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• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.
  24 KB (3,177 words) - 12:53, 20 February 2024
• 2015 IMO Problems
  ...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 ...ath>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined by
  3 KB (606 words) - 23:59, 1 July 2022
• Pigeonhole Principle
  ...ts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes and is that if <math> ...nhole principle is as follows: suppose for contradiction that there exists a way to place <math>n</math> balls into <math>k</math> boxes where <math>n>k
  11 KB (1,985 words) - 21:03, 5 August 2023
• Combination
  ...generally concerned with finding the number 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
  ...ant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM- ...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
• 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
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• 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
• Divisibility rules
  ...number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. 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
• Jensen's Inequality
  ...[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n 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
• Combinatorial identity
  ...ts from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> objects from group <math>m</math> and the remaining from group <math ...ing <math>k</math> men from the <math>m</math> total men and picking <math>r-k</math> women from the <math>n</math> total women. The left hand side and
  12 KB (1,996 words) - 12:01, 18 May 2024
• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
  8 KB (1,469 words) - 21:11, 16 September 2022
• Chain Rule
  ...> such that <math>h(x) = f(g(x)) \forall x \in I</math>. If <math>x_0 \in I</math>, <math>g</math> is differentiable at <math>{x_0}</math>, and <math>{ ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.)
  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
• 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
• 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
• Complex number
  ...ver, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
  5 KB (860 words) - 15:36, 10 December 2023
• Cis
  ...hrm{cis}</math> is made up by the first letter of <math>\cos</math>, <math>i</math>, and the first letter of <math>\sin</math>. ...most always preferred to write <math>re^{i\theta}</math> rather than <math>r\mathrm{cis }(\theta)</math>, as [[Euler's formula]] states that
  1 KB (171 words) - 20:59, 11 July 2023
• Polar form
  ...n an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...s notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the origin, and <math>\theta</math> [[radian]]s coun
  633 bytes (105 words) - 13:35, 1 April 2022
• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we e ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
  3 KB (558 words) - 21:36, 11 December 2011
• 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
  6 KB (1,181 words) - 22:37, 22 January 2023
• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. So <math>z=re^{i\theta}</math> looks like:
  1 KB (238 words) - 22:51, 20 February 2022
• Pascal's triangle
  ...ns the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. ...2 is <math>{4 \choose 2} = 6</math>. Pascal's Triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the character
  5 KB (838 words) - 17:20, 3 January 2023
• 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 ..._1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math
  5 KB (912 words) - 20:06, 14 March 2023
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (827 words) - 17:30, 21 February 2024
• 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 Suppose that <math>P(z)</math> is a complex polynomial of degree <math>n</math> with no complex roots; without
  5 KB (832 words) - 14:22, 11 January 2024
• Majorization
  ...l for all <math> 1 \le k \le n </math>, <math> \sum_{i=1}^{k}a_i \ge \sum_{i=1}^{k}b_i </math>, with equality when <math>k = n </math>. If <math>\{a_i\ ...y if 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 cons
  2 KB (288 words) - 22:48, 5 July 2023
• Algebraic geometry
  ...>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''. ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
  2 KB (361 words) - 01:59, 24 January 2020
• Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (774 words) - 12:12, 29 October 2016
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• Modular arithmetic/Introduction
  ...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  15 KB (2,396 words) - 20:24, 21 February 2024
• Convex function
  A convex function is a continuous function whose value at the midpoint of every interval in its do ...vex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math>,
  2 KB (417 words) - 00:10, 20 February 2016
• Uncountable
  ...example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==
  2 KB (403 words) - 20:53, 13 October 2019
• 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:
  1 KB (217 words) - 09:32, 13 August 2011
• 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
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 13
  ...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...> is even, hence <math>k+n-1</math> is odd, and <math>S_n</math> cannot be a perfect square. Hence <math>k</math> must be even. In particular, as <math>
  10 KB (1,702 words) - 00:45, 16 November 2023
• 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
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AIME I Problems/Problem 10
  ...are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> int i,j;
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 9
  ...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}) \\
  4 KB (651 words) - 18:27, 22 May 2021
• 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 ..., D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);
  5 KB (730 words) - 15:05, 15 January 2024
• 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
  3 KB (439 words) - 18:24, 10 March 2015
• 2022 AIME I Problems/Problem 1
  .../math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. R(16) &= P(16)+Q(16) &&= 54+54 &&= 108, \\
  4 KB (670 words) - 13:03, 13 November 2023
• 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
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2000 AMC 12 Problems
  ...cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 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
• 2022 USAJMO Problems/Problem 1
  ...exist integers <math>a,d,r</math> with <math>m \nmid d</math> and <math>m|a+(n-1)d-gr^{n-1}</math> for all integers <math>n>1</math>. <cmath>m | a+nd-gr^n \; (1),</cmath>
  4 KB (792 words) - 00:29, 13 April 2024
• Prime ideal
  ...I,J\subseteq R</math> with <math>IJ\subseteq P</math> we have either <math>I\subseteq P</math> or <math>J\subseteq P</math>. ...nd for any <math>a,b\in R</math> if <math>ab\in P</math> then either <math>a\in P</math> or <math>b\in P</math>.
  967 bytes (176 words) - 18:08, 7 April 2012
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one r
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 12
  ...math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...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 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 [[Image:2005 AIME I Problem 1.png]]
  1 KB (213 words) - 13:17, 22 July 2017
• 2005 AIME I Problems/Problem 4
  ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 6
  ...ath>1 + ir</math> and <math>1 - ir</math>. Their product is <math>P = 1 + r^2 = 1 + \sqrt{2006}</math>. <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> 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 9
  ...ath> 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 11
  A [[semicircle]] with [[diameter]] <math> d </math> is contained in a [[square (geometry) | square]] whose sides have length 8. Given the maximum 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 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 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 Let <math>w_3</math> have center <math>(x,y)</math> and radius <math>r</math>. Now, if two circles with radii <math>r_1</math> and <math>r_2</mat
  12 KB (2,000 words) - 13:17, 28 December 2020
• 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 11
  ...<math> C </math> and a [[frustum]]-shaped solid <math> F, </math> in such a way that the [[ratio]] between the [[area]]s of the painted surfaces of <ma ...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 9
  ...ivide rectangle <math> DEFG </math> into a triangle <math> U_2 </math> and a trapezoid <math> V_2 </math> such that <math> U_1 </math> is similar to <ma .../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 7
  ...and so <math>P(x)\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>. ...1 - 4x^2)\cdots(1 - 225x^2)</math> <math>= 1 - (1 + 4 + \ldots + 225)x^2 + R(x)</math>.
  6 KB (941 words) - 11:37, 27 May 2024
• 2004 AIME II Problems/Problem 14
  Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> ...</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
• 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. ...ath> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each interior angle of <math>P_1^{}</math> is
  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
• 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
  ...math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...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
  ...rline{AB}</math> at <math>Q</math> and <math>\overline{DB}</math> at <math>R.</math> Given that <math>PQ = 735</math> and <math>QR = 112,</math> find < ...,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math> Find <math>\frac n{100}.</math>
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2000 AIME I Problems
  {{AIME Problems|year=2000|n=I}} [[2000 AIME I Problems/Problem 1|Solution]]
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  {{AIME Problems|year=2001|n=I}} [[2001 AIME I Problems/Problem 1|Solution]]
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  {{AIME Problems|year=2002|n=I}} ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  ...a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...<math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 4
  ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< real r=10;
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 8
  ...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>. The polynomial <math>x^9-1</math> has all of its roots with [[a ...\circ </math>, then <math>r^6+r^3+1=3</math>. (When we multiplied by <math>r^3 - 1</math> at the beginning, we introduced some extraneous solutions, and
  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
• 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 ...h> where <math>r</math> is the magnitude of <math>N</math> such that <math>r\geq0,</math> and <math>\theta</math> is the argument of <math>N</math> such
  7 KB (965 words) - 10:42, 12 April 2024
• 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>. 0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1 &\Longrightarrow (a
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1989 AIME Problems/Problem 14
  ...math>s</math> are integers, can be uniquely expressed in the base <math>-n+i</math> using the integers <math>0,1,2,\ldots,n^2</math> as digits. That is, <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
• 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 i=0; i<12; ++i)
  4 KB (740 words) - 17:46, 24 May 2024
• 1992 AIME Problems/Problem 10
  ...1</math>, inclusive. What is the integer that is nearest the area of <math>A</math>? ...q 1</math> we have the inequality <cmath>0\leq a,b \leq 40</cmath>which is a square of side length <math>40</math>.
  2 KB (323 words) - 12:05, 16 July 2019
• 1994 AIME Problems/Problem 14
  A beam of light strikes <math>\overline{BC}\,</math> at point <math>C\,</math ...,(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
• 1995 AIME Problems/Problem 5
  ...math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math> <cmath>m\cdot n = 13 + i,m' + n' = 3 + 4i\Longrightarrow m'\cdot n' = 13 - i,m + n = 3 - 4i.</cmath>
  3 KB (451 words) - 15:02, 6 September 2021
• 1996 AIME Problems/Problem 15
  ...ce as large as angle <math>DBA</math>, and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find <math>\lfloor 1000r \ ...=(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);
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 11
  ...rm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>. This is just a slight variation of Solution 1.
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1997 AIME Problems/Problem 15
  ...</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...{a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 12
  ...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which r
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 2
  ...rectangle]]s, of which <math>s</math> are [[square]]s. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...<math>{9\choose 2}</math> ways to pick the vertical sides, giving us <math>r = 1296</math> rectangles.
  3 KB (416 words) - 21:09, 27 October 2022
• 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
• 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
  In the middle of a vast prairie, a firetruck is stationed at the intersection of two [[perpendicular]] straigh ...,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 8
  ...a right circular [[cone]] is <math>12</math> inches tall and its base has a <math>5</math>-inch [[radius]]. The liquid that is sealed inside is <math>9 ...lume of the liquid can be found by <math>\frac{\pi}{3}r^2h - \frac{\pi}{3}(r')^2h'</math>.
  4 KB (677 words) - 16:33, 30 December 2023
• 2000 AIME I Problems/Problem 7
  Let <math>r = \frac{m}{n} = z + \frac {1}{y}</math>. (5)(29)(r)&=\left(x + \frac {1}{z}\right)\left(y + \frac {1}{x}\right)\left(z + \frac
  5 KB (781 words) - 15:02, 20 April 2024
• 2001 AIME I Problems/Problem 13
  .../math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, whe 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
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</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
• 2001 AIME I Problems/Problem 9
  ...r</math> are positive and satisfy <math>p+q+r=2/3</math> and <math>p^2+q^2+r^2=2/5</math>. The ratio of the area of triangle <math>DEF</math> to the are /* -- arbitrary values, I couldn't find nice values for pqr please replace if possible -- */
  4 KB (673 words) - 20:15, 21 February 2024
• 2001 AIME I Problems/Problem 7
  ...), 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); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D,NW));
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2001 AIME I Problems/Problem 6
  A fair die is rolled four times. The [[probability]] that each of the final t ...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 3
  From [[Vieta's formulas]], in a [[polynomial]] of the form <math>a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0 ...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
• 2002 AIME I Problems/Problem 15
  ...h>r</math> is not divisible by the square of any prime, find <math>p + q + r.</math> 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 7
  ...are not integers. That is, for all real numbers <math>x,y</math> and <math>r</math> with <math>|x|>|y|</math>, ...x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots</cmath>
  2 KB (316 words) - 19:54, 4 July 2013
• 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 Let the [[radius]] of the circles be <math>r</math>. The longer dimension of the rectangle can be written as <math>14r</
  2 KB (287 words) - 19:54, 4 July 2013
• 2003 AIME I Problems/Problem 13
  In base-<math>2</math> representation, all positive numbers have a leftmost digit of <math>1</math>. Thus there are <math>{n \choose k}</math> ...}</math>, so the sum of these elements is <math>\sum_{i=0}^{5} {2i \choose i} = 1 + 2 +6 + 20 + 70 + 252 = 351</math>.
  4 KB (651 words) - 19:42, 7 October 2023
• 2002 AIME II Problems/Problem 12
  ...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...ne <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2000 AIME II Problems/Problem 13
  ...<math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>. Thus <math>r=\frac{-1+\sqrt{161}}{40}</math>, and so the final answer is <math>-1+161+40
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 9
  Given that <math>z</math> is a complex number such that <math>z+\frac 1z=2\cos 3^\circ</math>, find the le ...we have <math>z = \frac{2\cos 3 \pm \sqrt{4\cos^2 3 - 4}}{2} = \cos 3 \pm i\sin 3 = \text{cis}\,3^{\circ}</math>.
