 1998 AIME Problems/Problem 5
  ...ath>n \in \mathbb{Z}</math> is 1 if <math>n</math> is even and -1 if <math>n</math> is odd. <math>\frac {k(k-1)}2</math> will be even if <math>4|k</math ...c{(n)(n-1)}{2} + \frac{(n+1)(n)}{2} = \left(\frac n2\right)(n+1 - (n-1)) = n</math>. So the first two fractions add up to <math>19</math>, the next two
  1 KB (225 words) - 02:20, 16 September 2017
• 1998 AIME Problems/Problem 4
  ...ath>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
• 1998 AIME Problems/Problem 2
  <div style="text-align:center;"><math>A = I + \frac B2 - 1</math></div> ..., whose area is <math>\frac 12</math> of the 30 by 30 [[square]] it is in. A simple way to see this is to note that the two triangles outside of the qua
  6 KB (913 words) - 16:34, 6 August 2020
• 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
  ...n_{}</math> are [[relatively prime]] positive integers. Find <math>\log_2 n.</math> ...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 12
  pair A=(0,0),B=(50,0),C=IP(circle(A,23+245/2),circle(B,27+245/2)), I=incenter(A,B,C); path P = incircle(A,B,C);
  3 KB (472 words) - 15:59, 25 February 2022
• 1999 AIME Problems/Problem 11
  ...rime 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
  ...nd <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 7
  ...step 1000 has been completed, how many switches will be in position <math>A</math>? ...}{d}</math> must be a multiple of 4 to ensure that a switch is in position A:
  3 KB (475 words) - 13:33, 4 July 2016
• 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 3
  ...[[positive integer]]s <math>n</math> for which <math>n^2-19n+99</math> is a [[perfect square]]. ...h> for some positive integer <math>x</math>, then rearranging we get <math>n^2-19n+99-x^2=0</math>. Now from the quadratic formula,
  2 KB (296 words) - 01:18, 29 January 2021
• 1999 AIME Problems/Problem 2
  ...}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n</math>. ...\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 15
  A stack of <math>2000</math> cards is labelled with the integers from <math>1 ...r logic as Solution 1, we find that 1999 has position <math>1024</math> in a <math>2048</math> card stack, where the fake cards towards the front.
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 14
  ...r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. pair A,B,C,P,Q;
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2000 AIME I Problems/Problem 13
  ...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 12
  Given a [[function]] <math>f</math> for which ...46</math> and <math>x = 200, 201, ... 351</math> are repeated. This gives a total of
  1 KB (238 words) - 18:50, 10 March 2015
• 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
  ...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 9
  Since <math>\log ab = \log a + \log b</math>, we can reduce the equations to a more recognizable form: Let <math>a,b,c</math> be <math>\log x, \log y, \log z</math> respectively. Using [[Sim
  4 KB (623 words) - 15:56, 8 May 2021
• 2000 AIME I Problems/Problem 8
  ...>p</math> is not divisible by the cube of any prime number. Find <math>m + n + p</math>. ...{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
  ...<math>n</math> are [[relatively prime]] positive integers. Find <math>m + n</math>. ..., <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 6
  ...e <math>y>x</math>, it follows that each ordered pair <math>(x,y) = (n^2, (n+2)^2)</math> satisfies this equation. The minimum value of <math>x</math> i .../math> = <math>x</math> and <math>b^2</math> = <math>y</math>, where <math>a</math> and <math>b</math> are positive.
  6 KB (966 words) - 21:48, 29 January 2024
• 2000 AIME I Problems/Problem 5
  ...ath>n</math> are [[relatively prime]] positive integers. What is <math>m + n</math>? ...quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle of Inclusion-Exclusion]] still requires us to find
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2000 AIME I Problems/Problem 4
  The diagram shows a [[rectangle]] that has been dissected into nine non-overlapping [[square]]s ...the resulting sides would not be integers and we would need to scale up by a factor of <math>2</math> to make them integers; if we started with <math>a_
  3 KB (485 words) - 00:31, 19 January 2024
• 2000 AIME I Problems/Problem 3
  ...ient]]s of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>. ...heorem]], <math>\binom{2000}{1998} b^{1998}a^2 = \binom{2000}{1997}b^{1997}a^3 \Longrightarrow b=666a</math>.
