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• Resources for mathematics competitions
  ...itions]] leads to links for many of these competition homepages. Here are a few examples: * [https://brilliant.org/ Brilliant] is a website where one can solve problems to gain points and go to higher levels
  16 KB (2,152 words) - 21:46, 6 May 2024
• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...3265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Chemistry competitions
  This is a resources page for students interested in '''Chemistry competitions'''. Pl The U.S. National Chemistry Olympiad national exam (USNCO) is a 3-part, 4 hour and 45 minute exam administered in mid or late April by ACS
  2 KB (258 words) - 19:31, 8 March 2023
• Inequality
  Inequalities are arguably a branch of [[elementary algebra]], and relate slightly to [[number theory]]. For two [[number]]s <math>a</math> and <math>b</math>:
  12 KB (1,798 words) - 16:20, 14 March 2023
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• Proof writing
  ...ance, when a mathematician uses the word "trivial" in a proof, they intend a different meaning to how the word is understood by the wider population. St ...n, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible.
  3 KB (502 words) - 18:16, 18 January 2016
• Simon's Favorite Factoring Trick
  ...of the equation and a product of variables with each of those variables in a linear term on the other side. An example would be: <cmath>xy+66x-88y=23333 ...Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
  7 KB (1,107 words) - 07:35, 26 March 2024
• 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
• Cauchy-Schwarz Inequality
  ..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...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
• Brute forcing
  ...e, through bashing calculations, and can actually sometimes be faster than a more creative approach, and is thus an important tool to have. ...te force would be to list all 91 possibilities (although this would not be a smart time to use brute force).
  1 KB (190 words) - 13:22, 5 May 2023
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...rrational number]]s (sometimes, but not universally, denoted <math>\mathbb{J}</math>). In addition <math>\mathbb{Z}^{+}</math> means positive integers a
  3 KB (496 words) - 23:22, 5 January 2022
• 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
  ...h> be an [[open interval]], and suppose <math>g:I \to J</math> and <math>f:J \to \mathbb{R}</math>. Let <math>h:I \to \mathbb{R}</math> such that <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
• 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
• 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
• Improper fractional base
  An '''improper fractional base''' is a type of [[number base]]. Instead of using an [[integer]] for the base in o ...ntation in an improper fractional base. (Note that this means there is not a unique representation for each number in an improper fractional base.)
  787 bytes (118 words) - 19:20, 23 October 2010
• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
  4 KB (690 words) - 13:11, 20 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
• 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
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math ...the numbers in this list up in the following way: Whenever a positive and a negative number are adjacent in this progression, pair them up and remove t
  6 KB (910 words) - 19:31, 24 October 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]]. Given <math>g : x \mapsto \max_{j : 2^j | x} 2^j</math>, consider <math>S_n = g(2) + \cdots + g(2^n)</math>. Define <math>S
  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 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 ....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
• 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
• 2006 AMC 12B Problems
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2001 AMC 12 Problems
  <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D) ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2006 AMC 12B Problems/Problem 14
  ...ll the sandwiches is <math> 2.53</math>. Assume that <math>B</math>, <math>J</math> and <math>N</math> are all positive integers with <math>N>1</math>. \mathrm{(A)}\ 1.05
  1 KB (227 words) - 17:21, 8 December 2013
• 2006 AMC 12B Problems/Problem 25
  A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by \mathrm{(A)}\ 165
  5 KB (924 words) - 12:02, 15 June 2022
• 2005 AMC 12B Problems/Problem 24
  ...<math>2</math>. The <math>x</math>-coordinates of the three vertices have a sum of <math>m/n</math>, where <math>m</math> and <math>n</math> are relati <math> \mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15}}}\qquad\mathrm{(C)}\ {{{16}}}\qquad\m
  4 KB (761 words) - 09:10, 1 August 2023
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...ath> with all <math>c_j \in \mathbb C</math> and <math>c_n \neq 0</math>, a general degree-<math>n</math> polynomial. The degree of <math>P(x)</math> i
  8 KB (1,427 words) - 21:37, 13 March 2022
• Prime ideal
  ...ath>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/Problem 12
  ...> AB </math> with <math> AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </m ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME 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 14
  pair A = origin; pair C = rotate(15,A)*(A+dir(-50));
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 7
  ...he product of each pair of distinct terms, or <math>C = \sum_{1 \le i \neq j}^{15} S_iS_j</math>. Also, we know that ...\right)^2 &= \left(\sum_{i=1}^{n} S_i^2\right) + 2\left(\sum_{1 \le i \neq j}^{15} S_iS_j\right)\\ (-8)^2 &= \frac{15(15+1)(2\cdot 15+1)}{6} + 2C\end{al
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME II Problems/Problem 10
  ...{40} </math> whose binary expansions have exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] that ...\leq 39\}</math>. (The second condition ensures simultaneously that <math>j \neq k</math> and that each such number less than <math>2^{40}</math> is co
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2019 AIME I Problems/Problem 2
  ...>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the form <math>\frac{m}{n}</math> wher ...\ne 0</math> because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry.
  5 KB (830 words) - 22:15, 28 December 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. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 1998 AIME Problems
  ...allelogram]]. Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overli ...egative]] term encountered. What positive integer <math>x</math> produces a sequence of maximum length?
  7 KB (1,084 words) - 02:01, 28 November 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
• 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
• 1985 AIME Problems/Problem 10
  where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[ ...oor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</math> values for <math>b<
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 5
  A [[sequence]] of [[integer]]s <math>a_1, a_2, a_3, \ldots</math> is chosen s ...math>a_{j + 6n} = a_j</math> for all [[integer]]s <math>n</math> and <math>j</math>.
  2 KB (410 words) - 13:37, 1 May 2022
• 1986 AIME Problems/Problem 11
  ...he <math>j</math>th column of <math>P_{B \to C}</math> is <math>\delta_{i, j}</math> by using combinatorial identities, which is left as an exercise for {{AIME box|year=1986|num-b=10|num-a=12}}
  6 KB (872 words) - 16:51, 9 June 2023
• 1987 AIME Problems/Problem 7
  ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...one of <math>m, j</math> equal to 3. This gives 7 possible triples <math>(j, m, p)</math>: <math>(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4),
  3 KB (547 words) - 22:54, 4 April 2016
• 1989 AIME Problems/Problem 5
  ...heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j</math>. ...{3}\right)^3\left(\frac{2}{3}\right)^2 = \frac{40}{243}</math>, so <math>i+j=40+243=\boxed{283}</math>.