  4 KB (675 words) - 13:42, 4 April 2024
• 2006 USAMO Problems/Problem 3
  Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with Let <math>g(n) = f(n^2)</math>, then <math>g(n)</math> is a polynomial of degree <math>2</math> or
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  .../math> is greater than the minimum number of jumps needed to reach <math>2^i</math>. ...the minimum number of jumps needed to reach the integer <math>n_{i,k} = 2^i k</math>. We must prove that
  7 KB (1,280 words) - 17:23, 26 March 2016
• Inradius
  ...s [[incircle]] (assuming an incircle exists). It is commonly denoted <math>r</math>. pair A=(0,0),B=(4,0),C=(1.5,2),I=incenter(A,B,C),F=foot(I,A,B);
  2 KB (336 words) - 00:44, 23 April 2024
• Ring
  ...lgebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multipl * <math>(R,+)</math> is an [[abelian group]];
  6 KB (994 words) - 06:16, 8 April 2015
• Holomorphic function
  A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a rather than at points, for when we consider the behavior of a function
  9 KB (1,537 words) - 21:04, 26 July 2017
• Quadratic reciprocity
  Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer. Then we can define the [[Legendre symbol]] <cmath> \genfrac{(}{)}{}{}{a}{p} =\begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p, \\
  7 KB (1,182 words) - 16:46, 28 April 2016
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 9
  Let <math>a_{n}</math> be a [[geometric sequence]] of [[complex number]]s with <math>a_{0}=1024</math> Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</m
  5 KB (744 words) - 19:46, 20 October 2020
• Mock AIME 1 2006-2007 Problems/Problem 10
  ..., and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive ...s <math>b=\frac{6}{11}</math>, radius <math>c=\frac{2}{5}</math> and <math>r=1</math>, see picture.
  1 KB (236 words) - 23:58, 24 April 2013
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...<math>n</math> are positive integers. Construct circle <math>O</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>. Let ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• 1959 IMO Problems/Problem 5
  (a) Prove that the points <math>N </math> and <math>N' </math> coincide. (b) Prove that the straight lines <math>MN </math> pass through a fixed point <math>S </math> independent of the choice of <math>M </math>.
  4 KB (729 words) - 08:23, 23 May 2024
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 3
  ...scribe]]d in a [[circle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</ma (a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
  6 KB (1,080 words) - 19:28, 21 September 2014
• 2005 Canadian MO Problems/Problem 3
  Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. ...math> on the circle such that <math>a</math> is (strictly) closer to <math>A</math> than any other point in <math>S</math>, <math>b</math> is closer to
  2 KB (460 words) - 13:35, 9 June 2011
• Addition
  ...ign) which combines two quantities. The result of addition is called [[sum|a sum]]. For example, the sum of 3 and 2 is 5 because <math>3+2=5</math>. ...h>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]])
  2 KB (309 words) - 20:34, 4 July 2019
• Volume
  The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volu The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math>
  3 KB (523 words) - 20:24, 17 August 2023
• Zero (constant)
  ...to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.
  2 KB (371 words) - 12:55, 21 June 2023
• 1970 IMO Problems/Problem 1
  ...math> of <math>\triangle ABC</math>. Let <math>r_1, r_2</math>, and <math>r</math> be the inscribed circles of triangles <math>AMC, BMC</math>, and <ma <math>\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.
  2 KB (380 words) - 22:12, 19 May 2015
• Metric space
  ...es S \to \mathbb{R}_{\geq 0}</math>. The metric <math>d</math> represents a distance function between pairs of points of <math>S</math> which has the f ...a that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.
  2 KB (324 words) - 01:19, 22 December 2012
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• 1981 IMO Problems/Problem 2
  ...n \} </math>. Each of these subsets has a smallest member. Let <math>F(n,r) </math> denote the arithmetic mean of these smallest numbers; prove that F(n,r) = \frac{n+1}{r+1}.
  5 KB (879 words) - 11:18, 27 June 2020
• 1987 IMO Problems/Problem 6
  ..., then we can conclude that <math>m</math> is a prime. Since there must be a factor of <math>m</math> less than <math>\sqrt{m}</math>. ...t let <math>r=\lfloor\sqrt{n/3}\rfloor</math>, then we can write <math>n=3(r+\epsilon)^2(0\leq\epsilon< 1)</math>, so <math>h=6r\epsilon+3\epsilon^2\leq
  2 KB (430 words) - 13:03, 24 February 2024
• 1981 IMO Problems/Problem 5
  ...nt <math>O</math> and lie inside a given [[triangle]]. Each circle touches a pair of sides of the triangle. Prove that the [[incenter]] and the [[circum ...vely, and let the centers of the circles inscribed in the [[angle]]s <math>A,B,C</math> be denoted <math>O_A, O_B, O_C </math>, respectively.
  2 KB (373 words) - 23:09, 29 January 2021
• Cube (geometry)
  ...] to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. All edges of cubes are equal to each other. ...on a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
  2 KB (244 words) - 11:04, 24 April 2023
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• Matrix
  ...<math>F</math>, where <math>A</math> and <math>B</math> are the sets <math>A=\{1,2,\ldots,m\}</math> and <math>B=\{1,2,\ldots,n\}</math>. ...>F^m</math>, where <math>m</math> is the number of rows. If a matrix <math>A</math> has <math>m</math> rows and <math>n</math> columns, its order is sai
  4 KB (856 words) - 15:29, 30 March 2013
• 1985 IMO Problems/Problem 1
  A [[circle]] has center on the side <math>AB</math> of the [[cyclic quadrilat ...\frac{ \pi - \angle DAB}{2} = \angle DCO</math>, so <math>DCOT </math> is a cyclic quadrilateral and <math>T </math> is in fact the <math>T</math> of t
  4 KB (684 words) - 07:28, 3 October 2021
• 2001 AMC 8 Problems
  ...class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ball. If it takes him <math>2</math> seconds to paint one dimple, how <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D)
  13 KB (1,994 words) - 13:04, 18 February 2024
• De Moivre's Theorem
  ...<math>n\in\mathbb{Z}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>. (\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \\
  2 KB (363 words) - 01:42, 11 January 2024
• Quaternion
  ...icative [[inverse with respect to an operation | inverse]]; alternatively, a non[[commutative]] [[field]]) which generalize the [[complex number]]s. ...<math>a, b, c, d</math> are any [[real number]]s and the behavior of <math>i, j, k</math> is "as you would expect," with the properties:
  1,007 bytes (155 words) - 20:47, 14 October 2013
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha returns the complex number <math>e^{i\theta}</math>, i.e. (cos(theta),sin(theta)) where theta is measured in radians.
  7 KB (1,205 words) - 21:38, 26 March 2024
• Asymptote: Logical Operators and Loops
  real r=5; size(r*cm);
  2 KB (324 words) - 16:50, 2 October 2016
• 2006 Canadian MO Problems/Problem 2
  Let <math>ABC</math> be an [[acute angle]]d [[triangle]]. Inscribe a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is ...n by [[similarity]], <math>\frac{AE}{AC} = \frac{GH}{BH} = \frac{FH}{CH} = r</math>.
  2 KB (416 words) - 20:00, 21 September 2014
• 2007 AIME II Problems/Problem 9
  ...thing for <math>F</math>, we find that <math>FD = 105</math> as well. Draw a line through <math>E,F</math> parallel to the sides of the rectangle, to in ...tor, <math>7</math>? Then we're left with much simpler numbers which saves a lot of time. In the end, we will multiply by <math>7</math>.
  5 KB (818 words) - 11:05, 7 June 2022
• 2007 AIME II Problems/Problem 10
  .../math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0 or 6 elements, probability: <math>\frac{2}{2^6} =
  8 KB (1,367 words) - 11:48, 23 October 2022
• Euclidean domain
  A '''Euclidean domain''' (or '''Euclidean ring''') is a type of [[ring]] in which the [[Euclidean algorithm]] can be used. Formally we say that a ring <math>R</math> is a Euclidean domain if:
  2 KB (357 words) - 15:28, 22 August 2009
• Mock AIME 1 Pre 2005 Problems
  <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that < ...th>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</ma
  6 KB (1,100 words) - 22:35, 9 January 2016
• 2007 AIME I Problems
  {{AIME Problems|year=2007|n=I}} [[2007 AIME I Problems/Problem 1|Solution]]
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 8
  We can see that <math>Q_1</math> and <math>Q_2</math> must have a [[root]] in common for them to both be [[factor]]s of the same cubic. Let this root be <math>a</math>.
  4 KB (728 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 9
  ...>. Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>. [[Point]]s <math>O_1</math> and <math>O_2</math [[Image:AIME I 2007-9.png]]
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2007 AIME I Problems/Problem 12
  ...+ s</math>, where <math>p,q,r,s</math> are integers. Find <math>\frac{p-q+r-s}2</math>. ...sectorpoint(A,B)), Cp=rotate(theta,A)*C, Bp=rotate(theta,A)*B, X=extension(A,Bp,B,C), Y=extension(B,C,Bp,Cp);
  10 KB (1,458 words) - 20:50, 3 November 2023
• 2007 AIME I Problems/Problem 13
  ...math> and vertex <math>E</math> has eight edges of length <math>4</math>. A plane passes through the midpoints of <math>AE</math>, <math>BC</math>, and [[Image:AIME I 2007-13.png]]
  7 KB (1,034 words) - 21:56, 22 September 2022
• 2007 AIME I Problems/Problem 15
  ...an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math>r</math>. [[Image:AIME I 2007-15.png]]
  4 KB (673 words) - 22:14, 6 August 2022
• 2007 AIME II Problems/Problem 15
  .../math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides pair A,B,C,X,Y,Z,P,Q,R;
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems/Problem 14
  Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> ...2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1</math>. Because <math>f(0) = 1</math>, the product of all the roots
  7 KB (1,335 words) - 17:44, 25 January 2022
• 2007 AIME II Problems
  ...rs among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains ...of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 USAMO Problems/Problem 1
  ...m Vandervelde'') Let <math>n</math> be a [[positive]] [[integer]]. Define a [[sequence]] by setting <math>a_1 = n</math> and, for each <math>k>1</math> ...integers, so <math>b_{k+1} \le b_k</math>. As the <math>b_k</math>'s form a [[non-increasing]] sequence of positive integers, they must eventually beco
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2007 USAMO Problems/Problem 4
  (''Reid Barton'') An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized [[Square (geometr ...two or more dinosaurs. Find with proof the [[maximum]] number of cells in a primitive dinosaur.
  10 KB (1,878 words) - 14:56, 30 June 2021
• 2007 USAMO Problems/Problem 6
  ...[[internally tangent|tangent internally]] to <math>\Omega</math> at <math>A</math> and tangent internally to <math>\omega</math>. Let <math>P_A</math> <cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath>
  7 KB (1,274 words) - 15:11, 31 August 2017
• Circumradius
  ...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==
  4 KB (729 words) - 16:52, 19 February 2024
• LaTeX:Layout
  ...his allows you to use the same code in many source files by just including a single line in each source file. ...font size, which is 10pt by default but can be increased to 11pt or 12pt. A reference on other options for this command can be found [http://www.nada.k
  30 KB (5,171 words) - 10:16, 4 April 2021
• LaTeX:Commands
  | <math>\textstyle 2^{23}</math>||2^{23}||<math>\textstyle n_{i-1}</math>||n_{i-1} | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2}
  12 KB (1,898 words) - 15:31, 22 February 2024
• Dodecagon
  A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\ci A regular dodecagon can be seen below:
  1 KB (219 words) - 13:08, 15 June 2018
• Continuous
  A property of a function. ...<math>x\in I\cap (a-\delta, a+\delta) -\{a\}</math> we have <math>|f(x)-f(a)| < \varepsilon</math>.
  630 bytes (114 words) - 12:52, 15 May 2022
• 2006 AIME II Problems/Problem 11
  A [[sequence]] is defined as follows <math> a_1=a_2=a_3=1, </math> and, for a ~ I-_-I
  3 KB (417 words) - 10:07, 12 October 2023
• 2006 AIME II Problems
  ...vex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \ang ...\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1999 PMWC Problems
  label("$A$", (3.9,6.75), NE * labelscalefactor); pair P,Q,R,S;
  6 KB (703 words) - 21:21, 21 April 2014
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2002 PMWC Problems
  for(int i = 0; i < nrows; ++i) for(int j = 0; j <= i; ++j)
  5 KB (725 words) - 16:07, 23 April 2014
• 1989 USAMO Problems/Problem 4
  ...lengths satisfy the inequalities <math>AB < AC < BC</math>. If point <math>I</math> is the center of the inscribed circle of triangle <math>ABC</math> a ..., <math>CO</math> to <math>D</math>,<math>E</math>,<math>F</math> on <math>a</math>,<math>b</math>,<math>c</math>, respectively.
  2 KB (290 words) - 19:11, 18 July 2016
• Factor Theorem
  ...theorem''' is a theorem regarding the relationships between the factors of a polynomial and its roots. ...y, you can determine whether a number in the form <math>f(a)</math> (<math>a</math> is constant, <math>f</math> is polynomial) is <math>0</math> using p
  3 KB (508 words) - 12:31, 2 May 2024
• Summation
  A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written using sigma notatio ...+c_{b}</math>. Here <math>i</math> refers to the index of summation, <math>a</math> is the lower bound, and <math>b</math> is the upper bound.