  679 bytes (98 words) - 00:51, 2 November 2023
• 2000 AIME I Problems/Problem 2
  ...<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 pair A=(11,10), B=(10,11), C=(-10, 11), D=(-10, -11), E=(10, -11);
  3 KB (434 words) - 22:43, 16 May 2021
• 2000 AIME I Problems/Problem 1
  ...he least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least ...th>s separated, so we need to find the first power of 2 or 5 that contains a 0.
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 15
  ...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
  ...e <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
  ...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 11
  ...th> and <math>x_5 = y_3.</math> Find the smallest possible value of <math>N.</math> <cmath>\begin{align*}x_i &= (i - 1)N + c_i\\
  3 KB (493 words) - 13:51, 22 July 2020
• 2001 AIME I Problems/Problem 10
  ...and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> ...rac {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
  ...nd <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 8
  ...math> number that is twice <math>N</math>. For example, <math>51</math> is a 7-10 double because its base-<math>7</math> representation is <math>102</ma We let <math>N_7 = \overline{a_na_{n-1}\cdots a_0}_7</math>; we are given that
  3 KB (502 words) - 11:28, 9 December 2023
• 2001 AIME I Problems/Problem 7
  ...h> 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
  ...</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
  ...h> 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
• 2001 AIME I Problems/Problem 4
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. First, draw a good diagram.
  3 KB (534 words) - 03:22, 23 January 2023
• 2001 AIME I Problems/Problem 3
  ...1} + \cdots + a_0 = 0</math>, then the sum of the roots is <math>\frac{-a_{n-1}}{a_n}</math>. ...ath>1000</math> pairs of roots that sum to <math>\frac{1}{2}</math> to get a sum of <math>\boxed{500}</math>.
  2 KB (335 words) - 18:38, 9 February 2023
• 2001 AIME I Problems/Problem 2
  A finite [[set]] <math>\mathcal{S}</math> of distinct real numbers has the fo Let <math>x</math> be the mean of <math>\mathcal{S}</math>. Let <math>a</math> be the number of elements in <math>\mathcal{S}</math>.
  1 KB (225 words) - 07:57, 4 November 2022
• 2001 AIME I Problems/Problem 1
  ...math> (note that if <math>b = 10a</math>, then <math>b</math> would not be a digit). ...= a</math>, we have <math>n = 11a</math> for nine possibilities, giving us a sum of <math>11 \cdot \frac {9(10)}{2} = 495</math>.
  4 KB (687 words) - 18:37, 27 November 2022
• 2002 AIME I Problems/Problem 15
  ...CD</math> is a square with <math>AB = 12;</math> face <math>ABFG</math> is a trapezoid with <math>\overline{AB}</math> parallel to <math>\overline{GF},< triple A=(-6,-6,0), B = (-6,6,0), C = (6,6,0), D = (6,-6,0), E = (2,0,12), H=(-6+2*s
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 14
  A set <math>\mathcal{S}</math> of distinct positive integers has the followin .../math>, the <math>(n+1)</math>th positive integer congruent to 1 mod <math>n</math>.
  2 KB (267 words) - 19:18, 21 June 2021
• 2002 AIME I Problems/Problem 13
  ...d <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 12
  ...ath>, where <math>a</math> and <math>b</math> are real numbers, find <math>a+b</math>. Thus <math>a+b = 1+274 = \boxed{275}</math>.
  1 KB (217 words) - 22:39, 21 November 2018
• 2002 AIME I Problems/Problem 11
  ...d <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 10
  In the diagram below, angle <math>ABC</math> is a right angle. Point <math>D</math> is on <math>\overline{BC}</math>, and <ma ...> is <math>10/3</math> that of triangle <math>AEG</math>, since they share a common side and angle, so the area of triangle <math>AGF</math> is <math>10
  4 KB (643 words) - 22:44, 8 August 2023
• 2002 AIME I Problems/Problem 9
  Harold, Tanya, and Ulysses paint a very long picket fence. ...nerality]] consider <math>\text{gcd}\,(h,t) = 1</math>; then there will be a common solution <math>\pmod{h \times t}</math>).