  2 KB (258 words) - 00:07, 25 June 2023
• 1990 AIME Problems/Problem 9
  ...air]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>\frac{i}{j}^{}_{}</math>, in lowest terms, be the [[probability]] that heads never occ ...3} + {9\choose2} + {{10}\choose1} + {{11}\choose0} = 144</math>. There are a total of <math>2^{10}</math> possible flips of <math>10</math> coins, makin
  3 KB (425 words) - 12:36, 12 May 2024
• 1991 AIME Problems/Problem 3
  ...this last equation and using the well-known fact <math>\log(a_{}^{}b)=\log a + \log b</math> (valid if <math>a_{}^{},b_{}^{}>0</math>), we have ...{j=1}^{k}\frac{(N-j+1)x}{j}\right]=\sum_{j=1}^{k}\log\left[\frac{(N-j+1)x}{j}\right]\, .
  5 KB (865 words) - 12:13, 21 May 2020
• 2006 AMC 10B Problems/Problem 22
  ...the sandwiches is <math> \$ 2.53</math>. Assume that <math>B</math>, <math>J</math>, and <math>N</math> are positive integers with <math>N>1</math>. Wha <math> \mathrm{(A) \ } \$ 1.05\qquad \mathrm{(B) \ } \$ 1.25\qquad \mathrm{(C) \ } \$ 1.45\qq
  2 KB (394 words) - 00:51, 25 November 2023
• 1993 AIME Problems/Problem 13
  ...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? Let <math>A</math> and <math>B</math> be Kenny's initial and final points respectively
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1994 AIME Problems/Problem 4
  ...x+1}</math> for some <math>x\in\mathbb{Z}</math>, then <math>\lfloor\log_2{a}\rfloor=\log_2{2^{x}}=x</math>. ...th>. So the sum of <math>\lfloor\log_2{a}\rfloor</math> for all such <math>a</math> is <math>x\cdot2^x</math>.
  2 KB (264 words) - 13:33, 11 August 2018
• 1995 AIME Problems/Problem 12
  // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1996 AIME Problems/Problem 12
  <cmath>2\sum\limits_{k = 1}^{9}\sum\limits_{j = 1}^{k}j = 330</cmath> ...her way to think about this is that the total sum is 330. Because you have a total of <math>10 \cdot 9</math> possible "pairs" or "differences", the ave
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 7
  ...r schemes are equivalent if one can be obtained from the other by applying a [[rotation]] in the plane board. How many inequivalent color schemes are po for(int j=0;j<7;++j)
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 3
  This is equivalent to starting off with a <math>x^0y^0</math> terms and choosing between <math>4</math> options <math ...ent (which we don't care about) and <math>0\le i,j\le n</math> and <math>i,j \in \mathbb{N}</math>.
  3 KB (515 words) - 04:29, 27 November 2023
• 1998 AIME Problems/Problem 15
  ...and <math>(j,i)</math> do not both appear for any <math>i</math> and <math>j</math>. Let <math>D_{40}</math> be the set of all dominos whose coordinate We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected w
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1999 AIME Problems/Problem 7
  ...step 1000 has been completed, how many switches will be in position <math>A</math>? ...}{d}</math> must be a multiple of 4 to ensure that a switch is in position A:
  3 KB (475 words) - 13:33, 4 July 2016
• 2001 AIME I Problems/Problem 15
  ...ritten on the faces of a regular [[octahedron]] so that each face contains a different number. The [[probability]] that no two consecutive numbers, whe ...hedron <math>DBEG</math>, and four are <math>\textit{long}</math>, joining a vertex of one tetrahedron to the diagonally opposite point from the other.
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 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
• 2003 AIME I Problems/Problem 3
  ...]] <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math> ...ok at the all of the next elements in the array, basically the whole <math>j = i + 1</math> shebang. Then, we see that if we set the sum of the whole ar
  2 KB (317 words) - 00:09, 9 January 2024
• 2003 AIME II Problems/Problem 13
  ...ne of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bu ...o return to the original vertex, the net number of clockwise steps must be a multiple of 3, i.e., <math>\#CW - \#CCW \equiv 0 \pmod{3}</math>. Since <m
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2001 AIME II Problems/Problem 15
  ...th>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <ma ...ath>E</math>, where the drilling starts, is at <math>(8,8,8)</math>. Using a little visualization (involving some [[similar triangles]], because we have
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 10
  ..., where <math>i</math> and <math>j</math> are integers and <math>0\leq i < j \leq 99</math>? ...mplies that <math>10^{6} - 1 | 10^{6k} - 1</math>, and so any <math>\boxed{j-i \equiv 0 \pmod{6}}</math> will work.
  4 KB (549 words) - 23:16, 19 January 2024
• 2001 AIME II Problems/Problem 6
  [[Square]] <math>ABCD</math> is inscribed in a [[circle]]. Square <math>EFGH</math> has vertices <math>E</math> and <math> .../math>. By the [[Pythagorean Theorem]], the radius of <math>\odot O = OC = a\sqrt{2}</math>.
  4 KB (772 words) - 19:31, 6 December 2023
• 2000 AIME II Problems/Problem 14
  ...!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>. f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j &= (-1)^{17}\cdot 1 + \sum_{m=1}^{62}\sum_{k=32m}^{32m+15}(-1)^{k+1}k
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2006 USAMO Problems/Problem 1
  (''Kiran Kedlaya'') Let <math>p</math> be a prime number and let <math>s</math> be an integer with <math>0 < s < p </ma if and only if <math>s </math> is not a divisor of <math>p-1 </math>.
  3 KB (506 words) - 17:54, 22 June 2023
• 2006 USAMO Problems/Problem 3
  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
  ...ow that if <math>k\ge 2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math ...igher power of 2). Thus we must have <math>M < i_0</math>, since otherwise a number divisible by <math>2^{i_0}</math> is visited before <math>2^{i_0} k_
  7 KB (1,280 words) - 17:23, 26 March 2016
• Multinomial Theorem
  Note that this is a direct generalization of the [[Binomial Theorem]], when <math>k = 2</math> ...{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j}
  3 KB (476 words) - 19:37, 4 January 2023
• 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
• 2006 AMC 10B Problems
  <math> \mathrm{(A) \ } -2006\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{( <math> \mathrm{(A) \ } -72\qquad \mathrm{(B) \ } -27\qquad \mathrm{(C) \ } -24\qquad \mathrm{
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2000 AMC 12 Problems/Problem 8
  <math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{ ...qual to <math>4</math>. Thus, the original sequence can be generated from a quadratic function.
  8 KB (1,202 words) - 16:17, 10 May 2024
• 2005 AMC 10A Problems/Problem 1
  <math> \textbf{(A) } 2\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 10\qqu Let <math>m</math> be Mike's bill and <math>j</math> be Joe's bill.