  3 KB (482 words) - 16:39, 8 October 2023
• 1962 IMO Problems
  == Day I == (a) Its decimal representation has 6 as the last digit.
  3 KB (431 words) - 21:17, 20 August 2020
• Relation
  ...e structure that can be placed on a pair or collection of [[set]]s to give a notion of "relatedness" to the [[element]]s of those sets. ...al number]]s that indicates that a first number is exactly one larger than a second number, or the relation betweem the [[integer]]s and the set <math>\
  2 KB (445 words) - 08:16, 21 August 2009
• Ideal
  In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorp ...xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as
  8 KB (1,389 words) - 23:44, 17 February 2020
• Binet's Formula
  To derive a general formula for the Fibonacci numbers, we can look at the interesting q ...onacci recursive relation is <math>F_n = F_{n-1} + F_{n-2}.</math> This is a constant coefficient linear homogenous recurrence relation. We also know th
  6 KB (953 words) - 21:37, 30 May 2024
• 1976 USAMO Problems/Problem 5
  If <math>P(x)</math>, <math>Q(x)</math>, <math>R(x)</math>, and <math>S(x)</math> are all [[polynomial]]s such that <cmath> P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x), </cmath>
  3 KB (572 words) - 17:14, 16 August 2015
• Graph (graph theory)
  ...[[empty set | nonempty]] [[set]] of [[vertex|vertices]] that are joined by a number (possibly zero) of [[edge]]s. Graphs are frequently represented gra ...' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For graphs with multiple
  8 KB (1,428 words) - 10:26, 27 August 2020
• 1995 AHSME Problems
  ...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1997 PMWC Problems
  ...represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{ ...&\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array} </cmath>
  15 KB (2,057 words) - 19:13, 10 March 2015
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  <math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1974 USAMO Problems/Problem 1
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. ...P</math> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</cmath> (Why?)
  7 KB (1,291 words) - 20:30, 27 April 2020
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. <i><b>Isogonal lines definition</b></i>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 1976 IMO Problems/Problem 2
  ...th>2^{n-1}</math> are negative. Also, for ever root <math>r</math>, <math>|r|<2</math>. ...<s<2</math>, so <math>0<r^2<4</math>, so <math>r</math> is real and <math>|r|<2</math>. Therefore all of the roots of <math>P_{k+1}</math> are real and
  3 KB (596 words) - 16:19, 28 July 2015
• 2007 AMC 12B Problems/Problem 24
  ...</math> are there such that <math>\text{gcd}(a,b)=1</math> and <math>\frac{a}{b} + \frac{14b}{9a}</math> is an integer? <math>\mathrm {(A)}\ 4\quad\mathrm {(B)}\ 6\quad\mathrm {(C)}\ 9\quad\mathrm {(D)}\ 12\quad\m
  9 KB (1,522 words) - 22:46, 12 May 2022
• 1992 USAMO Problems/Problem 3
  ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible ...ere <math>\forall i<n</math>, <math>i\in \mathbb{Z}^+</math>, <math>a_i<a_{i+1}</math>, and for each <math>x\le p</math>, <math>x\in \mathbb{Z}^+</math>
  5 KB (858 words) - 07:52, 19 July 2016
• 2004 AMC 12A Problems/Problem 19
  ...th>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has radius <math>1</math> and passes through the center of <math>D</ pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
  5 KB (785 words) - 00:29, 31 July 2023
• 2008 AMC 12B Problems
  A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AIME I Problems
  {{AIME Problems|year=2008|n=I}} Of the students attending a school party, <math>60\%</math> of the students are girls, and <math>40\%</
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  ...te which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin ...by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> cm and is cut parallel to one face of the cheese. T
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 5
  ...ble after making <math>17</math> complete rotations. The value of <math>h/r</math> can be written in the form <math>m\sqrt {n}</math>, where <math>m</m ...t {r^{2} + h^{2}}</math>. Thus, the length of the path is <math>2\pi\sqrt {r^{2} + h^{2}}</math>.
  1 KB (230 words) - 20:18, 4 July 2013
• 2008 AIME I Problems/Problem 6
  ...he first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of th ...h row be <math>a(n,k)</math>. Writing out some numbers, we find that <math>a(n,k) = 2^{n-1}(n+2k-2)</math>.{{ref|1}}
  3 KB (509 words) - 17:21, 22 March 2018
• 2008 AIME I Problems/Problem 11
  ...math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? ...nding two <math>B</math>s to a string of length <math>n-2</math> ending in a <math>B</math>. Thus, we have the [[recursion]]s
  7 KB (1,173 words) - 22:39, 28 November 2023
• 2008 AIME I Problems/Problem 15
  .... When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of t real r=(sqrt(51)+s)/sqrt(2);
  6 KB (1,041 words) - 00:54, 1 February 2024
• 2008 AIME I Problems/Problem 14
  ...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...= O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);
  8 KB (1,333 words) - 00:18, 1 February 2024
• Mock AIME 5 Pre 2005 Problems
  &\text {i. In form of abcde and fghij such that fghij = 2(abcde)}\\ &\text {ii. all ten digits, a through j are all distinct.}\\
  6 KB (909 words) - 07:27, 12 October 2022
• Mock AIME 1 2007-2008 Problems
  A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange Let <math>S = (1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.
  6 KB (992 words) - 14:15, 13 February 2018
• 2008 AIME II Problems/Problem 9
  ...ckwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-directi ...the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> an
  5 KB (725 words) - 22:37, 28 January 2024
• 2008 AIME II Problems/Problem 10
  The diagram below shows a <math>4\times4</math> rectangular array of points, each of which is <math>1 int i, j;
  4 KB (569 words) - 09:44, 25 November 2019
• 2008 AIME II Problems/Problem 11
  real r=44-6*35^.5; pair A=(0,96),B=(-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M=foot(Q,X,P);
  6 KB (1,065 words) - 20:12, 9 August 2022
• 2005 USAMO Problems/Problem 1
  ...sible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. ...>D_n</math> is placed exactly one time, and any two adjacent elements have a common prime factor. Hence this arrangement has the desired property.
  4 KB (650 words) - 13:40, 4 July 2013
• Cone
  ...three-dimensional [[solid]]. It consists of a [[circle | circular]] base, a [[point]] (called the ''vertex''), and all the points that lie on [[line se path3 rightanglemark(triple A, triple B, triple C, real s=8) { // olympiad package
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2008 AIME II Problems/Problem 13
  ...where <math>a</math> and <math>b</math> are positive integers. Find <math>a + b</math>. ...its conjugate but we can do that in the Cartesian plane too (just reflect a point in the Cartesian plane over the x-axis)! If you're familiar with inve
  6 KB (894 words) - 18:56, 25 December 2022
• 2008 AMC 10A Problems/Problem 15
  <math>\mathrm{(A)}\ 120\qquad\mathrm{(B)}\ 130\qquad\mathrm{(C)}\ 140\qquad\mathrm{(D)}\ 150 Set the time Ian traveled as <math>I</math>, and set Han's speed as <math>H</math>. Therefore, Jan's speed is <m
  6 KB (1,033 words) - 15:19, 1 July 2021
• 2006 IMO Problems/Problem 5
  ...gree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x ...ose on the other hand that <math>a_i \neq b_i</math>, for some index <math>i</math>. In this case, we claim that <math>a_j + b_j</math> is constant for
  3 KB (704 words) - 14:42, 7 September 2021
• 2008 USAMO Problems/Problem 1
  <cmath> k_0 k_1 \dotsc k_n = a^2+a+1, </cmath> ...on <math>n</math>. For our base case, <math>n=0</math>, we may let <math>a</math> be positive integer.
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 USAMO Problems/Problem 4
  ...- 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles tria ..., P_2, \ldots, P_n</math>. Consider a diagonal <math>d = \overline{P_a\,P_{a+k}},\,k \le n/2</math> in the triangulation. We show that <math>k</math> mu
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2008 USAMO Problems/Problem 3
  (''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) .../math> paths (a partition of <math>S_n</math> into <math>m</math> paths is a set <math>\mathcal{P}</math> of <math>m</math> nonempty paths such that eac
  9 KB (1,585 words) - 01:00, 14 August 2014
• Normal subgroup
  ...al subgroup''' <math>{\rm H}</math> of a [[group]] <math>{\rm G}</math> is a [[subgroup]] of <math>{\rm G}</math> for which the relation "<math>xy^{-1} <math>N</math> is said to be a normal subgroup of a group <math>G</math> if <math>aNa^{-1}=N</math>.Note that this means <math>
  15 KB (2,840 words) - 12:22, 9 April 2019
• Free magma
  ...'free magma''' is [[magma]] structure that is as general as possible&mdash;a magma generated from an initial set with no constraints or relations. The free magma generated from a [[set]] <math>X</math> is constructed as follows.
  4 KB (887 words) - 13:19, 6 July 2016
• KGS math club/solution 6 1
  ...ing to write a solution for general <math>n</math> (number of the balls in a group, set to 5 if you like). The problem statement is thus following: ...oof that they all must weigh the same. Can you find a proof? Can you find a way to extend the result to the general case where the balls have real weig
  4 KB (759 words) - 06:27, 18 July 2009
• 1998 AHSME Problems
  ...ides of five congruent rectangles is labeled with an integer. In rectangle A, <math>w = 4, x = 1, y = 6, z = 9</math>. In rectangle B, <math>w = 1, x = <math> \mathrm{(A)\ } A \qquad \mathrm{(B) \ }B \qquad \mathrm{(C) \ } C \qquad \mathrm{(D) \ } D
  15 KB (2,222 words) - 10:40, 11 August 2020
• Routh's Theorem
  ...ath> be the intersection of <math>BE</math> and <math>CF</math>, and <math>I</math> be the intersection of <math>CF</math> and <math>AD</math>. Then, '' <cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath>
  2 KB (267 words) - 00:02, 24 March 2021
• Carnot's Theorem
  ...dicular]] distances from the [[circumcenter]] <math>O</math> to the sides (i.e., signed lengths of the pedal lines from <math>O</math>) is: <math>OO_A+OO_B+OO_C=R+r</math>
  4 KB (723 words) - 01:45, 18 February 2021
• 2008 IMO Problems/Problem 2
  '''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real num Consider the transormation <math>f:\mathbb{R}/\{1\} \rightarrow \mathbb{R}/\{-1\}</math> defined by <math>f(u) = \frac{u}{1-u}</math> and put <math>\
  3 KB (593 words) - 01:09, 19 November 2023
• 2000 USAMO Problems
  Call a real-valued function <math>f</math> very convex if ...{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},</cmath>
  3 KB (495 words) - 19:02, 18 April 2014
• 2008 iTest Problems
  ...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 80
  and let <math>r(x)</math> be the polynomial remainder when <math>p(x)</math> is divided by the remainder when <math>|r(2008)|</math> is divided by <math>1000</math>.
  3 KB (560 words) - 19:49, 23 November 2018
• Van Aubel's Theorem
  ...: <math>ABA'B'</math>, <math>BCB'C'</math>, <math>CDC'D'</math>, <math>DAD'A'</math>; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van pair A, B, C, D, O, P, Q, R, SS;
  2 KB (410 words) - 14:01, 4 March 2023
• 2002 USAMO Problems
  Let <math>S </math> be a set with 2002 elements, and let <math>N </math> be an integer with <math> 0 (a) the union of any two red subsets is red;
  3 KB (486 words) - 09:21, 14 May 2021
• 1992 USAMO Problems/Problem 4
  ...<math>C</math>, and <math>P</math> is tangent to the sphere through <math>A'</math>, <math>B'</math>, <math>C'</math>, and <math>P</math>. Prove that ...ause the base angles are equal. Thus, <math>AP=BP</math>. Similarly, <math>A'P=B'P</math>. Thus, <math>AA'=BB'</math>. By symmetry, <math>BB'=CC'</math>
  5 KB (807 words) - 18:37, 25 June 2021
• 1972 USAMO Problems/Problem 1
  ...or and least common multiple, respectively, of the positive integers <math>a,b,\ldots, g</math>. For example, <math>(3,6,18)=3</math> and <math>[6,15]=3 <center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center>
  5 KB (1,018 words) - 11:14, 6 October 2023
• 1974 USAMO Problems
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. Prove that if <math>a</math>, <math>b</math>, and <math>c</math> are positive real numbers, then
  3 KB (427 words) - 18:55, 3 July 2013
• 1962 IMO Problems/Problem 6
  Consider an isosceles triangle. Let <math>r</math> be the radius of its circumscribed circle and <math>\rho</math> the <center><math>d=\sqrt{r(r-2\rho)}</math>.</center>
  2 KB (308 words) - 06:29, 16 December 2023
• 1976 USAMO Problems
  int i; for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);}
  3 KB (510 words) - 19:01, 3 July 2013
• 1999 AHSME Problems/Problem 21
  ...ath> thus dividing the interior of the circle into four regions. Let <math>A,B,</math> and <math>C</math> be the areas of the non-triangular regions, wi ...2+B^2=C^2 \qquad \mathrm{(D) \ }20A+21B=29C \qquad \mathrm{(E) \ } \frac 1{A^2}+\frac 1{B^2}= \frac 1{C^2}</math>
  2 KB (396 words) - 18:47, 7 September 2023
• 2004 AMC 10B Problems/Problem 22
  A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those cir <math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{
  5 KB (700 words) - 13:46, 6 April 2024
• 1971 IMO Problems/Problem 6
  ...r an element <math>a_{ij} = 0</math>, the sum of the elements in the <math>i</math>th row and the <math>j</math>th column is <math>\geq n</math>. Prove ...S</math> (because the sum of the elements of a row and a column meeting in a zero is <math> \ge n</math>) plus the number of nonzero elements times <mat
  6 KB (1,192 words) - 14:14, 29 January 2021
• 2009 AMC 12A Problems/Problem 15
  For what value of <math>n</math> is <math>i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i</math>? Note: here <math>i = \sqrt { - 1}</math>.