  4 KB (749 words) - 19:44, 25 April 2024
• 2002 AIME I Problems/Problem 8
  (1) <math>a_1,a_2,a_3\cdots</math> is a nondecreasing sequence of positive integers (2) <math>a_n=a_{n-1}+a_{n-2}</math> for all <math>n>2</math>
  1 KB (205 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 7
  <math>1^n</math> will always be 1, so we can ignore those terms, and using the defini ...cimal of <math>\dfrac{10}{7}</math> repeats every 6 digits, we can cut out a lot of 6's from <math>858</math> to reduce the problem to finding the first
  2 KB (316 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 6
  Let <math>A=\log_{225}x</math> and let <math>B=\log_{64}y</math>. From the first equation: <math>A+B=4 \Rightarrow B = 4-A</math>.
  1 KB (194 words) - 19:55, 23 April 2016
• 2002 AIME I Problems/Problem 5
  Let <math>A_1,A_2,A_3,\cdots,A_{12}</math> be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon ...ide only form 3 squares, and all 12 combinations of two vertices that form a square diagonal only form 6 squares. So in total, we have overcounted by <m
  1 KB (220 words) - 20:50, 12 November 2022
• 2002 AIME I Problems/Problem 4
  ...gers <math>m</math> and <math>n</math> with <math>m<n</math>, find <math>m+n</math>. ...-\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 3
  ...e Dick's present age. How many ordered pairs of positive integers <math>(d,n)</math> are possible? ...ath>10b+a>10a+b</math>, then <math>b>a</math>. The possible pairs of <math>a,b</math> are:
  2 KB (246 words) - 17:02, 21 May 2023
• 2002 AIME I Problems/Problem 2
  ...shows twenty congruent [[circle]]s arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r {{AIME box|year=2002|n=I|num-b=1|num-a=3}}
  2 KB (287 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 1
  ...h> 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
  ...d <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
  ...</math> consecutively and in that order. Find the smallest value of <math> n </math> for which this is possible. To find the smallest value of <math>n</math>, we consider when the first three digits after the decimal point are
  3 KB (477 words) - 14:23, 4 January 2024
• 2003 AIME I Problems/Problem 13
  ...<math>1</math>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</math>. ...</math>. Thus there are <math>{n \choose k}</math> numbers that have <math>n+1</math> digits in base <math>2</math> notation, with <math>k+1</math> of t
  4 KB (651 words) - 19:42, 7 October 2023
• 2003 AIME I Problems/Problem 12
  ...f <math> ABCD </math> is <math> 640 </math>. Find <math> \lfloor 1000 \cos A \rfloor. </math> (The notation <math> \lfloor x \rfloor </math> means the g pair A=(0,0),B=(1.8,0),D=IP(CR(A,x),CR(B,BD)),C=OP(CR(D,1.8),CR(B,2.80 - x));
  3 KB (487 words) - 22:14, 24 November 2019
• 2003 AIME I Problems/Problem 11
  ...re [[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
  pair A=(0,0), B=(2,0), C=(1,Tan(37)), M=IP(A--(2Cos(30),2Sin(30)),B--B+(-2,2Tan(23))); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); D(A--D(MP("M",M))--B); D(C--M);
  7 KB (1,058 words) - 01:41, 6 December 2022
• 2003 AIME I Problems/Problem 9
  ...th pairs. This gives <math>\sum_{n = 10}^{18} (19 - n)^2 = \sum_{n = 1}^9 n^2 = 285</math> balanced numbers. Thus, there are in total <math>330 + 285 ...uare|sum of consecutive squares]], namely <math>\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}</math>.
  4 KB (696 words) - 11:55, 10 September 2023

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