  909 bytes (134 words) - 19:05, 25 December 2022
• Navier-Stokes Equation
  ...ate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of ...odel weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design
  3 KB (553 words) - 22:08, 2 May 2022
• 2015 IMO Problems/Problem 6
  (i) <math>1\le a_j\le2015</math> for all <math>j\ge1</math>,<br /> ...integers <math>b</math> and <math>N</math> for which<cmath>\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2</cmath>for all integers <math>m</math> an
  4 KB (833 words) - 01:33, 31 December 2019
• Volume
  The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volu The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math>
  3 KB (523 words) - 20:24, 17 August 2023
• Divisibility rules/Rule for 3 and 9 proof
  A [[number]] <math>N</math> is [[divisible]] by 3 or 9 if the sum of its [[di where <math>a_i</math> is a digit for each <math>i</math>. Then the numerical value of <math>N</math>
  2 KB (316 words) - 20:29, 6 March 2014
• AwesomeMath
  ...ek to improve their performance on contests such as AMC10/12, AIME, or USA(J)MO. Step 1. Create a Student Account
  1 KB (195 words) - 11:19, 30 November 2023
• Stirling number
  ...<math>n</math>-[[element]] [[set]] with exactly <math>k</math> [[cycle of a permutation | cycles]]. ...can easily be shown by considering the cases that <math>n + 1</math> is in a part of size <math>1</math> or size larger than <math>1</math>.) They can
  2 KB (253 words) - 00:50, 15 May 2024
• 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
• 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
• Pick's Theorem
  ...[[polygon]], all of whose [[vertex | vertices]] are [[lattice points]] in a [[coordinate plane]], in terms of the number of lattice points inside the p <math>A = I + \frac{1}{2}B - 1</math>
  2 KB (301 words) - 13:08, 20 February 2024
• Fermat point
  ...>\triangle ABC</math> (with no angle more than <math>120^{\circ}</math> is a point <math>P</math> which has the minimum total distance to three [[vertic pair A=(2,4), B=(1,1), C=(6,1);
  4 KB (769 words) - 16:07, 29 December 2019
• Chinese Remainder Theorem
  The '''Chinese Remainder Theorem''' is a [[number theory | number theoretic]] result. ...esidue class mod <math>m</math> with a residue class mod <math>n</math> is a residue class mod <math>mn</math>.
  6 KB (1,022 words) - 14:57, 6 May 2023
• Matrix
  ...<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 3
  ...t) = w \left( (1+x)^k \sum_{j=1}^{n}Q_{i_j - k} \right) = 2 w \left( \sum_{j=1}^{n}Q_{i_j - k} \right)</math>, which is greater than or equal to <math> ..._{j=0}^{k-1} b_j x^j \equiv \sum_{j=0}^{k-1}\left[ (a_j + b_j)x^j + b_j x^{j+k} \right] \pmod{2}
  2 KB (354 words) - 04:56, 11 March 2023
• 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
• 2003 AMC 12A Problems/Problem 12
  <math> \mathrm{(A) \ } 8\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ ...nate the red card numbered <math>i</math> and the blue card numbered <math>j</math>, respectively.
  2 KB (368 words) - 18:04, 28 December 2020
• 2001 IMO Shortlist Problems/A2
  ...math>. Therefore, define <math>b_n=a_{n+N}</math>. If there does not exist a sequence <math>b_0, b_1, \ldots</math> of positive integers for which <math ...h>1+b_n\le b_{n-1}\sqrt[n]{2}</math>. Again, suppose there does exist such a sequence. Then, define <math>x_0=b_0</math> and <math>x_n=x_{n-1}\sqrt[n]{2
  2 KB (411 words) - 04:38, 17 June 2019
• Quaternion
  ...icative [[inverse with respect to an operation | inverse]]; alternatively, a non[[commutative]] [[field]]) which generalize the [[complex number]]s. ...th>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
• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} <math>\textbf{(A)}\ {-}\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Logical Operators and Loops
  for (int j=0; j<5; ++j) if (floor((i-j)/2)==((i-j)/2))
  2 KB (324 words) - 16:50, 2 October 2016
• Pigeonhole Principle/Solutions
  The Martian must pull 5 socks out of the drawer to guarantee he has a pair. Also, note that it is possible to pull out 4 socks without obtaining a pair.
  10 KB (1,617 words) - 01:34, 26 October 2021
• 2002 USAMO Problems/Problem 3
  Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real ...th <math>x_i \neq x_j </math> for all <math>i \neq j </math>, there exists a unique monic real polynomial <math>P(x) </math> of degree <math>n </math>
  4 KB (688 words) - 13:38, 4 July 2013
• 2007 AIME I Problems/Problem 10
  In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so th ...nother row that is complementary to the first. We remove those two and use a similar argument again to show that every group of <math>6</math> rows can
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2004 USAMO Problems
  Let <math>ABCD </math> be a [[quadrilateral]] circumscribed about a circle, whose interior and exterior angles are at least <math>60 ^{\circ} < ...> are integers whose greatest common divisor is 1. Let <math>S </math> be a set of integers with the following properties:
  3 KB (474 words) - 09:20, 14 May 2021
• 2007 AIME II Problems/Problem 3
  pair A, B, C, D, E, F; A = (0,13);
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 13
  ...>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? for (int j=0; j<i; ++j){
  3 KB (600 words) - 11:10, 22 January 2023
• 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
• 2003 USAMO Problems/Problem 1
  ...+ a </math> is divisible by 5, which is true when <math> k \equiv -3^{n-1}a \pmod{5} </math>. Since there is an odd digit in each of the residue class ...even digits. We could then halve all of the digits in that number to get a second multiple of <math> 5^n</math> with at most n digits that only uses t
  4 KB (736 words) - 22:17, 3 March 2023
• 2003 USAMO Problems/Problem 3
  A = a_0,a_1,a_2,\dots, a_n t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)
  3 KB (636 words) - 13:39, 4 July 2013
• Pentagon
  ...metry]], a '''pentagon''' is a [[polygon]] with 5 sides. Each [[angle]] of a [[regular polygon | regular]] pentagon is <math>108^{\circ}</math>. The sum It is possible to construct a regular pentagon with compass and straightedge:
  1 KB (191 words) - 09:59, 6 June 2022
• 2004 IMO Shortlist Problems/N2
  a) Prove that <math> \displaystyle \psi(mn) = \psi(m)\psi(n) </math> for ever ...thbf{N} </math> the equation <math> \displaystyle \psi(x) = ax </math> has a solution. <br>
  6 KB (1,007 words) - 09:10, 29 August 2011
• Parallelepiped
  ...that has [[parallelograms]] for its faces. Similarly, a parallelepiped is a [[hexahedron]] with six parallelogram faces. Specific parallelepipeds inclu ...ensional parallelepiped, or as an <math>n-</math>parallelepiped. A cube is a parallelepiped with all [[square]] faces.