  4 KB (634 words) - 16:34, 3 December 2020
• 2002 AMC 12A Problems/Problem 24
  ...d pairs of real numbers <math>(a,b)</math> such that <math>(a+bi)^{2002} = a-bi</math>. \text{(A) }1001
  4 KB (743 words) - 19:54, 14 March 2024
• 2007 AMC 10B Problems/Problem 25
  ...ny pairs of positive integers <math>(a,b)</math> are there such that <math>a</math> and <math>b</math> have no common factors greater than <math>1</math <cmath>\frac{a}{b} + \frac{14b}{9a}</cmath>
  8 KB (1,425 words) - 23:05, 21 June 2023
• 2009 AMC 12B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\t
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 20
  ...h> pyramids and a new polyhedron <math>R</math>. How many edges does <math>R</math> have? <math>\mathrm{(A)}\ 200\qquad
  9 KB (1,567 words) - 13:43, 19 August 2023
• 2009 AMC 10B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E
  15 KB (2,262 words) - 00:53, 18 June 2021
• 2009 AMC 10B Problems/Problem 24
  int i; real r=5, R=6;
  3 KB (546 words) - 04:26, 16 January 2023
• 2009 AIME I Problems/Problem 2
  ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that Let <math>z = a + 164i</math>.
  5 KB (850 words) - 12:53, 23 December 2023
• 2009 AIME I Problems/Problem 12
  ...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat First, note that <math>AB=37</math>; let the tangents from <math>I</math> to <math>\omega</math> have length <math>x</math>. Then the perimete
  12 KB (1,970 words) - 22:53, 22 January 2024
• 1966 AHSME Problems/Problem 4
  ...ath> is the ratio of the area of circle I to that of circle II, then <math>r</math> equals: <math>\text{(A) } \sqrt{2} \quad \text{(B) } 2 \quad \text{(C) } \sqrt{3} \quad \text{(D)
  1 KB (191 words) - 22:09, 14 January 2018
• Maximal ideal
  ...<math>I\neq R</math>, and there is no ideal <math>J</math> with <math>I<J<R</math>.) ...h>R/I</math> is a [[field]] iff <math>I</math> is a maximal ideal of <math>R</math>.
  968 bytes (183 words) - 19:50, 23 August 2009
• Polynomial ring
  ...polynomials in a [[commutative]] <math>x</math> with coefficients in <math>R</math>." That is, it is the ring of all sums of the form where <math>N</math> is a [[nonnegative integer]] that varies from sum to sum.
  12 KB (2,010 words) - 00:10, 3 August 2020
• Cauchy's Integral Formula
  '''Cauchy's Integral Formula''' is a fundamental result in [[complex analysis]]. It states that if <math>U</math> is a subset of
  4 KB (689 words) - 17:19, 18 January 2024
• Liouville's Theorem (complex analysis)
  In [[complex analysis]], '''Liouville's Theorem''' states that a [[Picard's Little Theorem]] is a stronger result.
  2 KB (412 words) - 20:30, 16 January 2024
• 2009 AIME II Problems
  ...ng a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finis ...ath>b</math>, and <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_
  8 KB (1,366 words) - 21:33, 3 January 2021
• 2000 AMC 8 Problems
  <math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qqu <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua
  15 KB (2,165 words) - 03:32, 13 April 2024
• 2009 AIME II Problems/Problem 13
  Let <math>A</math> and <math>B</math> be the endpoints of a semicircular arc of radius <math>2</math>. The arc is divided into seven co ...th>A</math> and <math>B</math>. Assume <math>C_1</math> is closer to <math>A</math> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac
  8 KB (1,279 words) - 20:27, 17 May 2024
• 2009 USAMO Problems/Problem 1
  ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...and <math>O_2</math> , <math>O_3</math> respectively. Hence, by [[power of a point]](the power of <math>O_1</math> can be expressed using circle <math>\
  2 KB (290 words) - 13:16, 17 April 2021
• Radical axis
  The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  12 KB (2,125 words) - 08:38, 23 May 2024
• KGS math club/solution 5 1
  ...nce P has nonzero degree, n>0. The Taylor polynomial of P about the point a is <math>F(x)=P(a)+P'(a)(x-a)+\dotsb +\dfrac{P^{(n)}(a)(x-a)^n}{n!}=P(x)</math>.
  2 KB (305 words) - 19:24, 21 December 2015
• Base introduction
  Bases. These confuse a lot of us. Hopefully this will make it better. ...as a subscript. Every number in base 10 (our normal numbers) does not have a subscript.
  3 KB (552 words) - 20:57, 9 July 2009
• 1965 IMO Problems/Problem 4
  Then, for <math>i = 1,2,3,4</math> we have: <math>x_i + \prod_{j \neq i}x_j = 2</math>. ...rightarrow x_i = 1 \pm t</math> where <math>t = \pm \sqrt{1-P} \in \mathbb{R}</math>.
  2 KB (382 words) - 12:51, 29 January 2021
• 2009 USAMO Problems
  ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.
  4 KB (718 words) - 18:16, 17 September 2012
• 2009 USAMO Problems/Problem 6
  ...i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>. ...ac{c_2}{d_2} - \frac{c_1}{d_1})</math> is an integer. Now we can set <math>r = \frac{b_1 b_2}{d_1 d_2}</math>, because the least common denominator of <
  2 KB (463 words) - 12:19, 21 August 2020
• Schroeder-Bernstein Theorem
  ...orem''' (sometimes called the '''Cantor-Schroeder-Bernstein Theorem''') is a result from [[set theory]], named for Ernst Schr&ouml;der and Felix Bernste ...ath> and <math>g : B \to A</math>, then there is a [[bijection]] <math>h : A \to B</math>.
  4 KB (805 words) - 13:09, 20 February 2024
• 1992 AJHSME Problems
  <math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad <math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfra
  17 KB (2,346 words) - 13:36, 19 February 2020
• Semisimple module
  A '''semisimple module''' is, informally, a [[module]] that is not Specifically, it is a module <math>M</math> with the following property:
  6 KB (1,183 words) - 15:02, 18 August 2009
• Constructible number
  ...can be constructed with a [[straight edge]] and [[compass]] starting with a segment of length <math>1</math>. ...e as a real number if and only if it is constructible as a complex number, i.e., our two definitions coincide in this case.
  8 KB (1,305 words) - 08:39, 21 August 2009
• Unique factorization domain
  ...ization domain if for any nonzero element <math>r\in R</math> which is not a [[unit (ring theory)|unit]]: ..._n</math> are (not necessarily distinct) [[irreducible element]]s in <math>R</math>.
  6 KB (1,217 words) - 23:05, 23 August 2009
• 1994 IMO Problems/Problem 3
  * (a) Prove that, for each positive integer <math>m</math>, there exists at leas '''a) Surjectivity of f'''
  7 KB (1,298 words) - 19:59, 9 February 2024
• 2010 AMC 12A Problems
  {{AMC12 Problems|year=2010|ab=A}} <math>\textbf{(A)}\ -4020 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 10A Problems
  {{AMC10 Problems|year=2010|ab=A}} \mathrm{(A)}\ 1
  13 KB (1,902 words) - 11:20, 5 March 2023
• Uniform convergence
  ...f(x)| < \varepsilon</math>. (More generally, we can replace <math>\mathbb{R}</math> with any [[metric space]] <math>Y</math>.) ...r condition, because (speaking informally) the function has to converge at a similar rate everywhere on its [[domain]].
  4 KB (622 words) - 19:20, 3 March 2010
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...ich is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identity matrix]].
  19 KB (3,412 words) - 14:57, 21 September 2022
• Proofs without words
  Fibonacci: sum_{i=1}^{2n-1} F_iF_{i+1} = F_{2n}^2 rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2)
  55 KB (7,986 words) - 17:04, 20 December 2018
• Pell equation
  ...e of [[diophantine equation]] in the form <math>x^2-Dy^2 = \pm1</math> for a [[natural number]] <math>D</math>. Generally, <math>D</math> is taken to be Note that if <math>D = d^2</math> is a perfect square, then this problem can be solved using [[difference of squar
  6 KB (1,076 words) - 10:20, 29 October 2023
• 2010 AIME I Problems
  {{AIME Problems|year=2010|n=I}} ...h>p</math> be the probability that exactly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 3
  ...ath> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>. Now we proceed by the logarithm rule <math>\log(ab)=\log a + \log b</math>. The equation becomes:
  3 KB (465 words) - 22:07, 17 December 2021
• 2010 AIME I Problems/Problem 6
  Let <math>P(x)</math> be a [[quadratic]] polynomial with real coefficients satisfying <math>x^2 - 2x + real R(real x) { return 2*(x-1)^2+1; }
  6 KB (1,019 words) - 20:39, 20 November 2023
• 2010 AIME I Problems/Problem 8
  ...is the union of a [[circle]] and its interior). The minimum value of <math>r</math> can be written as <math>\frac {\sqrt {m}}{n}</math>, where <math>m</ ...ac12,\frac12\right)</math> to the furthest point on a box in the middle of a quadrant, for instance <math>(5,4)</math>, is <math>\sqrt {\frac92^2 + \fra
  5 KB (891 words) - 00:39, 12 February 2021
• 2010 AIME I Problems/Problem 11
  ...- x| + y \le 10</math> and <math>3y - x \ge 15</math>. When <math>\mathcal{R}</math> is revolved around the [[line]] whose equation is <math>3y - x = 15 pair A = (8,10), B = (4.5,6.5), C= (9.75,8.25), F=foot(A,B,C), G=2*F-A; fill(A--B--C--cycle,rgb(0.9,0.9,0.9));
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME I Problems/Problem 13
  .../math>, respectively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose label("$A$",(-16.43287,-9.3374),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• 2010 AIME I Problems/Problem 15
  ...th>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the [[incircle]]s of <math>\t label("r",(10.19662,1.92704),SE);
  14 KB (2,210 words) - 13:14, 11 January 2024
• Mock AIME 2 2010 Problems
  ...{10})</math>, with <math>0 \le b_i \le 2, 3|a_i-b_i</math> for <math>1 \le i \le 10</math>. Given that <math>f</math> can take on <math>K</math> distinc ...where <math>a, b</math> are relatively prime positive integers. Find <math>a+b</math>.
  7 KB (1,150 words) - 09:10, 8 October 2018
• Mock AIME 2 2010 Answer Key
  ...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...{10})</math>, with <math>0 \le b_i \le 2, 3|a_i-b_i</math> for <math>1 \le i \le 10</math>. Given that <math>f</math> can take on <math>K</math> distinc
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2007 IMO Problems
  ..._1, a_2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define <cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath>
  3 KB (505 words) - 09:24, 10 September 2020
• 2010 AIME II Problems
  A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from ...where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such t
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 7
  ...w+9i</math>, and <math>2w-4</math>, where <math>i^2=-1</math>. Find <math>|a+b+c|</math>.<!-- don't remove the following tag, for PoTW on the Wiki front Set <math>w=x+yi</math>, so <math>x_1 = x+(y+3)i</math>, <math>x_2 = x+(y+9)i</math>, <math>x_3 = 2x-4+2yi</math>.
  4 KB (735 words) - 02:41, 27 June 2022
• 2010 AIME II Problems/Problem 9
  ...of the smaller hexagon to the area of <math>ABCDEF</math> be expressed as a fraction <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> pair A,B,C,D,E,F;
  5 KB (857 words) - 22:22, 27 August 2023
• 1992 USAMO Problems/Problem 5
  Let <math>\, P(z) \,</math> be a polynomial with complex coefficients which is of degree <math>\, 1992 \,</m if and only if <math>Q\left(z_{i}\right)=0</math> for every <math>i\in\left\{1;\;2;\;\ldots;\;1992\right\}</math>. So it is enough to show that
  5 KB (825 words) - 02:21, 2 September 2016
• 1993 USAMO Problems
  ...math>, determine, with proof, which of the two positive real numbers <math>a</math> and <math>b</math> satisfying <center><math>a^n = a + 1, \quad b^{2n} = b + 3a</math></center>
  3 KB (427 words) - 08:25, 19 July 2016
• 1993 USAMO Problems/Problem 3
  Consider functions <math>f : [0, 1] \rightarrow \mathbb{R}</math> which satisfy <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td>(i)</td><td><math>f(x)\ge0</math> for all <math>x</math> in <math>[0, 1]</math
  3 KB (523 words) - 17:03, 13 August 2023
• 2010 USAMO Problems
  ...diameter <math>AB</math>. Denote by <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> the feet of the perpendiculars from <math>Y</math> o ...to make more than <math>\binom{n}{3}</math> such switches before reaching a position in which no further switches are possible.