  2 KB (248 words) - 20:08, 17 August 2023
• Chicken McNugget Theorem
  ...ritten in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n
  17 KB (2,748 words) - 19:22, 24 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
• Triangular number
  A rather simple recursive definition can be found by noting that <math>T_{n} ...ecause you can make a triangle out of dots, and the number of dots will be a triangular number:
  2 KB (275 words) - 08:39, 7 July 2021
• 2007 AMC 12A Problems/Problem 24
  <math>\mathrm{(A)}\ 2014524\qquad\mathrm{(B)}\ 2015028\qquad\mathrm{(C)}\ 2015033\qquad\math ...me nonnegative integer <math>j</math>. Since <math>x\leq \pi</math>, <math>j\leq n/2</math>. So there are <math>1 + \lfloor n/2 \rfloor</math> solutions
  4 KB (588 words) - 14:40, 23 August 2023
• 2006 AIME II Problems/Problem 13
  ...</math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>? ...c{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)</math>. Thus, <math>j</math> is a factor of <math>N</math>.
  4 KB (675 words) - 10:40, 14 July 2022
• 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
• 1998 PMWC Problems
  for(int j = n-i; j > 0; --j){ dot((z.x+j -1 + i/2 ,z.y + i*sqrt(3)/2));
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  pair A,B,C,D,E,F,G,O; A=origin;
  4 KB (641 words) - 21:24, 21 April 2014
• 2002 PMWC Problems
  for(int j = 0; j <= i; ++j) for(int j = 0; j <= i; ++j)
  5 KB (725 words) - 16:07, 23 April 2014
• 2003 PMWC Problems
  pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); draw((M--N)^^(origin--a)^^(origin--b));
  7 KB (918 words) - 16:15, 22 April 2014
• 1989 USAMO Problems/Problem 3
  ...</math> and <math>b</math> such that <math>P(a + bi) = 0</math> and <math>(a^2 + b^2 + 1)^2 < 4 b^2 + 1</math>. <cmath> P(z) = \prod_{j=1}^n (z- z_j) . </cmath>
  2 KB (340 words) - 19:11, 18 July 2016
• Cramer's Rule
  '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]]. ...the [http://en.wikipedia.org/wiki/Determinant matrix determinant] to solve a system of ''n'' linear equations in ''n'' variables.
  2 KB (352 words) - 18:22, 11 October 2023
• 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
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Let <math>A</math> be a [[partially ordered set]].
  9 KB (1,669 words) - 19:02, 1 August 2018
• 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
• Work
  ...er a certain distance. In the case of a constant force <math>F</math> over a distance <math>d</math>, the amount of work done is found using the equatio Where <math>W</math> is the work done (J), <math>F</math> is the force (N), <math>d</math> is the distance (m), and
  1 KB (192 words) - 00:46, 16 July 2018
• Celsius
  The '''Celsius''' (also '''Centigrade''') scale is a system for [[measure|measuring]] [[temperature]]. Originally, <math>0^\circ ...befvw efvn five fvwb ebfvw vnj e. vebr2hbwvi bviue ier erwb s achk er jhw j jhher hjh jfkhh eef hjh je vhj knefkh knvefk lkh evwk ewf k kvkj!!! buife
  4 KB (357 words) - 22:40, 17 April 2024
• 1999 USAMO Problems/Problem 4
  Since <math>-a_i</math> is a positive real for all <math>k+1 \le i \le n</math>, it follows that ...e 4</math>) <math>\sum_{i=1}^n a_i^2 < \sum_{i=1}^n 4 = 4n \le n^2</math>, a contradiction. Hence, assume <math>n \ge k \ge 1</math>. Then
  3 KB (499 words) - 09:47, 20 July 2016
• 1999 IMO Problems/Problem 2
  Let <math>n \geq 2</math> be a fixed integer. * (a) Find the least constant <math>C</math> such that for all nonnegative real
  4 KB (838 words) - 01:04, 17 November 2023
• Proofs of AM-GM
  with equality if and only if <math>a_i = a_j</math> for all <math>i,j</math> such that <math>\lambda_i, \lambda_j \neq 0</math>. We use [[Cauchy Induction]], a variant of induction in which one proves a result for <math>2</math>, all powers of <math>2</math>, and then that <mat
  12 KB (2,171 words) - 07:55, 11 May 2023
• 2008 AMC 12B Problems/Problem 23
  <math>\text{(A)}\ 11\qquad \text{(B)}\ 12\qquad \text{(C)}\ 13\qquad \text{(D)}\ 14\qquad ...h> is a perfect square. When these are multiplied, they equal <math>2^{a+n-a} \times 5^{b+n-b} = 10^n</math>. <math>\log 10^n=n</math> so the number of
  5 KB (814 words) - 18:02, 17 January 2023
• Vieta's formulas
  ...elementary symmetric polynomials]] of its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many ma
  3 KB (515 words) - 19:04, 19 May 2024
• 2000 AMC 12 Problems/Problem 16
  A checkerboard of <math>13</math> rows and <math>17</math> columns has a number written in each square, beginning in the upper left corner, so that <math>\text {(A)}\ 222 \qquad \text {(B)}\ 333\qquad \text {(C)}\ 444 \qquad \text {(D)}\ 5
  2 KB (310 words) - 11:28, 3 August 2021
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1990 USAMO Problems/Problem 4
  Let a <math>k</math>-good sequence be a sequence of distinct integers <math>\{ a_i \}_{i=1}^k</math> such that for ...a</math>, and furthermore, the terms of each of these sequences constitute a permutation of <math>k</math> consecutive integers.
  3 KB (529 words) - 19:15, 18 July 2016
• 2007 AMC 10A Problems/Problem 18
  ...f its sides has length <math>4</math>, and each two consecutive sides form a right angle. Suppose that <math>\overline{AG}</math> and <math>\overline{CH ..., B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
  6 KB (867 words) - 00:17, 20 May 2023
• 1974 USAMO Problems/Problem 1
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. ...P</math> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</cmath> (Why?)
  7 KB (1,291 words) - 20:30, 27 April 2020
• 1994 USAMO Problems
  ...the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in A convex hexagon <math>ABCDEF</math> is inscribed in a circle such that <math>AB = CD = EF</math> and diagonals <math>AD</math>, <
  2 KB (391 words) - 07:58, 19 July 2016
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• 1976 IMO Problems/Problem 2
  ...x^{2} - 2</math> and <math>P_{j}(x) = P_{1}(P_{j - 1}(x))</math> for <math>j= 2,\ldots</math> Prove that for any positive integer n the roots of the equ ...math>P_{k+1}(x)</math>. Let <math>P_1(r)=s</math>, where <math>s</math> is a real root of <math>P_k(x)</math>. We have that <math>-2<s<2</math>, so <mat
  3 KB (596 words) - 16:19, 28 July 2015
• 1976 IMO Problems/Problem 5
  ...om the set <math>\{ - 1,0,1\}</math><math>.</math> Prove that there exists a solution <math>x_{1}, \ldots,x_{q}</math> for the system with the propertie '''a.)''' all <math>x_{j}, j = 1,\ldots,q</math> are integers<math>;</math>
  2 KB (377 words) - 16:28, 29 January 2021
• Partition (combinatorics)
  ...partitions of 3: <math>3 = 2+1 =1+1+1</math>. Each of the [[summand]]s is a ''part'' of the partition. ...ence of a formula! No simpler formula is known, and the existence of such a formula is doubtful.
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2001 AMC 12 Problems/Problem 6
  A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each a different digit. The digits in each part of the number are in decreasing
  2 KB (340 words) - 03:02, 28 June 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 3
  ...be <math>b</math>, swimming rate be <math>s</math>, jogging rate be <math>j</math>, all in km/h. ...5s = 57</math>. Mod 4, we need <math>s\equiv1\pmod{4}</math>. Thus, <math>(j,s) = (13,1),(8,5),(3,9)</math>.
  3 KB (564 words) - 22:15, 28 November 2023
• Mock AIME 5 Pre 2005 Problems
  &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}</cmath>
  6 KB (909 words) - 07:27, 12 October 2022
• 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
• 2005 USAMO Problems/Problem 2
  ...th>157 \equiv -147 \equiv 5 \pmod{19}</math>. For reference, we construct a table of powers of five: Evidently, the order of 5 is 9. Hence 5 is the square of a multiplicative generator of the nonzero integers mod 19, so this table show
  7 KB (1,053 words) - 10:38, 12 August 2015
• 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
• 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 ...th>P(a_j) = b_j</math> and <math>P(b_j) = a_j</math> for all indices <math>j</math>. Let the set <math>\{ a_1, \dotsc, a_k, b_1, \dotsc, b_k \}</math>
  3 KB (704 words) - 14:42, 7 September 2021
• Schreier's Theorem
  '''Schreier's Refinement Theorem''' is a result in [[group theory]]. Otto Schreir discovered it in 1928, and used i ...ath>\Sigma_1</math> and <math>\Sigma_2</math> be [[composition series]] of a group <math>G</math>. Then there exist equivalent composition series <math
  2 KB (337 words) - 12:13, 9 April 2019
• 2008 Mock ARML 2 Problems
  <math>ABCD</math> is a convex quadrilateral such that <math>|AB| = 5</math>, <math>|BC| = 17</math ...which is less than <math>30</math> (a proper divisor of <math>n</math> is a positive integer that divides but is not equal to <math>n</math>).
  4 KB (582 words) - 21:57, 8 May 2019
• 2003 AMC 12A Problems/Problem 17
  ...h>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance fr pair A,B,C,D,M,P;
  6 KB (1,026 words) - 22:35, 29 March 2023
• 2003 AMC 12B Problems/Problem 18
  ...math>x</math> has a prime factorization <math>a^cb^d.</math> What is <math>a + b + c + d?</math> <math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\
  6 KB (914 words) - 11:07, 7 September 2023
• Jean-Victor Poncelet
  ...try]], though he has many other contributions in the field of geometry. As a mechanic and physicist, he improved the design of the waterwheel in his hom ...French academy called the Ecole Polytechnique in 1808, and then in 1810 at a local university before graduating.
  2 KB (253 words) - 11:41, 19 December 2018
• 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
• Legendre's Formula
  where <math>p</math> is a prime and <math>e_p(n!)</math> is the [[exponent]] of <math>p</math> in the We use a counting argument.
  4 KB (699 words) - 17:55, 5 August 2023
• 2001 IMO Shortlist Problems/C2
  ...>\{1,2,\ldots,n\}</math> such that <math>n!</math> is a divisor of <math>S(a) - S(b)</math>. ...ers <math>0\leq y\leq n-1</math>. This means that the sum of every <math>S(a)</math> is congruent to <math>\sum_{i=0}^{n!-1}=\frac{(n!-1)n!}{2}</math>.
  2 KB (364 words) - 08:55, 31 August 2011
• 2001 IMO Shortlist Problems/N6
  ...h>a_i</math>. This then each ai takes on the form, <math>j+b_1, j+b_2,..., j+b_{98}</math>. Then we must find the <math>98</math> <math>b</math> integer '''Lemma:''' We try to show that for a prime <math>p</math>, there are such <math>p</math> integers less than or e
  2 KB (443 words) - 13:08, 17 August 2011
• Millennium Problems
  ...set of seven problems for which the [[Clay Mathematics Institute]] offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new mi ...24, 2000 at the [[Collège de France]]. [[Timothy Gowers]] first presented a lecture titled ''The Importance of Mathematics'' as an introduction. After
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1989 AHSME Problems/Problem 29
  (A) <math>-2^{50}</math> (B) <math>-2^{49}</math> (C) 0 (D) <math>2^{49}</math By the [[Binomial Theorem]], <math>(1+i)^{99}=\sum_{n=0}^{99}\binom{99}{j}i^n =</math> <math>\binom{99}{0}i^0+\binom{99}{1}i^1+\binom{99}{2}i^2+\bino
  1 KB (173 words) - 19:52, 30 December 2020
• 2008 IMO Problems/Problem 5
  For convenience, let <math>A</math> denote the set <math>(1,2,\ldots n)</math> and <math>B</math> the se We can describe each sequences of switching the lamps as a <math>k</math>-dimensional vector
  4 KB (677 words) - 01:10, 19 November 2023
• 2000 USAMO Problems