  3 KB (525 words) - 13:44, 4 July 2013
• 2010 USAMO Problems/Problem 3
  the inequality <math>a_ia_j \le i+j</math> for all distinct indices <math>i, j</math>. <cmath> \prod_{i=1}^{1005}(4i-1) = 3\times 7 \times \ldots \times 4019. </cmath>
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2010 USAJMO Problems
  ...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. <li> <math>x_i + x_{n-i} = 2n</math> for all <math>i = 1, 2, \ldots, n - 1</math>;
  3 KB (538 words) - 13:55, 16 June 2020
• 2010 USAJMO Problems/Problem 1
  ...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...> be the set of positive perfect squares. We claim that the relation <math>R = \{(j, k)\in [n]\times[n]\mid jk\in S\}</math> is an equivalence relation
  12 KB (2,338 words) - 20:30, 13 February 2024
• 1984 USAMO Problems/Problem 2
  ...(\text{i})\quad</math> For which positive integers <math>n</math> is there a finite set <math>S_n</math> of <math>n</math> distinct positive integers su a) We claim that for any numbers <math>p_1</math>, <math>p_2</math>, ... <mat
  4 KB (855 words) - 03:12, 23 October 2022
• 1978 USAMO Problems
  Given that <math>a,b,c,d,e</math> are real numbers such that <math>a+b+c+d+e=8</math>,
  2 KB (372 words) - 19:06, 3 July 2013
• 1995 AJHSME Problems
  <math>\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 4 <math>\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qqu
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2010 AMC 12B Problems/Problem 21
  Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(1) = P(3) = P(5) = P(7) = a</math>,
  7 KB (1,166 words) - 00:04, 26 February 2024
• 2019 IMO Problems/Problem 6
  ...perpendicular to <math>EF</math> meets <math>\omega</math> again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumci ...t lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.
  5 KB (792 words) - 01:52, 19 November 2023
• 1996 AJHSME Problems
  <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare
  13 KB (1,880 words) - 13:35, 19 February 2020
• Python reference
  This page is intended to serve as a brief, beginners reference to the [[Python]] programming language. In Python, there is no need to specify a type when storing information in a variable. It is basic and straightforward to store information.
  28 KB (4,762 words) - 21:20, 12 June 2023
• 1974 IMO Problems
  ...ritten. These three numbers <math>p, q, r</math> satisfy <math>0 < p < q < r</math>. The three cards are shuffled and one is dealt to each player. Each ...and <math>C</math> has 9. At the last round <math>B</math> received <math>r</math> counters. Who received <math>q</math> counters on the first round?
  3 KB (459 words) - 15:53, 29 January 2021
• 2010 IMO Problems
  ...athbb{R}\rightarrow\mathbb{R}</math> such that for all <math>x,y\in\mathbb{R}</math> the following equality holds ...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math>
  4 KB (603 words) - 09:22, 10 September 2020
• 1974 IMO Problems/Problem 1
  ...ritten. These three numbers <math>p, q, r</math> satisfy <math>0 < p < q < r</math>. The three cards are shuffled and one is dealt to each player. Each ...and <math>C</math> has 9. At the last round <math>B</math> received <math>r</math> counters. Who received <math>q</math> counters on the first round?
  2 KB (418 words) - 15:46, 24 November 2010
• Fractional ideal
  ...t of <math>K</math> such that for any <math>x,y\in I</math> and <math>r\in R</math>: *<math>x+y\in I</math>
  2 KB (288 words) - 20:05, 23 January 2017
• Hölder's Inequality
  Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (762 words) - 00:11, 18 June 2023
• 1999 AMC 8 Problems/Problem 15
  ...letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qq
  3 KB (516 words) - 14:50, 21 December 2022
• 2011 AMC 12A Problems
  {{AMC12 Problems|year=2011|ab=A}} A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</m
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 25
  ...<math>BC=1</math>, and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter o \textbf{(A)}\ 60^{\circ} \qquad
  6 KB (1,046 words) - 13:05, 28 June 2022
• 2011 AMC 12A Problems/Problem 23
  ...a</math> and <math>b</math> are complex numbers. Suppose that <math>\left| a \right| = 1</math> and <math>g(g(z))=z</math> for all <math>z</math> for wh \textbf{(A)}\ 0 \qquad
  3 KB (519 words) - 15:49, 5 November 2023
• 1993 AHSME Problems
  ...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2011 AIME I Problems/Problem 4
  ...h>AC=117</math> and <math>BC=120</math>. The angle bisector of angle <math>A</math> intersects <math> \overline{BC} </math> at point <math>L</math>, and pair A,B,C,D,K,L,M,N,P,Q;
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems
  {{AIME Problems|year=2011|n=I}} ...olution in jar <math>C</math> is added to jar B. At the end both jar <math>A</math> and jar <math>B</math> contain solutions that are <math>50\%</math>
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2002 AMC 8 Problems
  A [[circle]] and two distinct [[Line|lines]] are drawn on a sheet of paper. What is the largest possible number of points of intersecti <math>\text {(A)}\ 2 \qquad \text {(B)}\ 3 \qquad {(C)}\ 4 \qquad {(D)}\ 5 \qquad {(E)}\ 6<
  15 KB (2,102 words) - 09:58, 5 May 2024
• 2011 AIME I Problems/Problem 11
  ...is divided by 1000. Let <math>S</math> be the sum of the elements in <math>R</math>. Find the remainder when <math>S</math> is divided by 1000. ...</math> and <math>j =3</math> are the first two integers such that <math>2^i \equiv 2^j \pmod{1000}</math>. All that is left is to find <math>S</math> i
  11 KB (1,668 words) - 22:10, 24 February 2023
• Bézout's Identity
  ...ath> such that <math>x\alpha+y\beta=g</math>. In other words, there exists a linear combination of <math>x</math> and <math>y</math> equal to <math>g</m ...math> is the smallest positive integer that can be expressed in this form, i.e. <math>g = \min\{x\alpha+y\beta|\alpha,\beta\in\mathbb Z, x\alpha+y\beta
  4 KB (768 words) - 16:50, 6 September 2023
• 2011 AIME I Problems/Problem 13
  ...ath>, <math>s</math>, and <math>t</math> are positive integers. Find <math>r+s+t</math>. ...h>a^2+b^2+c^2=1</math>. The distance from a point <math>(X,Y,Z)</math> to a plane with equation <math>Ax+By+Cz+D=0</math> is
  4 KB (613 words) - 11:37, 8 September 2022
• 2011 AIME II Problems/Problem 10
  A [[circle]] with center <math>O</math> has radius 25. [[Chord]] <math>\overl The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma
  11 KB (1,720 words) - 03:12, 18 December 2023
• 1999 AMC 8 Problems
  <math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock?
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2011 AMC 10B Problems
  <math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad <math> \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 85 \qquad\textbf{(D)}\ 90
  13 KB (2,090 words) - 18:05, 7 January 2021
• 2011 AMC 10B Problems/Problem 20
  ...h>B</math> than any of the other three vertices. What is the area of <math>R</math>? <math> \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad\textbf{
  6 KB (938 words) - 23:22, 31 August 2023
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton
  17 KB (2,261 words) - 00:30, 22 April 2024
• Mock Amc 12
  ...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 ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.
  18 KB (2,206 words) - 19:41, 24 December 2020
• 1999 AMC 8 Problems/Problem 4
  for (int a = 0; a < 6; ++a) dot((4*a,3*b));
  1 KB (196 words) - 21:02, 14 April 2017
• 2012 AMC 12B Problems/Problem 23
  ...integers, <math>0\le d\le c\le b\le a\le 4</math>, and the polynomial has a zero <math>z_0</math> with <math>|z_0|=1.</math> What is the sum of all val <math>\textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\q
  11 KB (1,979 words) - 17:25, 6 September 2021
• 1967 AHSME Problems
  ...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E
  20 KB (3,108 words) - 14:14, 20 February 2020
• 2003 AMC 8 Problems
  Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of <math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22
  16 KB (2,236 words) - 12:02, 19 February 2024
• 1997 USAMO Problems/Problem 6
  <math>a_i+a_j \le a_{i+j} \le a_i+a_j+1</math> ...h>i, j \ge 1</math> with <math>i+j \le 1997</math>. Show that there exists a real number <math>x</math> such that <math>a_n=\lfloor{nx}\rfloor</math> (t
  6 KB (1,107 words) - 14:12, 12 April 2023
• 2011 IMO Problems
  ...inct positive integers which achieve the largest possible value of <math>n(A)</math>. ...d the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.
  4 KB (682 words) - 00:12, 18 February 2021
• 2009 AMC 8 Problems
  Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\
  18 KB (2,551 words) - 18:46, 27 February 2024
• 1987 USAMO Problems
  ...sector of <math>\angle A</math>, find all possible values for <math>\angle A</math>. ...<math>X</math>, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to <math>X</math>.
  2 KB (356 words) - 18:42, 18 July 2016
• 1988 USAMO Problems
  ...e t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. ...ath>X</math>. Show that for any map <math>f: P\mapsto X</math> we can find a 10-element subset <math>Y</math> of <math>X</math>, such that <math>f(Y-\{k
  2 KB (319 words) - 16:33, 5 February 2018
• 2002 AMC 8 Problems/Problem 7
  ...a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose ca real[] r={6, 8, 4, 2, 5};
  1 KB (196 words) - 00:41, 5 July 2013
• 1999 AMC 8 Problems/Problem 25
  ...ht triangle <math>ACG</math>. Points <math>K</math>, <math>E</math>, <math>I</math> are midpoints of the sides of triangle <math>JDG</math>, etc. If the label("$A$",(0,0),SW);
  4 KB (655 words) - 05:21, 12 March 2024
• 1996 AJHSME Problems/Problem 5
  The letters <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the for (int i = -3; i <= 3; ++i)
  3 KB (506 words) - 00:24, 5 July 2013
• 1992 AJHSME Problems/Problem 11
  The bar graph shows the results of a survey on color preferences. What percent preferred blue? for (int a = 1; a <= 6; ++a)
  2 KB (226 words) - 00:09, 5 July 2013
• 1996 AHSME Problems/Problem 30
  A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive s <math>\textbf{(A)}\ 309 \qquad \textbf{(B)}\ 349 \qquad \textbf{(C)}\ 369 \qquad \textbf{(D)
  10 KB (1,515 words) - 13:09, 20 December 2023
• 2009 AMC 8 Problems/Problem 4
  real r=0.5, s=1.5; draw(shift(s,r)*p);
  5 KB (633 words) - 01:56, 26 November 2023
• 1996 AHSME Problems
  <math> \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E ...After <math>10</math> days of doing his chores daily, Walter has received a total of <math>36</math> dollars. On how many days did Walter do them excep
  15 KB (2,343 words) - 13:39, 19 February 2020
• 2010 PUMaC Problems
  == Division A == Assume that <math>f(a+b) = f(a) + f(b) + ab</math>, and that <math>f(75) - f(51) = 1230</math>. Find <mat
  22 KB (3,694 words) - 23:58, 3 June 2022
• 2003 AMC 8 Problems/Problem 18
  ...one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
  2 KB (250 words) - 22:17, 5 January 2024
• 2009 PUMaC Problems
  '''Algebra A''' ...a sequence of integers, such that <math>-1\leq x_i\leq 2</math>, for <math>i=1,2,\dots,n</math>, <math>x_1+x_2+\dots+x_n = 7</math> and <math>x_1^8+x_2
  25 KB (4,154 words) - 16:27, 2 September 2011
• 1985 AHSME Problems
  <math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{( In an arcade game, the "monster" is the shaded sector of a circle of radius <math>1</math> cm, as shown in the figure. The missing pie
  17 KB (2,488 words) - 03:26, 20 March 2024
• 2011 AMC 8 Problems
  ...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{
  16 KB (2,371 words) - 17:34, 9 January 2024
• 1985 IMO Problems/Problem 6
  ...ution, one can write <math>x_n=P_n(x_1)</math> , where <math>P_n</math> is a polynomial with non-negative coefficients and zero constant term. Thus, <ma (a) <math>\displaystyle x_n<x_{n+1} \Leftrightarrow x_n>1-\frac{1}{n} \Leftrig
  9 KB (1,788 words) - 00:02, 30 January 2021
• Barycentric coordinates
  ...triangle <math> \Delta{A_1}{A_2}{A_3} </math>. These masses then determine a point <math> P </math>, which is the geometric centroid of the three masses The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL
  25 KB (5,106 words) - 11:00, 24 May 2024
• Mathtime Version 1 Issue 2
  ...e the positive integer <math>n</math> such that <math>n^2+20n+40</math> is a perfect square. (Proposed by djmathman) ...umber of ordered integer pairs <math>(a,b)</math> such that <math>a^2+b^2|(a+b)^2</math>. (Proposed by djmathman)
  15 KB (2,444 words) - 21:46, 1 January 2012
• 2011 AMC 8 Problems/Problem 7
  ...2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0), V=(10,4), Z=(10,2), A_1=(8,4), B_1=(8,0), C_1=(6 draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V--Z--cycle,linewidth(1.6)); draw(G_1--K_
  4 KB (557 words) - 00:30, 4 January 2023
• Mock Geometry AIME 2011 Problems/Problem 9
  ...dge length 6, and <math>PA=PB=PC=PD=6\sqrt{2}.</math> The probability that a randomly selected point inside the pyramid is at least <math>\frac{\sqrt{6} [I believe solution is wrong - IP' is not sqrt(6)/3 as stated in the second pa
  3 KB (501 words) - 17:54, 16 February 2018
• 2012 AMC 10A Problems/Problem 25
  <math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\ ...0</math>. Then the set of points <math>(x,y,z)</math> is a tetrahedron, or a triangular pyramid. The point <math>(x,y,z)</math> distributes uniformly in
  13 KB (2,133 words) - 01:22, 6 February 2024
• 2012 AMC 12A Problems/Problem 18
  ...<math>AB=27</math>, <math>AC=26</math>, and <math>BC=25</math>. Let <math>I</math> be the intersection of the internal angle bisectors of <math>\triang <math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26
  4 KB (717 words) - 19:07, 28 July 2021
• 2012 AMC 12A Problems/Problem 22
  Distinct planes <math>p_1,p_2,....,p_k</math> intersect the interior of a cube <math>Q</math>. Let <math>S</math> be the union of the faces of <math> <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua
  5 KB (940 words) - 17:13, 4 April 2020
• 2012 AIME I Problems
  {{AIME Problems|year=2012|n=I}} ...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 12
  ...ath>n</math> are relatively prime positive integers, and <math>p</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p pair A,B,C,D,E0,F;
  9 KB (1,523 words) - 12:23, 7 September 2022
• 2012 AIME I Problems/Problem 14
  ...math>b,</math> and <math>c</math> in the complex plane are the vertices of a right triangle with hypotenuse <math>h.</math> Find <math>h^2.</math> ...9}\cdot\frac{x^2+y^2}{4}=\frac{x^2}{9}+\frac{y^2}{9}</math>. Hence, <math>|a|^2+|b|^2+|c|^2=\frac{6x^2+6y^2}{9}=\frac{2x^2+2y^2}{3}=250</math>. Therefor
  12 KB (2,228 words) - 23:59, 20 January 2024
• 1968 IMO Problems/Problem 3
  ...x_1, x_2, \cdots, x_n</math> where <math>a, b, c</math> are real and <math>a \neq 0</math>. Let <math>\Delta = (b - 1)^2 - 4ac</math>. Prove that for th (a) if <math>\Delta < 0</math>, there is no solution,
  3 KB (484 words) - 14:19, 20 March 2012
• 2007 iTest Problems/Problem 15
  Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and ...one of its sides coincides with a side of the square. Then "circumscribe" a circle around the
  3 KB (539 words) - 21:01, 29 July 2018
• 2012 AIME II Problems
  ...science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and mus ...rs with <math>p</math> and <math>q</math> relatively prime. Find <math>p+q+r</math>.