  Call a real-valued function <math>f</math> very convex if ...nction of <math>R, W,</math> and <math>B,</math> the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
  3 KB (495 words) - 19:02, 18 April 2014
• 2000 USAMO Problems/Problem 4
  ...re colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. for(int j = 0; j < 10; ++j){
  2 KB (382 words) - 13:37, 4 July 2013
• 2000 USAMO Problems/Problem 5
  ...<math>A_1A_2A_3</math> be a [[triangle]] and let <math>\omega_1</math> be a [[circle]] in its plane passing through <math>A_1</math> and <math>A_2.</ma pair A = (0,0), B=3*expi(1), C=(3.5)*expi(0); /* arbitrary points */
  3 KB (609 words) - 09:52, 20 July 2016
• 2001 USAMO Problems/Problem 1
  Suppose a configuration exists with <math>n \le 22</math>. ...re at least <math>n \ge 5 \cdot 5 + 1 = 26</math> balls, contradiction. If a ball appears <math>4</math> or more times, the remaining balls of the <math
  5 KB (841 words) - 17:19, 5 May 2022
• Karamata's Inequality
  .../math> [[Majorization|majorizes]] <math>(b_i)</math> and <math>f</math> is a [[convex function]], then .../math> is convex over the interval <math>(a, b)</math>, then <math>\forall a\leq x_1\leq x_2 \leq b</math> and <math>\Gamma(x, y):=\frac{f(y)-f(x)}{y-x}
  2 KB (370 words) - 03:39, 28 March 2024
• 1973 USAMO Problems/Problem 2
  ...re not congruent <math>\bmod{8}</math> when <math>i\geq 3</math> and <math>j\geq 2</math>. Thus after the "1", the terms of each sequence are not equal. {{USAMO box|year=1973|num-b=1|num-a=3}}
  2 KB (342 words) - 18:54, 3 July 2013
• 2004 IMO Problems/Problem 4
  ...r all <math>i</math>, <math>j</math>, <math>k</math> with <math>1 \leq i < j < k \leq n</math>. {{IMO box|year=2004|num-b=3|num-a=5}}
  2 KB (322 words) - 00:54, 19 November 2023
• 2002 AMC 10A Problems
  {{AMC10 Problems|year=2002|ab=A}} <math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qq
  11 KB (1,733 words) - 11:04, 12 October 2021
• 2002 AMC 10A Problems/Problem 20
  .... Point <math>H</math> lies on <math>\overline{GD}</math>, and point <math>J</math> lies on <math>\overline{GF}</math>. The line segments <math>\overlin <math>\text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3
  3 KB (439 words) - 22:15, 9 June 2023
• 1973 USAMO Problems/Problem 1
  Two points <math>P</math> and <math>Q</math> lie in the interior of a regular tetrahedron <math>ABCD</math>. Prove that angle <math>PAQ < 60^\cir Let the side length of the regular tetrahedron be <math>a</math>. Link and extend <math>AP</math> to meet the plane containing triang
  2 KB (276 words) - 08:10, 4 January 2022
• 1975 USAMO Problems/Problem 3
  If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that <cmath>P(k)=\frac{k}{k+1}</cm Let <math>Q(x) = (x+1)P(x) - x</math>, and clearly, <math>Q(x)</math> has a degree of <math>n+1</math>.
  3 KB (465 words) - 17:29, 8 January 2024
• 1962 IMO Problems/Problem 7
  (a) Prove that the tetrahedron <math>SABC</math> is regular. '''Part (a)'''
  11 KB (1,928 words) - 20:52, 21 November 2023
• 1963 IMO Problems/Problem 3
  We can obtain the <math>y</math>-coordinate of the top side in a different way by multiplying the lengths of the sides <math>a_{\frac{n}{2}+ ...left(\frac{2\pi}{n}i\right)}\vec{i}+\sin{\left(\frac{2\pi}{n}i\right)}\vec{j}</math>. Now rotate and translate the given polygon in the Cartesian Coordi
  4 KB (836 words) - 17:58, 7 December 2022
• 2001 AMC 10 Problems
  <math>\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 10 \qq A number <math>x</math> is <math>2</math> more than the product of its recipr
  14 KB (1,983 words) - 16:25, 2 June 2022
• 2000 AMC 10 Problems/Problem 16
  label("$A$",(0,3),NW); <math>\textbf{(A)}\ \frac{4\sqrt{5}}{3} \qquad\textbf{(B)}\ \frac{5\sqrt{5}}{3} \qquad\textb
  7 KB (966 words) - 23:22, 31 July 2023
• 2007 IMO Shortlist Problems/A1
  You are given a sequence <math>a_1,a_2,\dots ,a_n</math> of numbers. For each <math>i</math <center><math>d_i=\max\{a_j:1\leq j\leq i\}-\min\{a_j:i\leq j\leq n\}</math></center>
  3 KB (734 words) - 05:11, 26 March 2024
• 2004 AMC 12B Problems/Problem 14
  ...and <math>BC</math>, respectively, with <math>CM=CN=4</math>. Points <math>J</math> and <math>K</math> are on <math>AB</math> so that <math>MJ</math> an pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0);
  7 KB (1,083 words) - 22:41, 23 November 2020
• 1976 IMO Problems
  ...e area of <math>32 \text{ cm}^{2}</math> the sum of two opposite sides and a diagonal is <math>16 \text{ cm}</math>. Determine all the possible values t ...x^{2} - 2</math> and <math>P_{j}(x) = P_{1}(P_{j - 1}(x))</math> for <math>j= 2,\ldots</math> Prove that for any positive integer n the roots of the equ
  2 KB (409 words) - 16:25, 29 January 2021
• 1971 IMO Problems/Problem 6
  ...</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 that the sum of all the eleme ...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
• 2001 AMC 12 Problems/Problem 10
  for(int j=0; j<3; ++j) draw(shift(3*i,3*j)*p);
  2 KB (252 words) - 00:11, 15 August 2022
• 2001 AMC 12 Problems/Problem 22
  ...rline{EF}</math> at <math>H</math> and <math>\overline{EG}</math> at <math>J</math>. The area of the rectangle <math>ABCD</math> is <math>70</math>. Fin \text{(A) }\frac {5}{2}
  7 KB (1,120 words) - 02:27, 29 August 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
  ...inside or on <math>Q</math>. The cuts produce <math>n</math> 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 12B Problems/Problem 25
  int i,j; for (j=-7;j<8;++j) {
  15 KB (2,229 words) - 03:36, 4 September 2021
• Maximal ideal
  ...s, <math>I\neq R</math>, and there is no ideal <math>J</math> with <math>I<J<R</math>.) ...he [[quotient ring]] <math>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
  ...ring''' <math>R[x]</math> is, informally, "the ring of all polynomials in a [[commutative]] <math>x</math> with coefficients in <math>R</math>." That where <math>N</math> is a [[nonnegative integer]] that varies from sum to sum.
  12 KB (2,010 words) - 00:10, 3 August 2020
• 1989 AJHSME Problems
  <math>\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 <math>\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\
  13 KB (1,765 words) - 11:55, 22 November 2023
• Hilbert's Basis Theorem
  '''Hilbert's Basis Theorem''' is a result concerning [[Noetherian]] [[ring]]s. It states that if <math>A</math> is a (not necessarily [[commutative]])
  4 KB (617 words) - 19:59, 23 April 2023
• Determinant
  For an <math>n\times n</math> matrix <math>a = (a_{ij})</math>, the determinant is defined by \det a &= \sum_{\sigma \in S_n} \text{sign} (\sigma) a_{1\sigma(1)}
  8 KB (1,345 words) - 00:31, 9 May 2020
• 2004 USAMO Problems/Problem 2
  ...th> are integers whose greatest common divisor is 1. Let <math>S</math> be a set of integers with the following properties: (a) For <math>i = 1, \dots, n</math>, <math>a_i \in S</math>.
  3 KB (508 words) - 14:23, 17 July 2014
• 1968 AHSME Problems/Problem 35
  MP("a",(5,0),S); MP("J",(0,10),N);
  3 KB (597 words) - 01:52, 16 August 2023
• 1989 AJHSME Problems/Problem 19
  for (int a=1; a<13; ++a) draw((a,0)--(a,.5));
  2 KB (255 words) - 03:28, 28 November 2019
• 1989 AJHSME Problems/Problem 22
  ...th>, <math>9</math> are "cycled" separately as follows and put together in a numbered list: <math>\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qqu
  1 KB (177 words) - 16:03, 19 April 2021
• 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>. ...t=0</math>, then we have <math>(x_1,x_2,x_3,x_4)=(1,1,1,1)</math> which is a solution.
  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
  ...d <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>. ...all <math>i</math> with <math>1 \le i < m</math>, where <math>m</math> is a positive integer. At this point, we will have <math>r</math> = <math>\frac{
  2 KB (463 words) - 12:19, 21 August 2020
• 2009 IMO Problems/Problem 6
  ...ldots,a_n</math> in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in <math>M</math>. ...a_1</math> will work. We will assume that the order can be chosen in such a way for all integers <math>n < k</math> for <math>k \ge 3</math>.
  5 KB (1,055 words) - 01:18, 19 November 2023
• 1998 USAMO Problems/Problem 3
  *<math>1 + y_i\ge \sum_{j\neq i}{(1 - y_j)}</math> *<math>\frac {1 + y_i}{n}\ge \frac {1}{n}\sum_{j\neq i}{(1 - y_j)}</math>
  2 KB (322 words) - 13:31, 23 August 2023
• 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
• Tower law
  ...le m, 1\le j\le n\}</math> (which clearly has <math>mn</math> elements) is a basis for <math>L</math> over <math>F</math>. ...in K</math>. And now as <math>\{\beta_1,\beta_2,\ldots,\beta_n\}</math> is a basis for <math>K</math> over <math>F</math> we can write <math>a_i=\sum_jb
  3 KB (567 words) - 08:42, 21 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
• 1989 IMO Problems/Problem 6
  ...of the set<math> \{1,2, \ldots, 2n\}</math> where <math>\mbox{n}</math> is a positive integer, is said to have property <math>T</math> if <math>|x_i - x ...n,2), \cdots, (n,2n),(2n,n)</math> represent the adjacent pairs. There are a total of 2n pairs.
  11 KB (1,928 words) - 12:26, 26 July 2023
• Mock AIME 3 2006-2007/Problem 3
  Let <math>P(x) = 2x^{59} - x^2 - x - 6</math>. If <math>Q(x)</math> is a polynomial whose roots are the 59th powers of the roots of <math>P(x)</math ...<math>P</math>. Then, <math>\sum_{i} a_i = 0</math>, and <math>\sum_{i\neq j} = 0</math>. Therefore, <math>\sum_{i} a_i^2 = 0</math>.
  862 bytes (151 words) - 01:35, 30 December 2009
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...<math>A</math> is defined as <math>P_A(t) = \det(tI - A)</math>, which is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math>
  19 KB (3,412 words) - 14:57, 21 September 2022
• KGS math club/solution 9 1
  ...inciple (pigeonhole principle), some two of them S_i and S_j, where i > 1, j > 1, are congruent modulo N, so |S_i - S_j| = 0 modulo N. ...vely prime with 10^(min(i, j)). Then, since i >= 1 and j >= 1, N must have a common factor with 10.
  629 bytes (125 words) - 06:42, 4 March 2010
• Proofs without words
  rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2) ...ck(pair A, pair B, pair ticklength = (0.15,0)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); }
  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/Problem 11
  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)); draw(Arc((8,7.67),A,G,CW),dark,EndArrow(8)); draw(B--C--G--cycle,linetype("4 4"));
  4 KB (636 words) - 16:46, 25 November 2023
• 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 ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2007 IMO Problems
  <cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath> (a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>,
  3 KB (505 words) - 09:24, 10 September 2020
• 2007 IMO Problems/Problem 1
  <cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath> (a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>,
  4 KB (768 words) - 23:15, 31 March 2010
• 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 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
• LaTeX:LaTeX on AoPS
  LaTeX uses a special "math mode" to display mathematics. There are two types of this "ma Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that <code>$
  8 KB (1,356 words) - 22:35, 26 June 2020
• 2010 USAMO Problems
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter <math>AB</math>. Denote by <math>P</math>, <math>Q</ ...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>. If we can demonstrate a sequence in which for all <math>1 \le i \le 1005</math> the product
  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> given any two indices <math>i</math> and <math>j</math> (not necessarily distinct) for which <math>x_i + x_j < 2n</math>, th
  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>. ...e 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 on <mat
  12 KB (2,338 words) - 20:30, 13 February 2024
• 2010 USAJMO Problems/Problem 2
  <li> (a). <math>x_1 < x_2 < \cdots <x_{n-1}</math>; <li> (c). given any two indices <math>i</math> and <math>j</math> (not necessarily distinct)
  6 KB (1,177 words) - 23:09, 20 November 2022
• 2010 USAMO Problems/Problem 1
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter Since <math>XY</math> is a chord of the circle with diameter <math>AB</math>,
  13 KB (2,178 words) - 14:14, 11 September 2021
• 2010 USAMO Problems/Problem 2
  There are <math>n</math> students standing in a circle, one behind the other. The students have heights <math>h_1 < h_2 < \ldots < h_n</math>. If a
  5 KB (823 words) - 19:20, 3 October 2017
• Scholarships by State