  7 KB (1,228 words) - 12:16, 13 March 2020
• 2012 AIME II Problems/Problem 13
  ...circle centered at <math>A</math> with radius <math>\sqrt{111}</math> and a circle centered at <math>B</math> with radius <math>\sqrt{11}</math> inters There are two equilateral triangles with <math>\overline{AD_1}</math> as a side; let <math>E_1</math> be the third vertex that is farthest from <math>
  13 KB (2,052 words) - 18:02, 5 February 2024
• 2017 AMC 8 Problems
  <math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo
  12 KB (1,771 words) - 21:13, 20 January 2024
• Mock AIME I 2012 Problems
  ...<math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? [[Mock AIME I 2012 Problems/Problem 1 |Solution]]
  7 KB (1,309 words) - 11:13, 8 April 2012
• Mock AIME I 2012 Problems/Problem 8
  ...an be expressed in the form <math>\dfrac{r+\sqrt{s}}{t}</math>. Find <math>r+s+t</math>. ...fourth power, the desired answer is <math>\dfrac{7+\sqrt{45}}{2} \implies r+s+t=\boxed{054}</math>.
  3 KB (543 words) - 18:51, 7 May 2020
• 1979 USAMO Problems
  ...cts the angle <math>ACB</math> in the spherical triangle <math>ABC</math> (a spherical triangle has great circle arcs as sides). ..._1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at
  2 KB (321 words) - 13:04, 24 December 2015
• 2012 USAJMO Problems/Problem 3
  Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove tha ...} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).</cmath>
  6 KB (1,135 words) - 20:20, 17 June 2022
• 2015 IMO Problems/Problem 1
  ...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 (773 words) - 08:14, 19 July 2016
• 2002 Romanian NMO Problems
  Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.
  10 KB (1,695 words) - 10:03, 10 May 2012
• 1988 USAMO Problems/Problem 4
  ...ta IAB</math>, <math>\Delta IBC</math>, and <math>\Delta ICA</math> lie on a circle whose center is the circumcenter of <math>\Delta ABC</math>. ...>, and <math>O_b</math>, respectively. It then suffices to show that <math>A</math>, <math>B</math>, <math>C</math>, <math>O_a</math>, <math>O_b</math>,
  3 KB (504 words) - 19:25, 14 October 2021
• 1974 AHSME Problems
  <math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } ...h that <math> x_1\not=x_2 </math> and <math> 3x_i^2-hx_i=b </math>, <math> i=1, 2 </math>. Then <math> x_1+x_2 </math> equals
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1974 AHSME Problems/Problem 16
  ...radius <math> R </math> is circumscribed about the triangle. Then <math> R/r </math> equals <math> \mathrm{(A)\ } 1+\sqrt{2} \qquad \mathrm{(B) \ }\frac{2+\sqrt{2}}{2} \qquad \mathrm{(C
  2 KB (226 words) - 14:11, 3 March 2018
• 2006 SMT/Geometry Problems
  Given a cube, determine the ratio of the volume of the octahedron formed by connect ...ath> and <math> X </math> is on the same side of line <math> AB </math> as a given point <math> C </math>.
  5 KB (837 words) - 12:22, 27 May 2012
• 1980 AHSME Problems
  <math>\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is
  15 KB (2,302 words) - 10:47, 30 April 2021
• 1961 AHSME Problems/Problem 40
  <math>\textbf{(A)}\ \frac{60}{13}\qquad Let <math>x^2 + y^2 = r^2</math>, so <math>r = \sqrt{x^2 + y^2}</math>. Thus, this problem is really finding the shorte
  3 KB (436 words) - 22:31, 11 June 2020
• 1998 USAMO Problems/Problem 6
  ...h exactly <math>k</math> of the quadrilaterals <math>A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.) ...> for all <math>i</math>, then quadrilateral <math>A_{i+1}A_{i+2}A_{i+3}A_{i+4}</math> is not tangential.
  5 KB (871 words) - 18:59, 10 May 2023
• 1999 USAMO Problems/Problem 1
  (a) every square that does not contain a checker shares a side with one that does; ...ven squares, such that every two consecutive squares of the sequence share a side.
  4 KB (747 words) - 05:23, 13 May 2023
• 1978 USAMO Problems/Problem 2
  ...d <math>O'</math> each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for <math>O</math>. real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
  4 KB (712 words) - 21:57, 12 November 2023
• 2020 Mock Combo AMC 10
  Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000. ...friend, Sara, along with eight other classmates, are randomly seated along a row of ten chairs. What is the probability that Fred sits next to Sara?
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2012 AMC 8 Problems
  ...many pounds of meat does she need to make <math> 24 </math> hamburgers for a neighborhood picnic? <math> \textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}6\frac{2}3\qquad\textbf{(C)
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 8 Problems/Problem 5
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0);
  2 KB (208 words) - 20:16, 3 January 2024
• 2013 AMC 12A Problems/Problem 13
  ...</math>, where these fractions are in lowest terms. What is <math> p + q + r + s </math>? <math> \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D
  3 KB (447 words) - 19:29, 4 September 2022
• 2013 AMC 10A Problems/Problem 18
  ...ght)</math>, where these fractions are in lowest terms. What is <math>p+q+r+s</math>? <math> \textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qqu
  4 KB (592 words) - 22:19, 2 November 2023
• 2013 AMC 10A Problems/Problem 20
  A unit square is rotated <math>45^\circ</math> about its center. What is the <math> \textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \f
  4 KB (701 words) - 17:55, 23 July 2021
• 2013 AMC 12A Problems/Problem 25
  <math> \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{ ...rname{Im}(z)>0</math> such that <math>a,b</math> are integers where <math>|a|, |b|\leq 10</math>.
  4 KB (766 words) - 14:22, 8 October 2023
• 2013 AIME I Problems
  {{AIME Problems|year=2013|n=I}} ...les twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
  9 KB (1,580 words) - 13:07, 24 February 2024
• 2013 AIME I Problems/Problem 1
  ...les twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have
  824 bytes (134 words) - 20:03, 24 January 2021
• 2013 AIME I Problems/Problem 5
  ...ath>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. ...3]{9}+1</math> to rationalize the denominator, and we therefore have <math>r=\frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</math>, and the answer is <math>\boxed
  3 KB (573 words) - 21:11, 27 August 2023
• 2013 AIME I Problems/Problem 10
  ...sum of the <math>{p}_{a,b}</math>'s for all possible combinations of <math>a</math> and <math>b</math>. ...<math>r+si</math> is a root, by the Complex Conjugate Root Theorem, <math>r-si</math> must be the other imaginary root. Using <math>q</math> to represe
  2 KB (389 words) - 12:05, 10 August 2021
• 2013 AIME I Problems/Problem 12
  ...ively prime, and c is not divisible by the square of any prime. Find <math>a+b+c+d</math>. First, find that <math>\angle R = 45^\circ</math>.
  9 KB (1,490 words) - 02:25, 2 May 2024
• 2013 AIME II Problems
  ...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. Positive integers <math>a</math> and <math>b</math> satisfy the condition
  8 KB (1,402 words) - 12:17, 13 March 2020
• 2013 AIME II Problems/Problem 4
  ...integer that is not divisible by the square of any prime. Find <math>p+q+r</math>. ...th>xy = \dfrac{25\sqrt{3}}{12}</math> yielding an answer of <math> p + q + r = 25 + 2 + 12 = \boxed{040}</math>.
  8 KB (1,268 words) - 14:10, 31 January 2024
• 2013 AIME II Problems/Problem 12
  ...all polynomials of the form <math>z^3 + az^2 + bz + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Find the number of ...: <math>f(z)=(z-r)(z-\omega)(z-\omega^*)</math>, where <math>r\in \mathbb{R}</math>, <math>\omega</math> is nonreal, and <math>\omega^*</math> is the
  8 KB (1,393 words) - 19:00, 24 May 2023
• 2012 IMO Problems/Problem 5
  ...th> be the foot of the altitude from <math>C</math>. Let <math>X</math> be a point in the interior of the segment <math>CD</math>. Let <math>K</math> be ...umcircle of triangle <math>ABC</math>, the circle with its center as <math>A</math> and radius as <math>AC</math>, and the circle with its center as <ma
  2 KB (437 words) - 08:30, 20 November 2023
• Eisenstein's criterion
  <math> 1) p</math> is a prime which divides each of <math>a_0,a_1,a_2,...,a_{n-1} </math> ...h>, so <math>p|f(x)</math>. However, we know that <math>p\nmid a_n</math>, a contradiction. Therefore, <math>f(x)</math> is irreducible.