  ...ships.asp Junior League of Rockford Outstanding Volunteer Scholarship] for a high school senior in Winnebago County. * Senator George J. Mitchell Scholarship - for seniors of Maine public schools for <dollar/>1,
  9 KB (1,107 words) - 15:32, 16 July 2010
• 2008 iTest Problems/Problem 77
  ...home from vacation, Wendy looks for something to do. She starts working on a project for the math team. ...cluding Wendy, who are about to be sophomores on the math team. Elected as a math team officer,
  5 KB (788 words) - 05:15, 28 January 2019
• 2010 IMO Problems/Problem 2
  ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE ...rove alternatively that if <math>EI</math> meets the circle again at <math>J</math> and <math>JD</math> meets <math>IF</math> at <math>G</math>, then <m
  3 KB (525 words) - 14:52, 16 July 2023
• 2010 IMO Problems/Problem 5
  ...math>, remove one coin from <math>B_j</math> and add two coins to <math>B_{j+1}</math>; Type 2) Choose a non-empty box <math>B_k</math>, <math>1\leq k \leq 4</math>, remove one coi
  3 KB (411 words) - 06:25, 25 June 2018
• 2010 IMO Problems/Problem 6
  ...ldots</math> be a sequence of positive real numbers, and <math>s</math> be a positive integer, such that So for solving This Problem, we need to take a assumption that,
  4 KB (786 words) - 08:46, 12 March 2024
• 1971 IMO Problems
  ...; let <math>P_i</math> be the polyhedron obtained from <math>P_1</math> by a translation that moves vertex <math>A_1</math> to <math>A_i (i = 2, 3, \cdo ...ows: <math>X</math> is a point on edge <math>AB</math> distinct from <math>A</math> and <math>B</math>; similarly, <math>Y, Z, T</math> are interior poi
  3 KB (495 words) - 13:52, 29 January 2021
• 2010 IMO Problems
  ...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math> ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE
  4 KB (603 words) - 09:22, 10 September 2020
• 2004 AMC 8 Problems
  ...ter length represents <math>72</math> kilometers. How many kilometers does a <math>17</math>-centimeter length represent? <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\q
  13 KB (1,860 words) - 19:58, 8 May 2023
• Fractional ideal
  ...> for some nonzero <math>\alpha\in R</math>. Explicitly, <math>I</math> is a subset of <math>K</math> such that for any <math>x,y\in I</math> and <math> ...h> (simply take <math>\alpha = 1</math>). Moreover, it is easy to see that a fractional ideal, <math>I</math>, of <math>R</math> is an integral ideal of
  2 KB (288 words) - 20:05, 23 January 2017
• 1997 AJHSME Problems
  <math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{ Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is
  12 KB (1,702 words) - 12:35, 6 November 2022
• 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
• Mobius function
  The '''Möbius function''' is a multiplicative number theoretic function defined as follows: <cmath>\mu(n) = \begin{cases} 1 &\text{if}~n=1\\ 0 & \text{if}~n~\text{has a square factor} \\ (-1)^k & n = p_1p_2\cdots{p_k} .\end{cases}</cmath>
  5 KB (910 words) - 02:58, 1 March 2022
• 2012 IMO Problems/Problem 1
  ...int <math>J</math> is the centre of the excircle opposite the vertex <math>A.</math> This excircle is tangent to the side <math>BC</math> at <math>M</ma ...> and <math>AL</math> are tangents from <math>A</math> to the circle <math>J</math>, <math>AK = AL</math>. Then, we have <math>AK = AL</math>, which be
  7 KB (1,189 words) - 01:22, 19 November 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
• 1994 USAMO Problems/Problem 4
  ...\,</math> be a sequence of positive real numbers satisfying <math>\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,</math> for all <math>\, n \geq 1</math>. Prove <cmath>\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n}
  1 KB (210 words) - 13:30, 4 July 2013
• 2011 AIME I Problems/Problem 11
  ...ible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by 1000. Let <math>S</math> be the sum of t ...<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> in mod <math
  11 KB (1,668 words) - 22:10, 24 February 2023
• 2011 AIME I Problems/Problem 7
  ...for any power of <math>x</math>. Consider <math>m=2</math>. We can achieve a sum of <math>1-\left(\frac{1}{2}\right)^x</math> by doing <math>\frac{1}{2} ...2^{2008} = 2^{2009} + 2^{2009} = 2^{2010}</cmath>. Thus <math>2</math> is a possible value of <math>m</math>. For other values, for example <math>m = 3
  10 KB (1,673 words) - 19:27, 8 April 2024
• 2011 AIME II Problems/Problem 8
  ..._j</math>. Then the maximum possible value of the real part of <math>\sum_{j=1}^{12} w_j</math> can be written as <math>m+\sqrt{n}</math> where <math>m< ...e write <math>z=a+bi</math>, then the real part of <math>z</math> is <math>a</math> and the real part of <math>iz</math> is <math>-b</math>. The blue do
  5 KB (805 words) - 18:46, 27 January 2024
• 2011 AIME II Problems
  Gary purchased a large beverage, but only drank <math>m/n</math> of it, where <math>m</math> The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer
  8 KB (1,301 words) - 08:43, 11 October 2020
• 2011 AIME II Problems/Problem 14
  ...then the two numbers in these positions must be equivalent mod 2. If <math>j</math> is the same, then the two numbers must be equivalent <math>\mod 3</m ...d triple (or position) in which 1 can be placed has 2 options for i, 3 for j, and 5 for k, resulting in 30 different positions of placement.
  10 KB (1,581 words) - 22:09, 27 August 2023
• 2011 AIME II Problems/Problem 9
  ...eed to maximize involves products <math>x_i x_j</math> with <math>i \equiv j \pmod 3</math>. Factoring out say <math>x_1</math> and <math>x_4</math> we ..._4 = x_6</math>, at which point we know that <math>a+b = 1/3</math>, <math>a^3+b^3 \geq 1/540</math>, and we are trying to maximize <math>3a^2b+3ab^2</m
  3 KB (466 words) - 15:06, 16 January 2023
• 1998 USAMO Problems
  ...rpendicular bisectors of <math>DE</math> and <math>CF</math> intersect at a point <math>M</math> on <math>AB</math>. Find, with proof, the ratio <math ...e with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white,
  3 KB (486 words) - 06:11, 24 November 2020
• 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
• KGS math club/solution 11 5
  We'll start with a small solution (e.g. three horses or two horses), and show how to build up ...divides <math>s_j</math> exactly an integer number of times. We will find a list of speeds with instead <math>n+1</math> different horse speeds.
  4 KB (712 words) - 10:51, 11 May 2011
• 2011 USAJMO Problems/Problem 3
  For a point <math>P = (a, a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line ...th>a_n^3-\frac{3ka_n^2}2-\frac{3a_n}4+\frac k8=0.</cmath>Note that this is a cubic, and the roots are <math>a_1,a_2</math> and <math>a_3</math> which ar
  15 KB (2,593 words) - 13:37, 29 January 2021

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