  2 KB (475 words) - 14:47, 14 August 2018
• 2014 AIME II Problems/Problem 11
  ...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. ...ft(0, \frac{\sqrt{3}}{2}\right).</math> <math>M =</math> midpoint<math>(D, R) = \left(\frac{1}{4}, \frac{\sqrt{3}}{4}\right)</math> and the slope of <ma
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• 2014 AIME II Problems/Problem 10
  Let <math>z</math> be a complex number with <math>|z|=2014</math>. Let <math>P</math> be the polygo Now, let <math>w = r_w e^{i \theta_w}</math> and likewise for <math>z</math>. Consider circle <math>O</
  6 KB (1,045 words) - 13:08, 21 January 2024
• 2014 AIME II Problems/Problem 8
  ...}</math>. Circle D is internally tangent to circle <math>C</math> at <math>A</math>. Circle <math>E</math> is internally tangent to circle <math>C</math label("$A$", (5.812051070003994,-1.276266740352400), NE * labelscalefactor);
  6 KB (826 words) - 21:31, 9 January 2024
• 2013 USAJMO Problems
  Are there integers <math>a</math> and <math>b</math> such that <math>a^5b+3</math> and <math>ab^5+3</math> are both perfect cubes of integers? ...n cells that share only a corner are not adjacent). The filling is called a ''garden'' if it satisfies the following two conditions:
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• ASIA TEAM Problems
  ...th> such that Kelvin will reach <math>1</math> for the first time in <math>r</math> jumps. In particular, <math>o(1)=0</math> because Kelvin is already ...t, when he catches a fly, he places it on <math>a_i</math> for which <math>i</math> is the least such number satisfying the following rules:
  10 KB (1,710 words) - 23:23, 10 January 2020
• Mock AIME I 2015 Problems
  ...ath>. Finally, Palmer announces the product of the numbers in boxes D and A. If <math>k</math> is the number that Palmer says, what is <math>20k</math [[Mock AIME I 2015 Problems/Problem 1|Solution]]
  9 KB (1,463 words) - 14:48, 12 February 2017
• 1971 IMO Problems/Problem 5
  ...th>m</math> points in <math>S</math> which are at unit distance from <math>A</math>. ...ean distance between the points is <math>1</math>) and this will follow as a consequence.
  4 KB (749 words) - 14:09, 29 January 2021
• 1975 IMO Problems/Problem 6
  (i) for a positive integer <math>n</math> and all real <math>t, x, y</math> <cmath>P( (ii) for all real <math>a, b, c</math>, <cmath>P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,</cmath>
  3 KB (563 words) - 16:21, 29 January 2021
• 1962 AHSME Problems/Problem 21
  ...math>, with <math>r</math> and <math>s</math> real numbers, is <math>3+2i (i = \sqrt{-1})</math>. The value of <math>s</math> is: <math> \textbf{(A)}\ \text{undetermined}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\text
  833 bytes (135 words) - 22:19, 3 October 2014
• 1962 AHSME Problems
  <math>\textbf{(A)}\ \frac{4y-1}{8} \qquad <math>\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad
  17 KB (2,459 words) - 22:40, 10 April 2023
• 2013 AMC 8 Problems
  <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \ ...lf-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)
  15 KB (2,162 words) - 20:05, 8 May 2023
• 1994 AHSME Problems
  <math> \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\t A large rectangle is partitioned into four rectangles by two segments paralle
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1981 AHSME Problems
  <math> \textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\ <math> \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ \sqrt{5}\qquad\textbf{(C)}\ 3\qquad\textbf{(
  17 KB (2,633 words) - 15:44, 16 September 2023
• 1968 AHSME Problems
  Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter <math>\text{(A) } \frac{1}{\pi}\quad\text{(B) } \pi\quad\text{(C) } \frac{\pi^2}{2}\quad\t
  16 KB (2,571 words) - 14:13, 20 February 2020
• 1994 AHSME Problems/Problem 30
  ...1994 is greater than zero and is the same as the probability of obtaining a sum of <math>S</math>. The smallest possible value of <math>S</math> is <math> \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\
  2 KB (328 words) - 18:43, 28 May 2021
• 1983 IMO Problems/Problem 3
  ...ath>b</math> and <math>c</math> be positive integers, no two of which have a common divisor greater than <math>1</math>. Show that <math>2abc - ab - bc- ...prove <math>2abc-ab-bc-ca</math> is un-achievable. Also, assume WLOG <math>a\ge b\ge c</math>.
  4 KB (684 words) - 23:37, 29 January 2021
• 2014 USAJMO Problems/Problem 6
  Let <math>ABC</math> be a triangle with incenter <math>I</math>, incircle <math>\gamma</math> and circumcircle <math>\Gamma</math>. (a) Prove that <math>I</math> lies on ray <math>CV</math>.
  7 KB (1,273 words) - 18:17, 28 August 2021
• 2018 AIME I Problems/Problem 14
  Let <math>SP_1P_2P_3EP_4P_5</math> be a heptagon. A frog starts jumping at vertex <math>S</math>. From any vertex of the heptag ...row P_3 \leftrightarrow E</math>. We can count the number of left/right (L/R) paths of length <math>\le 11</math> that start at <math>S</math> and end a
  4 KB (649 words) - 17:29, 22 December 2023
• 2014 USAMO Problems/Problem 6
  ...>1</math> for all <math>i, j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.</cmath> ...math> and in each cell place a prime <math>p</math> dividing <math>\gcd (a+i, b+j)</math>.
  2 KB (361 words) - 11:55, 25 June 2020
• 1960 AHSME Problems
  If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 10B Problems
  <math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) <math>\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D)
  13 KB (2,011 words) - 21:54, 8 November 2022
• 2014 AIME II Problems
  ...s Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is pai ...has none of the three risk factors given that he does not have risk factor A is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are re
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• 2014 AIME I Problems
  {{AIME Problems|year=2014|n=I}} ...each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
  9 KB (1,472 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 3
  ...tten as a fraction in lowest terms, the numerator and the denominator have a sum of 1000. ...</math> numbers relatively prime to 1000 from 1 to 1000. Recall that <math>r=\frac{x}{y}<1</math> and note by Euclidean algorithm <math>\gcd(1000,1000-x
  6 KB (965 words) - 14:17, 27 December 2023
• 2014 AIME I Problems/Problem 4
  Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west dir ...arly, the second train has to cover a distance equal to its own length, at a rate of <math>r_2 + \dfrac{1}{3}</math>. Since the times are equal and <mat
  5 KB (784 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 11
  ...ns and independently of the other moves. The probability the token ends at a point on the graph of <math>|y|=|x|</math> is <math>\frac{m}{n}</math>, whe ...sformation <math>(x, y)\rightarrow (x+y, x-y)</math>. Then we can see that a movement up, right, left, or down in the old coordinates adds the vectors <
  11 KB (1,677 words) - 23:54, 4 February 2022
• 2014 AIME I Problems/Problem 13
  ...nts <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, pair A = (0,sqrt(850));
  9 KB (1,404 words) - 21:07, 13 October 2023
• 2014 AIME I Problems/Problem 15
  ...is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>. ...=90^\circ</math>. But <math>DF=FE</math>, so <math>\triangle DEF</math> is a 45-45-90 triangle. Letting <math>DG=3x</math>, we have that <math>EG=4x</ma
  10 KB (1,643 words) - 22:30, 28 January 2024
• 1973 AHSME Problems
  A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1992 AHSME Problems
  <math>\text{(A) } 2P\quad <math>\text{(A) } 40\%\quad
  16 KB (2,548 words) - 13:40, 19 February 2020
• 1990 AHSME Problems/Problem 25
  Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of t <math>\text{(A) } 1-\frac{\sqrt{3}}{2}\quad
  1 KB (229 words) - 19:36, 6 August 2019
• 1990 AHSME Problems/Problem 22
  ...nd <math>b</math> are real, then the product of those solutions with <math>a>0</math> is <math>\text{(A) } -2\quad
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• 1969 AHSME Problems/Problem 11
  ...h>-plane; point <math>R(1,m)</math> is taken so that <math>PR+RQ</math> is a minimum. Then <math>m</math> equals: <math>\text{(A) } -\tfrac{3}{5}\quad
  2 KB (294 words) - 04:13, 7 June 2018
• 1952 AHSME Problems/Problem 48
  ...th> miles apart, and starting at the same time, would be together in <math>r</math> hours if they traveled in the same direction, but would pass each ot <math>\text{(A) } \frac {r + t}{r - t} \qquad
  2 KB (370 words) - 01:27, 31 January 2023
• 1970 AHSME Problems
  <math>\textbf{(A) }\sqrt{2}+\sqrt{3}\qquad A square and a circle have equal perimeters. The ratio of the area of the circle to the ar
  15 KB (2,366 words) - 07:52, 26 December 2023
• 1980 AHSME Problems/Problem 28
  <math>\text{(A)} \ 17 \qquad ...<cmath>(w-1)(w^2+w+1) = w^3 - 1 = 0.</cmath> These roots are <math>w = e^{i \pi /3}</math> and <math>w = e^{2i \pi /3}</math>.
  3 KB (472 words) - 20:04, 30 October 2021
• 1983 AHSME Problems
  <math>\textbf{(A)}\ 8\qquad <math>\textbf{(A)} \ 1 \qquad
  15 KB (2,309 words) - 23:43, 2 December 2021
• 1982 AHSME Problems
  <math>\textbf{(A)} \ 2 \qquad If a number eight times as large as <math>x</math> is increased by two, then one
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1979 AHSME Problems
  label("$A$",(-2,1),W); <math>\textbf{(A) }8\qquad
  17 KB (2,664 words) - 01:34, 19 March 2022
• 1978 AHSME Problems
  <math>\textbf{(A) }-1\qquad If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
  15 KB (2,432 words) - 01:06, 22 February 2024
• 1977 AHSME Problems
  <math>\text{(A)}\ x \qquad <math>\textbf{(A)}\ \text{ equiangular}\qquad
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1976 AHSME Problems
  <math>\textbf{(A) }-2\qquad For how many real numbers <math>x</math> is <math>\sqrt{-(x+1)^2}</math> a real number?
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1975 AHSME Problems
  <math>\textbf{(A)}\ 3/4 \qquad <math>\textbf{(A)}\ \text{all }m\qquad
  16 KB (2,512 words) - 04:48, 27 November 2021
• 1972 AHSME Problems
  The lengths in inches of the three sides of each of four triangles <math>I, II, III</math>, and <math>IV</math> are as follows: \hbox{I}& 3,\ 4,\ \hbox{and}\ 5\qquad &
  17 KB (2,725 words) - 00:21, 26 May 2024
• 2007 iTest Problems/Problem 19
  ...rechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" ...son a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.
  3 KB (397 words) - 01:18, 29 October 2023
• 2007 iTest Problems/Problem 18
  ...ic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers. If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?
  2 KB (393 words) - 17:01, 10 June 2018
• 1969 IMO Problems/Problem 4
  ...Prove that <math>\gamma_1, \gamma_2</math>, and <math>\gamma_3</math> have a second tangent in common. ...ath>K_2</math> is the intersection of the inversion circle <math>(B, r_B = a)</math> with the line <math>AB</math>.
  5 KB (904 words) - 13:42, 29 January 2021
• 2007 iTest Problems/Problem 21
  James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  <math>\text{(A) } 0 \quad \text{(I) } 8 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  <math>\text{(A) } 0\qquad \text{(I) } 8\qquad</math>
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 52
  ...n of the [[2007 iTest]], where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substitute Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesi
  2 KB (340 words) - 16:54, 10 July 2018
• 2014 Canadian MO Problems
  ...red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red Let <math>p</math> be a fixed odd prime. A <math>p</math>-tuple <math>(a_1,a_2,a_3,\ldots,a_p)</math> of integers is s
  3 KB (463 words) - 19:07, 22 January 2017
• 2007 iTest Problems/Problem 23
  <math>\text{(A) } 0\quad \text{(I) } -4\quad</math>
  1 KB (202 words) - 16:48, 24 November 2018
• 2007 iTest Problems/Problem 24
  ...such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube. <math>\text{(A) }0 \quad
  2 KB (340 words) - 19:49, 30 June 2018
• 2007 IMO Problems/Problem 4
  ...tor of <math>\angle{BCA}</math> intersects the circumcircle again at <math>R</math>, the perpendicular bisector of <math>BC</math> at <math>P</math>, an real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  8 KB (1,480 words) - 14:52, 5 August 2022
• 1965 AHSME Problems
  <math>\textbf{(A)}\ 0 \qquad A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1964 AHSME Problems
  <math>\textbf{(A)}\ \log_{10}50 \qquad <math>\textbf{(A)}\ \text{a parabola} \qquad
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1961 AHSME Problems
  <math>\textbf{(A)}\ \frac{1}{25} \qquad An automobile travels <math>a/6</math> feet in <math>r</math> seconds. If this rate is maintained for <math>3</math> minutes, how
  18 KB (2,905 words) - 18:33, 5 April 2023
• 1957 AHSME Problems
  ...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2013 UNCO Math Contest II Problems
  ...equal to <math>4^3</math>, so <math>64</math> is both a perfect square and a perfect cube. (a) Find the smallest positive integer multiple of <math>12</math> that is a perfect square.
  11 KB (1,648 words) - 09:55, 20 December 2021
• 2013 UNCO Math Contest II Problems/Problem 10
  Dav designs a robot, which he calls FrankenCoder, to print nonsense text by scrambling el 1st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline
  4 KB (536 words) - 03:30, 13 January 2019
• 2015 AMC 12A Problems
  {{AMC12 Problems|year=2015|ab=A}} <math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 12A Problems/Problem 25
  ...p_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is <cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath>
  8 KB (1,340 words) - 14:24, 23 November 2023
• 2015 AMC 12B Problems
  <math>\textbf{(A)}\; -2 \qquad\textbf{(B)}\; \dfrac{1}{16} \qquad\textbf{(C)}\; \dfrac{7}{4} Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AIME I Problems/Problem 8
  ...en to carry the digits. Note that if you don't understand any of the steps I take, just try adding any 3-digit number to 864 regularly (using the old-fa .... The smallest value we can get with this is 695. Let's see if we can find a smaller one:
  10 KB (1,797 words) - 14:06, 21 January 2024
• 2015 AIME I Problems/Problem 10
  Let <math>f(x)</math> be a third-degree polynomial with real coefficients satisfying Since <math>f(x)</math> is a third degree polynomial, it can have at most two bends in it where it goes
  8 KB (1,474 words) - 10:00, 10 November 2023
• 2015 AIME II Problems
  ...ect to take Latin. The probability that a randomly chosen Latin student is a sophomore is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</ma ...ers <math>a</math> and <math>c</math>. Can you tell me the values of <math>a</math> and <math>c</math>?"
  8 KB (1,326 words) - 19:15, 13 January 2024
• 1998 IMO Problems/Problem 5
  Let I be the incenter of triangle ABC. Let the incircle of ABC touch the sides to M K meets the lines LM and LK at R and S, respectively. Prove that angle
  335 bytes (61 words) - 23:50, 18 November 2023
• 2016 AIME I Problems
  {{AIME Problems|year=2016|n=I}} ...and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>.
  8 KB (1,360 words) - 12:19, 29 January 2022
• 2016 AMC 10A Problems/Problem 24
  A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}</math>. Three of the sides of this quadr <math>\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\te
  24 KB (3,861 words) - 14:17, 26 April 2024
• 2016 AMC 10A Problems/Problem 25
  <math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qqu ...,2),(1,2),(0,2)</math>. Thus our answer is <math>5\times 3=\boxed{\textbf{(A) }15}.</math>
  7 KB (1,135 words) - 09:28, 5 September 2022
• Aops font
  The AoPS font is a font that can be used to make certain special symbols in the forum. You enc | <span class="aops-font">a</span>
  6 KB (924 words) - 21:31, 22 February 2024
• 1993 IMO Problems/Problem 1
  ...<math>f\left(x\right)=\sum_{i=0}^{n}\left(\sum_{j=0}^{i}b_jc_{i-j}\right)x^i</math>. ...t least <math>n-1</math>. However, because <math>h\left(x\right)</math> is a non-constant factor of <math>f\left(x\right)</math>, we have that the degre
  3 KB (536 words) - 11:26, 21 November 2023
• 1997 JBMO Problems
  ...y 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>. Hyy I am anshu Kumar anahu from https://t.me/AOPS01
  2 KB (245 words) - 22:44, 14 September 2023
• 2017 AIME I Problems
  {{AIME Problems|year=2017|n=I}} ...points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2019 AMC 12A Problems
  {{AMC12 Problems|year=2019|ab=A}} ...us <math>4</math> inches is <math>N</math> percent larger than the area of a pizza with radius <math>3</math> inches. What is the integer closest to <ma
  14 KB (2,180 words) - 22:25, 25 April 2024
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems
  ...> to <math>9</math>. If <math>\sqrt{STARS} = SAT</math>, what are <math>A, R, S</math>, and <math>T</math>? (a) Determine <math>f\circ g(x)</math> and <math>g\circ f(x)</math>.
  5 KB (846 words) - 03:36, 19 January 2019
• 2016 AMC 10B Problems/Problem 25
  ...r \rfloor</math> denotes the greatest integer less than or equal to <math>r</math>. How many distinct values does <math>f(x)</math> assume for <math>x <math>\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qqu
  12 KB (1,951 words) - 18:23, 4 April 2023
• 2016 AMC 12B Problems
  ...is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \frac{1}{2}</math>? <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME I Problems/Problem 11
  Let <math>P(x)</math> be a nonzero polynomial such that <math>(x-1)P(x+1)=(x+2)P(x)</math> for every r ...constant <math>C</math> (<math>Q(x)</math> cannot be periodic since it is a polynomial), so <math>P(x) = Cx(x-1)(x+1)</math>. We know that <math>\left(
  8 KB (1,415 words) - 14:00, 22 December 2021
• 2016 AIME I Problems/Problem 10
  A strictly increasing sequence of positive integers <math>a_1</math>, <math>a We first create a similar sequence where <math>a_1=1</math> and <math>a_2=2</math>. Continuin
  5 KB (883 words) - 15:24, 6 January 2024
• 2016 AIME I Problems/Problem 6
  In <math>\triangle ABC</math> let <math>I</math> be the center of the inscribed circle, and let the bisector of <math real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  14 KB (2,397 words) - 20:04, 27 August 2023
• 2016 AIME I Problems/Problem 1
  ...and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>. ...c{144}{1-a^2}=2016</math>. <math>\frac{12}{1-a}+\frac{12}{1+a}=\frac{24}{1-a^2}</math>, so the answer is <math>\frac{2016}{6}=\boxed{336}</math>.
  736 bytes (117 words) - 07:53, 4 November 2022
• 2016 AIME I Problems/Problem 9
  Triangle <math>ABC</math> has <math>AB=40,AC=31,</math> and <math>\sin{A}=\frac{1}{5}</math>. This triangle is inscribed in rectangle <math>AQRS</ma ...by drawing triangle ABC, where <math>A</math> is obtuse. Therefore, angle A is acute. Let angle <math>CAS=n</math> and angle <math>BAQ=m</math>. Then,
  9 KB (1,526 words) - 02:31, 29 December 2021
• 2016 AIME I Problems/Problem 12
  ...ast positive integer <math>m</math> such that <math>m^2 - m + 11</math> is a product of at least four not necessarily distinct primes. ...-1)^2\equiv -43\pmod{p}</math>. We can verify that <math>-43</math> is not a perfect square mod <math>p</math> for each of <math>p=3,5,7</math>. Therefo
  3 KB (498 words) - 20:32, 22 November 2022
• 2016 AIME II Problems/Problem 14
  ...planes). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>. ...ngle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)=
  15 KB (2,560 words) - 01:44, 1 July 2023
• 2016 AIME II Problems/Problem 9
  We have <math>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100< ...r values of <math>r</math> such that <math>j(r) -i(r)>1</math>. Let's make a table:
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• 2006 iTest Problems
  <math>\mathrm{(A)}\, 8</math> <math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAMO Problems
  ...+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldots, 99\}</math>. Find the smallest possible number of elements i Find all functions <math>f:\mathbb{R}\rightarrow \mathbb{R}</math> such that for all real numbers <math>x</math> and <math>y</math>, <
  4 KB (608 words) - 13:49, 22 November 2023
• 2016 USAJMO Problems
  ...on the arc <math>\stackrel{\frown}{BC}</math> that does not contain <math>A</math>, and let <math>I_B</math> and <math>I_C</math> denote the incenters ...the circumcircle of triangle <math>\triangle PI_BI_C</math> passes through a fixed point.
  3 KB (414 words) - 16:43, 5 August 2023
• 2016 USAJMO Problems/Problem 1
  ...on the arc <math>\stackrel{\frown}{BC}</math> that does not contain <math>A</math>, and let <math>I_B</math> and <math>I_C</math> denote the incenters ...the circumcircle of triangle <math>\triangle PI_BI_C</math> passes through a fixed point.
  5 KB (1,004 words) - 15:15, 28 June 2020
• 2016 USAJMO Problems/Problem 5
  ...umcenter. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to line <math>\overleftrightarrow{BC}</math>, and points <math>P</ma pair A=dir(110);
  10 KB (1,733 words) - 19:15, 14 June 2020
• 2016 USAMO Problems/Problem 2
  ...^{2}}{m}}\rfloor\right\rceil \geq 0</math>, as the term in each summand is a sum of floors also and is clearly an integer. <cmath>N = \frac{\left(k^2\right)!}{\prod_{1\le i, j\le k}(i+j-1)}.</cmath>
  4 KB (732 words) - 20:59, 8 April 2021
• 2016 USAMO Problems/Problem 3
  1. Let <math>I_A</math> be the <math>A</math>-excenter, then <math>I_A,O,</math> and <math>P</math> are colinear. ..._1BCY_1,</math> <math>AT</math> is tangent to <math>\omega</math> at <math>A.</math>
  6 KB (998 words) - 21:36, 17 October 2022
• 2016 USAMO Problems/Problem 5
  ...<math>O</math> is the circumcenter of triangle <math>ABC,</math> and <math>I</math> is the incenter of triangle <math>ABC.</math> ...BC</math>. Let <math>R</math> be the point such that <math>NPQR</math> is a rhombus. It follows that <math>OD\perp QR</math>.
  8 KB (1,449 words) - 00:09, 12 October 2023
• 1984 USAMO Problems/Problem 5
  <math>P(x)</math> is a polynomial of degree <math>3n</math> such that ...\sum_{p=1}^{n}\left ( \prod_{0\leq r\neq3p-2\leq 3n}^{{}} \frac{x-r}{3p-2-r}\right )</math>
  3 KB (443 words) - 21:54, 22 November 2023
• 1975 Canadian MO Problems
  A sequence of numbers <math>a_1, a_2, a_3, \dots</math> satisfies <div class=li><span class=num>(i)</span> <math>a_1 = \frac{1}{2}</math></div>
  3 KB (499 words) - 12:17, 11 August 2016
• Mock AIME 4 Pre 2005/Problems
  ...</math> and <math>b_{2005} = 6.</math> If <math>a_i = b_i</math> for <math>i = 1, 2, 3</math> and <math>a_1 = 3,</math> then compute the value of <math> ...nt in its interior such that <math>ABP</math> is equilateral. There exists a unique pair <math>\{m, n\}</math> of relatively prime positive integers suc
  7 KB (1,094 words) - 15:39, 24 March 2019
• Mock AIME 3 2010 Problems
  ...can distribute 1 piece of candy among 243 hungry schoolchildren sitting in a row -3. We have a triangle abc with sides ab, bc, ca with lengths α, β, and γ, respectivel
  14 KB (2,904 words) - 18:24, 16 May 2017
• 2017 AMC 12A Problems/Problem 17
  ...such that <math>z^{24}=1</math>. For how many of these is <math>z^6</math> a real number? <math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qq
  3 KB (459 words) - 04:19, 2 February 2021
• 2017 AMC 10A Problems
  {{AMC10 Problems|year=2017|ab=A}} <math>\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\q
  15 KB (2,285 words) - 18:02, 28 October 2023
• 2017 AMC 12A Problems/Problem 25
  The vertices <math>V</math> of a centrally symmetric hexagon in the complex plane are given by ...+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.</cmath>
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 10B Problems/Problem 21
  <math>\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 2\sqrt{2}\qq ...<math>[ADB] = [ADC] = 12</math>. Since <math>A = rs,</math> we have <math>r = \frac As</math>, so the inradius of <math>\triangle ADB</math> is <math>\
  4 KB (601 words) - 00:34, 8 August 2023
• 2017 AIME I Problems/Problem 2
  ...inder is always the positive integer <math>s \neq r</math>. Find <math>m+n+r+s</math>. ...of the problem separately. First, <cmath>855 \equiv 787 \equiv 702 \equiv r \pmod{m}.</cmath> We take the difference of <math>855</math> and <math>787<
  4 KB (580 words) - 03:08, 5 January 2024
• 2017 AIME I Problems/Problem 4
  A pyramid has a triangular base with side lengths <math>20</math>, <math>20</math>, and <ma ...<math>O</math>. Note that <math>O</math> is equidistant from each of <math>A</math>, <math>B</math>, and <math>C</math>. Then,
  7 KB (1,177 words) - 15:55, 5 January 2024
• 2017 AIME I Problems/Problem 8
  ...measures of <math>\angle POQ</math> and <math>\angle POR</math> are <math>a</math> and <math>b</math> respectively, and <math>\angle OQP</math> and <ma ...th diameter <math>\overline{OP}</math> and points <math>Q</math> and <math>R</math> on the semicircle. Since the radius of the semicircle is <math>100</
  9 KB (1,539 words) - 15:47, 17 February 2024
• 2017 AIME I Problems/Problem 10
  ...ies that <math>\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}</math> is a real number and the imaginary part of <math>z</math> is the greatest possib ...that. But there’s no fun in that, and let’s see if we can come up with a slicker solution that will be more enjoyable.
  13 KB (2,252 words) - 15:46, 6 January 2024
• 2017 AIME I Problems/Problem 15
  pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); pair R = A*t+B*(1-t), P=B*s;
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 AIME II Problems
  ...h>T_j</math>, the probability that <math>T_i</math> wins is <math>\frac{i}{i+j}</math>, and the outcomes of all the matches are independent. The probabi ...he triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>,
  7 KB (1,200 words) - 15:02, 8 September 2020
• 2017 USAJMO Problems/Problem 6
  ...unterclockwise, and writing the color of the points in that order - either R or B. For example, possible sequences include <math>RB</math>, <math>RBBR</ ...h> from my sequence when I choose them. I define this to be a move. Hence, a possible move sequence of <math>BBRRRB</math> is:
  8 KB (1,465 words) - 15:30, 12 June 2020

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